METHODS FOR CONSTRUCTING AND DECODING BLOCK ...

METHODS FOR CONSTRUCTING AND DECODING BLOCK

ERROR-CORRECTING CODES

By

Abdullah Abdulmutlib Hashim, B.Sc. (Eng.), M.Sc.

July, 1974

A thesis submitted for the degree of Doctor

of Philosophy of the University of London

and for. the Diploma of Imperial College

Electrical Engineering Department,

Imperial College of Science and Technology,

London, S. W. 7.

ABSTRACT

This thesis contains the results of an investigation

into that branch of algebraic coding theory which is concerned

with linear block error-correcting codes. The classical problems

of coding theory are: firstly, the discovery of "good" codes,

where "good" is used with reference to optimality of the code's

rate and error-correcting capability; and secondly, the ease

with which the encoding and decoding of codes may be carried

out. This thesis considers both these problems and as a result

of several new procedures which are introduced for constructing

and decoding these codes, a large number of new codes are

presented, many of these being "good" codes.

A computerised search procedure based on certain

properties of the parity-check matrix of block codes is described

which yields new "good" codes. Also, by employing a sub-set of

Walsh functions to construct the parity-check matrix, a class

of binary codes is obtained which can be decoded by a simple

one-step majority decoding algorithm. Another possibility for

finding new codes is brought about by a mathematical analysis

of the concept of anticodes which yields a new systematic

procedure for their generation. A fourth procedure, concerned

with the modification of existing codes by puncturing and

lengthening, is shown to produce other new families of codes,

the decoding procedures for which are outlined.

A description of an important new class of codes is

given, referred to here as "nested codes". The codes of this

class cover a wide range of rate and error-correcting capability,

and possess a useful mathematical structure in that their

syndromes corresponding to errors in the information digits

of the codeword are themselves codewords of another code

having the same properties. These nested codes are found to

be decodable by a very simple decoding algorithm which results

2.

in the decoder complexity increasing only linearly with the

code length and number of errors that may be corrected.

Moreover, the decoder is shown to be capable of correcting

some errors with weight greater than the error-correcting

capability of the code.

The properties of the parity-check matrix of codes

used for compound channels have also been found to yield a

computerised search procedure for finding new codes capable

of correcting either both random and burst errors or multi-

burst errors. A considerable number of codes have been found

by this procedure which are capable of correcting errors of

this type. Moreover, the techniques and methods of constr-

ucting nested codes have been modified to establish a new

class of burst-and-random-error correcting codes. Codes of

this class and their decoding algorithm have been found to

exhibit the same properties as nested codes.

Finally, the lower bounds on minimum-Hamming distance

for linear block codes are examined. It is shown that the

Varshamov-Gilbert bound may possibly be improved. Furthermore,

a lower bound on the maximum-Haunaing distance of linear anti-

codes is presented.

3.

ACKNOWLEDGEMENTS

I should like to express my grateful thanks to Dr.

A.G. Constantinides for his supervision, guidance and constant

encouragement during the course of this research; his

suggestions and many contributions are also much appreciated.

The work reported in this thesis has benefited in

many ways from the numerous discussions that I have had with

several persons, most notable among whom are Professor E.C.

Cherry of Imperial College; Dr. V.N. Nomokonov of the Leningrad

Institute of Technology, U.S.S.R.; Professor G. Longo of the

University of Trieste, Italy; Dr. P.G. Farrell of the University

of Kent, U.K.; Mrs. Z. Chiba of the GEC Hirst Research Centre,

U.K.; Dr. D.J. Goodman of the Bell Laboratories, U.S.A.;

Professor D.A. Bell of the University of Hull, U.K.; and my

colleague Mike Buckley who also reviewed the manuscript and

supplied many detailed corrections.

Thanks are also due to Messrs. P. Beevor, L.C.

Stenning, I. Colyer and R. Howie for the time they spent in

checking through the manuscript.

The help and encouragement of my colleagues Nelson

Esteves, Jide Olaniyan, Michael Lai and Majid Ahmadi has also

been very much appieciated.

Finally, I should like to express my gratitude to

Professor J. Brown for offering me the post of academic

visitor and providing research facilities at Imperial College;

to Baghdad University, College of Engineering Technology, for

allowing study leave for the period of this research; to

Miss Shelagh Jenkins for her impeccable typing of the manu-

script; and to my family and friends without whose help this

work would never have been accomplished.

4.

CONTENTS

Page

ABSTRACT 2

ACKNOWLEDGEMENTS 4

CONTENTS 5

LIST OF SYMBOLS AND ABBREVIATIONS 9

SECTION 1 INTRODUCTION

12

SECTION 2 FUNDAMENTALS OF CODING THEORY 17

2.1 The Analysis of Linear Block Codes 18

2.2 The Construction of Linear Block Codes 23

2.2.1 Repetition Codes 24

2.2.2 Hamming Codes 24

2.2.3 Golay Codes 24

2.2.4 Circulants Codes 25

2.2.5 Optimum Codes 25

2.2.6 Quasi-Perfect Codes 26

2.2.7 Reed-Muller Codes 27

2.2.8 Concatenated Codes 27

2.2.9 Cyclic Codes 28

2.2.10 The BCH Codes 30

2.2.11 Goppa Codes 31

2.2.12 Justesen Codes 32

2.3 Decoding of Linear Block Codes 33

2.3.1 Syndrome Decoding 33

2.3.2 Decoding of Binary Cyclic Codes 35

2.3.3 Error-Trapping for Binary Cyclic Codes 38

2.3.4 Majority Logic Decoding Algorithms 40

2.3.5 Decoding of BCH Codes 42

2.4 Minimum-Distance Bounds for Binary Linear

Codes 46

5.

Page

SECTION 3 NOVEL PROCEDURES FOR CONSTRUCTING

LINEAR BLOCK CODES 51

3.1 A Computerised Search for Linear Codes 52

3.2 Application of Walsh Functions in

constructing Linear Block Codes 60

3.2.1 Code Construction and Decoding 63

3.2.2 Remarks 65

3.3 Application of the Concept of Anticodes 67

3.3.1 Matrix Description of Linear 68

Anticodes

3.3.2 A Systematic Procedure of 70

generating Linear Anticodes

3.4 Two Procedures of Linear Block Code 71

Modification

3.4.1 Introduction 71

3.4.2 Linear Block Code Puncturing 74

Procedures

3.4.3 Encoding and Decoding of the 78

Punctured Codes

3.4.4 A Technique for Lengthening 79

Linear Block Codes

SECTION 4 A NEW CLASS OF NESTED LINEAR BLOCK CODES

AND THEIR NESTED DECODING ALGORITHM 88

4.1 Introduction 89

4.2 The Construction of Nested Codes 89

4.2.1 Double Error-Correcting Nested Codes 95

4.2.2 Multiple Error-Correcting Nested

Codes 98

4.3 The Nested Decoding Algorithm 102

6.

4.4 Features and Merits of the Nested

Decoding Algorithm

4.4.1

The Capability of Correcting

Pattern of Errors of Weight

greater than t.

4.4.2

The Complexity of the Nested

Decoder.

115

118

SECTION 5 LINEAR BLOCK CODES FOR NON-INDEPENDENT

ERRORS 121

5.1 Introduction 122

5.2 Block Codes for Non-Independent Errors,

Definition and Construction 124

5.3 Single-Burst-Error-Correcting Codes 131

5.4 Burst-and-Random-Error-Correcting Codes 138

5.5 Multiple-Burst-Error-Correcting Codes 152

SECTION 6 BOUNDS ON THE HAMMING DISTANCE OF LINEAR

CODES AND ANTICODES 157

6.1 Introduction 158

6.2 An Improvement on the Varshamov-Gilbert

Lower Bound on the Minimum Hamming Distance

of Linear Block Codes 160

6.3 The Weight Distribution of Linear Binary

Codes 165

6.4 Maximum Distance Bounds for Linear

Anticodes 171

SECTION 7 SUMMARY OF CONTRIBUTIONS AND SUGGESTIONS

FOR FURTHER RESEARCH 175

7.1 Summary of Contributions 176

7.2 Suggestions for Further Research 181

7.

Page

APPENDIX 1

Mathematical Background

186

APPENDIX 2

Computer Programmes 193

REFERENCES 204

8.

LIST OF SYMBOLS AND ABBREVIATIONS

Matrix X.

Transpose of the matrix EX]. txil Set x1, x2, •' xn (n is given in the text).

C, v011 Membership: v is an element of set fyl.

C,t1J1CV1 Membership: N1 is a sub-set of [V1.

)(11)(x) Set of all x that satisfy the condition P(x).

1:1

'1' Number of combinations of i out of n.

d Haunting distance of a code.

d(v,u) Hamming distance between two vectors v and u.

An error pattern.

Generator matrix of an (n,k) linear code.

GF(q) Galois field of q elements.

GF(Pn) Galois field of order Pn. •

[eJ CG]

g(x)

CH] DK]

K

Generator polynomial of cyclic code.

Parity-check matrix of an (n,k) linear code.

Identity matrix of order K.

Generator matrix of (m,k) linear anticode.

Number of information (data) digits in a block codeword.

(ICK1) First code of constructing an (n,k) nested code.

D.]

A parity-check matrix of linear anticode.

m Anticode length.

(m,k,6) A linear anticode of maximum Hamming distance

n Length of linear code.

9.

(n,k,d) A linear code that has a minimum Hamming distance d.

(n-k,k-k1) Second code of constructing an (n,k) nested code.

(n,k,t) A linear code capable of correcting t-random errors.

(n;k;t) A nest of t nested codes, corresponding to an (n,k)

nested code.

(nt,kt) The inner code of a nest of an (n,k,t) nested code.

q Number of code symbols.

r Number of parity-check digits in a block code.

R Code rate.

LSD Syndrome matrix.

Vn A vector space of dimension n.

V A subspace of the vector space Vn, used to indicate

the linear (n,k) block code .

V' A null space of subspace V.

w Weight of a codeword.

w(v) The Hamming weight of a vector v.

b Maximum distance of anticode.

BCH Bose-Chaudhuri-Hocquenghem code.

QLC Quasi-linear combinations.

B & R Burst-and-random.

tviveQLC(ni)(q-1)9 The set of vectors resulting from the quasi-

linear combinations of every i columns of a

parity-check matrix of an (n,k) block code

over a field of q. elements.

The largest integer, smaller than or equal to d/2. •

10.

EH/c] A matrix formed by placing the elements of matrix Da on the left and the elements of matrix EC] on the right.

A matrix formed by placing the elements of matrix CH] on the top and the elements of matrix [C] on the bottom.

II.

1. INTRODUCTION

12.

1. INTRODUCTION

During the last ten years the field of coding theory

has been extensively investigated and the original concepts put

forward in the early 1950's by Shannon, Hamming and others have

been expanded and developed. A considerable amount of material

has been published during this period in the form of research

reports and papers and several excellent textbooks have appeared

presenting, in a tutorial manner, the fundamentals underlying

coding theory and the recent discoveries in this subject. The

books of Berlekamp(3), Lin(5), Van Lint(6)

and Peterson and

Weldon() between them provide a comprehensive coverage of

algebraic coding theory, and summarize most of the work published

in the field during the last three decades, while other books

such as Massey(9) and Forney(10) have considered in detail

specific aspects of coding theory. Texts on general communic-

ation such as Lucky, Salz, and Weldon(1l), Gallager(12)

, and

Stiffler(13) have also devoted a great deal of space to the

problems of this field. Work in coding theory may therefore

be considered to be well advanced and the concepts and approach

to its problems to be well established. Some authorities such

as Wolf(1) and Chien(2) have consequently referred to the subject

as having reached a stage of maturity.

The remaining problems in the field can still, however,

be considered to be:

(a) Finding "good" codes:-

In spite of the fact that there are many known classes

of codes (such as Reed-Muller(25,26), Bose-Chaudhuri-Hocquenghem

(23,24), and Quadratic Residue(3,7) codes) the problem of

constructing arbitrarily long codes which meet or even come

close to the Varshamov-Gilbert(60,27) lower bound is an

important one, and is as yet unsolired except for codes with

rates of approximately 0 or 1.

13.

Justesen(59) in 1972 presented a class of codes air

any rate, whose ratio of minimum distance to block length

approaches a nonzero limit as the block length increases.

However, this limit is typically (.t rate 1/2) about 20 percent

of the Varshamov-Gilbert bound on achievable distance. The

class of Justesen codes is the only known class of asymptotically

good algebraic codes.

The gap between the.rates of known code and theoretically

achievable rates is still very wide, even for codes with moderate

length.

Since the early 1950's most of the effort in coding

theory has been directed towards constructing random-error-

correcting codes. The problems of treating burst-and-random-

error-correcting codes are, however, more complicated and have

still received considerably less attention.

(b) Finding practical decoding techniques:-

The most important known decoding algorithms are the

Viterbi(132-136) algorithm for maximum likelihood decoding of

convolutional codes, the Berlekamp(3) decoding algorithm for

BCH codes, and the majority logic decoding algorithm

Each of these decoding algorithms has unfortunately some severe

inherent limitations due to either the complexity of decoding

or its very restricted applicability. Other known decoding

algorithms (such as error-trapping techniques, and the Meggit(72'

73) decoding algorithm for cyclic codes) are simply implementable

for short codes but for long and high rate codes with large

error-correcting capabilities they are ineffective and impractical.

Under these circumstances a need exists for easily implementable

decoding algorithms.

14.

The solutions to the above classical problems have

been sought in two basic ways:

(1) Algebraic School (Hamming(30), Slepian(19), Reed-

Muller(25), Bose-Chaudhuri(23'24), ....). Here the approaches

are:-

(a) to find mathematical structures that yield

codes with desirable metric properties,

(b) to find specific encoding and decoding algorithms

that exploit the mathematical structure of the code.

(2) Probabilistic School (Shannon(103,104), Wozencraft(137,130

Gallager(12), Fano(139,140),...). Here the approaches are:-

(a) to find code ensembles with- good average

properties,

(b) to find encoding and decoding procedures applic-

able to the entire ensemble.

Our main concern in this thesis will be with that

branch of the algebraic school which is concerned with linear

block error-correcting codes. In Section 3 of this thesis the

problems of constructing and decoding block codes are

considered and several new procedures are introduced. The

problems of decoder complexity of block codes are investigated

in a detailed manner in Section 4 and a new class of codes is

described, which have been _ named "nested codes". The

codes of this class cover a wide range of rates and error-

correcting capabilities and possess a useful mathematical

structure which leads to a simple decoding algorithm whose

complexity increases only linearly with the code length and

number of errors that may be corrected. Moreover, the decoder

is shown to be capable of correcting some errors with weight

greater than the error-correcting capability of the code.

15.

Those techniques and methods introduced in Sections

3 and 4 are modified in Section 5 to enable the construction

and decoding of either random-and-burst or multiple-burst-

error-correcting codes. A number of codes are included as

examples of these procedures. In addition, a new class of

burst-and-random-error-correcting codes is introduced. Codes

of this class and their decoding algorithm are found to exhibit

the same properties as the nested codes introduced in Section 4.

In Section 6 the lower bounds on minimum Hamming

distances for linear block codes are examined. It is shown

that the Varshamov-Gilbert bound may possibly be improved.

Furthermore, a lower bound on the maximum Hamming distances of

linear anticodes is presented.

In view of the large volume of material that has been

published reviewing the fundamentals and concepts of coding

theory, it is felt that this thesis need not contain an

extensive introduction along these lines. However, a brief

review of those concepts of the field which are relevant to the

work of this thesis will be given. Section 2 therefore

contains a description of the classical techniques for the

construction, analysis, and implementation of linear block

codes and gives a brief outline of the bounds which are

eommonly applied to their performance. Other aspects of the

subjects which will be useful in the mathematical formulations

employed in the main body of the thesis are also outlined.

16.

♦ 2. FUNDAMENTALS OF CODING THEORY

2.1 The analysis of Linear Block Codes.

2.2 The Construction of Linear Block Codes.

2.3 Decoding of Linear Block Codes.

2.4 Minimum-Distance Bounds for Binary Linear Codes.

17.

2.1 THE ANALYSIS OF LINEAR BLOCK CODES

An (n,k) linear block code comprises a block of k

information digits and r=n-k redundant (or parity-check) digits.

It is a collection of M=qk distinct vectors called codewords.

Each codeword is an n-tuple vector of the vector space Vn over

a field F of q elements. A set of M such n-tuple vectors,

denoted by V, is a linear code, if and only if, it forms a

subspace of the vector space V over the field F of all qn (5) n

The rate of the code, R, is defined as(3):

R = (log M)/n = kin

For binary codes, q=2, while for nonbinary codes the

integer q>2. Usually q is chosen to be a prime number or a

prime raised to an integral power. If a set of the basis

vectors of the subspace V are considered as the rows of a k-by-n

matrix EC] then this matrix [G] is called the generator matrix of the code V. The reduced echelon form of the matrix DI generates the systematic form of the code. Its first k columns

make up a k-by-k identity matrix DO, whereas the remaining

(n-k) columns form a k-by-(n-k) arbitrary matrix [b].

n-tuples

VIM •

1- 0' . : : 0 b1,1 b1,2

. . .

b1,(n-k)

0 1 . . . 0 b2,1 b2,2 • •

. b2,(n-k)

• • • •

• • • • •

• • •

0 0 . . . 1 bk,1 bk,2 .' bk,(n-k)

EGKIkl bj=

A set of basis vectors from the null space V' of the

subspace V can be considered as the rows of a parity-check

matrix [H]. The reduced echelon form of [H] is an (n-k)-by-n

18.

matrix, the first k columns of which made up an (n-k)-by-k

matrix, and the remaining (n-k) columns an (n-k)-by-(n-k)

identity matrix The (n-k)-by-k matrix can be expressed

in terms of the negative transpose of the arbitrary matrix E].

DO= E-bT In-kJ

-b1,1 -b2,1 . . . .-bk,1 -b1,2 -b2,2 -bk,2

1 0 0

0 1 0

•

• •

-b1,(n-k) -b2,(n-k)* '-bk,(n-k) 0 0 ... 1

The parity-check matrix of a code can itself be a

generator matrix generating the so-called dual of the code

generated by EG-1 (7)

Since V is a rowspace of matrix [GJ, and V' is a

rowspace of matrix Da and a null space of V, an n-tuple

vector v is in V if and only if it is orthogonal to every row

of DO. That is to say, v[HT:1= 0

The above equation holds for every vector v4EV, and therefore

it follows that(7);

EG3 DT] = 0 As a consequence of the above results, for any n-tuple e that

is not a codeword(10;

e DT.3 0

The (n-k)-tuple vector (s) is referred to as the syndrome of

the n-tuple vector e.

The Hamming weight of a codeword vector is equal to

the number of non zero elements in that codeword vector.

19.

The weight distribution of a code is a list containing

the number of codewords of each possible weight (0,1,2,...,n).

The minimum weight of a code is the integer equal to

the smallest nonzero weight of a codeword in the code(3).

The Hamming distance between two n-tuple vectors is

equal to the number of components in which these vectors differ.

For linear codes, since the distance between any two codewords is

equal to the weight of another codeword in the the code, the

minimum Hamming distance in any (n,k) linear code therefore

coincides with the minimum Hamming weight of the nonzero

codewords.

The measure of minimum Halauting distance of a code drain man

provides important information regarding the capability of the

code to detect or correct random errors (or both)().

The codewords of a linear code are all the solutions to

a set of (n-k) homogeneous linear equations, called generalized

parity-check equations(). The coefficients of these equations

are elements from the appropriate field F. This implies that a

vector v of Hamming weight w, specifies a linearly dependent

set of w columns of the parity-check matrix DI conversely a

linear combination of w columns of [HJ resulting in the zero

vector, specifies a codeword v of weight w. It follows that

the (n,k) linear code V that has parity-check matrix [HJ will

correct all errors of weight t or less, if and only if every

2t columns from [HJ are linearly independent(7); and in any

(n,k) code the minimum Hamming distance is the number of

columns of [HJ in the smallest linearly dependent set.

Linear block codes, the class of codes with which

this thesis is exclusively concerned, form algebraic groups.

The mathematical decomposition of these groups as proposed by

Slepian(19)

is usually referred to' as the standard array. For

20.

an (n,k) linear code, the standard array is constructed by means

of the following procedure.

Place the M codeword vectors of the code in a row with

the zero vector as the leftmost element. From the remaining

(qn-qk) n-tuples, choose any n-tuple, e, and place it under the

zero codeword vector as the leader of the new row. The row is

then completed by placing under each codeword vector of the

first row its sum with e. This procedure is repeated until all

the qn-k rows of the array have been constructed. The set of

elements in a row of this array is called a left coset, and the

element appearing in the first column is called the coset leader.

No two n-tuples formed by the above process of adding different

codewords v1 and v2 to e can be identical because if (e+v1)=(e+v2),

then v1=v2' which is impossible. All the n-tuples of a left coset

have the same syndrome as the coset leader e since(5):

(e v) DT] e :HT]

Then, since every left coset has one syndrome and since

there are qn-k coset leaders, the left cosets are disjoint ().

A perfect linear block code is_defined as a linear code

that for some t has all patterns of weight t or less, and no

others, as coset leaders(5). Consider an (n,k) linear code over

GF(q). Each of these (n-k)-tuples indicates a particular error

pattern, including the no error pattern.

TherefEre in such a case the code can correct up to:

) (q - 1)

1=1 . . error patterns, and hence for a perfect code we must have ():

qn-k

(q - 1)i 1=1

Some examples of known perfect codes are as follows;

repetitive codes, the Hamming drain = 3 codes over any field;

21.

the two Golay codes (11,6), dmin = 5 code over GF(3), and

the(23,12)drain =7 code over GF(2).,. Van Lint(20) proved

that no other perfect codes exist over GF(Pa) for any a and P<:(d min - 1)/2, and Tietavainen

(21)(22) completed the

proof for P2>(dmin - 1)/2.

A quasi-perfect linear block code is defined as

a linear code for which some t has all coset leader patterns

of weight t or less, some of weight t+1, and none of greater

weight. In view of the previous arguments it can be seen that

a quasi-perfect code with elements from GF(q) is a block code

whose parameters satisfy the conditions

t+l

E(nixq _ i)i qn-k t

1=1 1=1

Note:

It is perhaps useful at this point to make a

distinction between good codes and optimum codes. In

general, a good code is that code which has the largest

number of information symbols for given values of code

length and minimum distance. On the other hand, an

optimum binary code is defined as the binary group code

for the binary symmetric channel, its probability of

error being as small as for any group code with the same

value of n, and k.

22.

2.2 THE CONSTRUCTION OF LINEAR BLOCK CODES

The major part of research effort in coding theory

has been directed towards finding ways and means of constructing

codes of high rates which exhibit a considerable mathematical

structure. The reason for the emphasis on the mathematical

structure is that it forms a basis for very simple coding and

decoding procedures. Because of this, most of the research in

block codes has been concentrated on a subclass of linear codes

known as cyclic codes, as is seen from the fact that the best

codes discovered during the past decade are cyclic. Bose-Chaudhuri-

Hocquenghem (BCH) code(23,24), quadratic residue (QR) codes (3,7)

(25' 26)

and shortened Reed-Muller (RM) codes are all cyclic.

However, it has been felt that the greater the mathematical

structure that a class or family of codes exhibits, the further

these codes are from the Varshamov(60) Gilbert(27) lower bound

on minimum Hamming distance; this is particularly so for large

values of n. Berlekamp(28) has recently shown that for BCH

codes we have for n tending to infinity:

d-f-1-(2nln R-1)/logILI

which is evidently an undesirable property. It is not yet

known, however, whether long cyclic codes are equally bad.

Kasami(29) has shown that good linear codes cannot be too

symmetric. This he achieved by showing that any code with given

d/n, which is invariant under the affine group, must have a rate

R tending to zero as n tends to infinity (this includes BCH

codes).

In this section a brief review is given of those

procedures for constructing codes that have led to some of the

best known linear block codes.

23.

2.2.1 Repetition Codes

This is the simplest example of binary codes. A

repetition code has one information digit and an arbitrary

number(r)of check digits. The value of each check digit is

identical to the value of the information digit. It is evident

that this class of codes is capable of correcting up to r/2

random errors. Repetition codes are perfect, and have the

lowest rate and the highest error-correcting capabilities.

2.2.2 Hamming Codes

Hamming(30) presented a class of perfect single-error-

correcting binary group codes, described through the parity-

check matrix. The columns of the parity-check matrix are all

the possible m-tuple vectors of the vector space Vm over the

field F of two elements. Since all columns of the H-matrix

are distinct and non zero, the null space of this matrix has

minimum weight 3 and is capable of correcting all patterns of

not more than one error. The code has m parity-check symbols,

and each code vector has a length of 2m - 1 and therefore 2m - 1-

m information symbols. A double-error--detecting and single-

error correcting Hamming code is obtained by adding an overall

parity check to a single-error-correcting Hamming code. This

code has the following parameters: n = 2m - 1, n-k = m+1, and

t = 1. The cyclic version of Hamming codes was recently

reported in references (8), (32), (33), (34) and (35).

2.2.3 Golay Codes

Golay(31) presented the only known multiple error-

correcting binary perfect code which is capable of correcting

up to three random errors. This code, (23,12), is cyclic

with many interesting properties,'as will be discussed later.

24.

C

The generator polynomial of this code is either:

L(x) = 4. x2 4_ x4 4 x5 .1. x6 4. x10 4 xll

4 x7 4. x9 4 x11 4x) = 1 + x + x5 + x6

Both gi(x) and g2(x) are factors of x23+1. The (24,12)

extended Golay codes have many important combinatorial properties

which have been studied by many authors (references (37), (38)

and (39)).

2.3.4 Circulants Codes

Leech(46) showed that the generator matrix of the

(23,12) Golay code can be written as:

1

where EC] is a circulant matrix, that is, each row is a cyclic shift of the previous row by one place. Karlin(47 '48) found a

large number of binary codes generated by circulants. Pless(49,50)

used the same properties of the circulant matrix to find some

new codes over GF(3),

2.2.5 Optimum Codes

A binary group code is called optimum for the binary

symmetric channel if its probability of error is as small as

for any group code with the same total number of symbols and

the same number of information symbols(7). Fontaine and

Peterson(56) presented a computer search procedure to find new

optimum linear block codes. The procedure was used to obtain

initially as good a code as possible and then to find all other

codes which have a smaller probability of error. Some of the

25.

new codes were also given. Tokura, Taniguchi and Kasami(53)

also found some new codes by using, a so called, systematic

computer search procedure for finding optimum error-correcting

linear block codes.

2.2.6 Quasi-Perfect Codes

The definition of the quasi-perfect code is given in

Section 2.2. The importance of such codes lies in the fact that

quasi-perfect codes by definition are optimum codes. Moreover,

Bose and Kuebler(81) were able to make limited progress in

finding optimum codes. They considered a code optimum if it

would correct all errors of weight t or less, with t as great

as possible, and would correct as many errors as possible of

weight t +1.

Wagner(51)(52) used the properties of the parity-check

matrix of binary linear codes to derive quasi-perfect codes. The

properties he employed are as follows:

1. Every 2t distinct colums from EEO are linearly

independent.

2. The set of colums from [HJ is (t+2) relatively

maximal in the space of all 2n-k possible columns. The

term "relatively maximal" is explained as follows. A

set 1A1 of vectors in a vector space S is called r-

relatively maximal in S, if every set of r-distinct

vectors from {Al is linearly independent and if every

vector ve:S is equal to a linear combination of r-1 vectors

from [Al (zero coefficients allowed). By computer search

using the above condition, many quasi-perfect codes were

found.

26.

2.2.7 Reed-Muller Codes

2m-r. The code is generated by a matrix EG: of k linearly d =2 min

independent rows. The first row is a vector vo whose 2m

components are all l's and m rows consisting of the vectors v1,

v2' .., vm of the characteristics illustrated for m=3 below.

The remaining k-m-1 rows are the modulo-2 product of the linear

combinations of every r vectors v1, v2, V.

vo 1 1 1 1 1 1 1 1

v1 0 0 0 0 1 1 1 1

v2 0 0 1 1 0 0 1 1

v3 0 1 0 1 0 1 0 1

Reed-Muller codes are equivalent to cyclic codes with

an overall parity check.

2.2.8 Concatenated Codes

Elias(61) presented the principle of the product of

two codes (n1, k1, d1) and (n2, k2, d2) to ) -t-o obtain a more

powerful (n1n2, k1k2, d1d2) code. The construction of product

codes may be best described by its codeword array. The inform-

ation digits k1k2 of the (n1n2, k1k2) code are placed in array

of k2 rows and k1 columns, the rows are extended by n1-k1 parity check digits, according to the encoding rules of the

(n1,

k1) code; the columns are extended by n2-k2 parity check

digits according to the encoding rules of the (n2, k2) code;

this is illustrated on the following page.

The resulting product codes are linear codes. Goldberg (15,62) showed by using new augmentation techniques that new

codewords may be added to a given product code while the

minimum distance is kept fixed.

Reed-Muller codes(25,26) are a class of binary group

codes with the following parameters. For any m and r <m there

is a Reed-Muller code for which, n= 2m, k = 1 + (m) + (2) + + (m) 1

27.

Checks

on

Rows

1

Information

Digits

Checks on

Columns

Checks on

Checks

The concatenated codes were presented by Forney(10)

In that procedure two codes are used, one called inner (n1,k1)

code with symbols from GF(q), and the other outer (n2,k2) code

with symbols from GF(qk ). The codewords of the concatenated

(n2n1, k2k1) codes are coded in two steps. First, the k1k2 information digits are divided into k2 bytes of k1 symbols.

These k2 bytes are encoded according to the rules for the outer

code. Second, each k1-digit byte is encoded into a code vector

in the inner code, resulting in a string of n2 code vectors of

the inner code, and hence a total of n2n1 digits.

2.2.9 Cyclic Codes

The study of general cyclic codes has been done

firstly by Prange(36) and references (3), (5), (7) and (8)

contain excellent expositions of these codes. Here we give

below a short exposition of cyclic codes and their salient

properties. Let T denote the cyclic operator for n-tuples, i.e.,

TCX .x x ,....,x 0- l' n-1 n-1' n-2]

A linear block code is said to be cyclic if the cyclic shift

Tx of every codeword x is again a.codeword. The components

•

28.

of an n-tuple codeword are treated as coefficients of a poly-

nomial as follows: n-1

a = E ao,ai, • .• a(x)

1=0

Ta =n-1Ya0' ...,an-2:] x a(x) an-1(xn- 1)

The structural properties of cyclic codes can be summarized as

follows. An (n,k) cyclic code over GF(q) is characterized by

a unique monic polynomial of degree n-k, called the generator

polynomial g(x) over GF(q), such that an n-tuple a=Ca 0 a' l' • • • ...,a11-1

is a codeword, if and only if, a(x) = a0+a1x +

+ an-lxn-1 is a multiple of g(x). Moreover, g(x) divides

(xn- 1). The set of codewords is the row spaces of the matrix

g0 gl g2 • gn-k-1 1 0 . . . 0

0 g0 g1 • . gn-k-2 gn-k-11. . . 0 G

.1.11111

o. . . 0 go g1

gn-k-1 1

The parity check matrix may be chosen as

1 hk-1 hk-2 . . . . h1 h0 0 0

0 1. . . . . h2 hl 0 h0

•

• 0 1 hk-1 h2 "- hi

where h(x)=(xn-3.)/g(x)=h0+11ix+ • • • • + hk_ix x k •

Conversely every monic polynomial g(x) which divides xn- 1

generates such an (n,k) cyclic code. The row space of the

parity-check matrix H of an (n,k) cyclic code is also a

cyclic code.

29.

An irreducible polynomial P(x) over GF(q) is said to

be primitive if its roots are primitive elements of GF(qm)

where m is the degree of P(x). This is fully equivalent to the

statement that the least integer n such that P(x) divides Kn - 1

is n = qm - 1 since this n is the multiplicative order of the

roots of P(x).

If g(x)= TT P.(x) is a product of distinct monic TT

irreducible polynomials over GF(q), then the cyclic code

generated by g(x) has length n equal to the least common multiple

of (m1, m2,

m/)wherem.is the multiplicative order of the

roots of P.(x), i.e. the least integer such that P.(x) divides

X

A cyclic code over GF(q) is called irreducible if its

check polynomial h(x) is irreducible over GF(q). The above

general properties of cyclic codes have been used to discover

many new codes during the last few years (see bibliography).

An (mn0, mk0 ) linear code is said to be quasicyclic

with basic block length n0 if every cyclic

by nn digits yields another codeword(54)

`' Weldon(55) gave a short computer generated

quasicyclic codes of rate 1/2. They also showed that there

exist a very long quasicyclic codes which meet the Gilbert bound.

Z. 2. 10 11-12Bose-cbudhuri-lcc 1...g isLLch2rri Codes

The BCH(23)'(24) codes are first of three of the most

important classes of linear codes discovered to date. These

codes are cyclic codes and are most conveniently described in

terms of the roots of g(x). Let q be a prime power, m be the

order of q modulo n, and 01 be a primitive nth root of unity in

GF(qm). Then the BCH code of length n, designed for a distance

d = dBCH and having its symbols from GF(q), will have the

parity-check matrix, [H3 below:

30. •

shift of a codeword

Chen, Peterson and

list of the best

EH] =

1 •

•

r

1

a

•

•

ad-1

a.2

oc4

ce(d-1)

CE 1-1 a2(n-1)

a(d-1)(n-1)

If n.= qn- 1 the code is called primitive and the BCH

bound of any such code has actual minimum distance

dmin. > dBCH

BCH codes have been the subject of much study owing

to their great generality. Berlekamp(3) gives an excellent

exposition of these codes. Kasami and Tokura(4o) showed that

there are binary primitive BCH codes of length n= 2m - 1 for

which dmin> dBcH. Extended binary BCH codes have also been

studied (references (41), (42), (43), (44) and (45)).

2.2.11 Goppa Codes

Goppa(57),(58) presents a new class of codes (see also

reference (82)). This class of codes is considered to be the

second of the three most important classes of linear codes

discovered to date. These codes are in general noncyclic. The

only cyclic codes in the class are the BCH codes. The basic

principles involved in the construction of this class of codes

are as follows:

Let integers m,t be given satisfying 3$;;m<i2m. Let:

Z = zeGF(2mt)I degree of minimal polynomial of

z is mtl

and let 0. be a primitive element of GF(2m). Then for any zaZ

the binary Goppa code (m,t,z) is the (n= 2m, k2m -mt) code

with the mt x 2m parity check matrix:

31.

1 1 1 [H] = 0 'z-1' z-Ot''''' z Wm-3

Goppa has shown that the minimum distance of (m,t,z) is

(i) at least 2t+ 1 for all zEZ, (ii) equal to that given by the

Gilbert bound for some zeZ. Unfortunately it is not as yet

known how to choose z4aZ so as to make this happen.

2.2.12 Justesen Codes

Justesen(59)

presented a constructive sequence of binary

codes for any rate R, 0<R<1, such that the minimum distance d is

given by:

— (1- r-1R) H

-1(1 - r) >0

Hence the codes are asymptotically good, when r is at its maximum

value of 1/2 and the corresponding rates in general given byl- 2

R = 1 + lo 1 - H-1(1- r)]

H(x) = -x log2 x-(1 - x)log2(1 - x) is the binary entropy

function.

The construction of these codes is based on Forney's(10)

concept of concatenated codes in which the m information digits

of an inner binary code are treated as single digits of an outer

Reed-Solomon(59) code over GF(2m), by generalizing the concept to

allow variation of the inner code. The inner codes are given by

a simple algebraic description and are shown to be equivalent to

the 2m - 1 distinct codes in the ensemble of randomly shifted

codes described by Massey(9) and attributed to Wozencraft.

Justesen codes are considered as the third of three most

important classes of linear codes. Justesen codes are the only known

asymptotically good codes; however, the ratio of minimum distance

to block length approaches a nonzero limit as the block length

increases, which is typically (at rate 1/2) about 20 percent of

the Gilbert bound on achievable distance.

• 32.

2.3 DECODING OF LINEAR BLOCK CODES

The construction of sufficiently simple error-correcting

devices is an important problem of the application of correcting

codes in data systems. The use of the algebraic features of some

classes of codes has opened up interesting possibilities in this

direction.

In this section we consider the principles involved in

the decoding of those classes of codes discussed in Section 2.2

and also the complexity of the associated decoding algorithms.

Most of these decoding algorithms are suitable for decoding

short random error-correcting codes. However, when they are

applied to long and high rate codes with large error-correcting

capabilities they become very ineffective and impractical.

There exist, however, two known methods for decoding long

random error-correcting codes that belong to the BCH class and

the class of majority logic decodable codes. Both of these

decoding algorithms have inherent limitations in that the

complexity of decoding is far from simple. We shall now

consider several decoding procedures that are in existence and

are applicable to the important classes of codes mentioned in

previous section. The complexity of decoding is generally

measured by the number of operations required for decoding a

single information symbol. The number of operations required

for decoding is the average number of times a sequence of

symbols at the channel output must be compared with code

combinations at the channel input to decode one information

symbol.

2.3.1 Syndrome Decoding

Basically all decoding algorithms of linear block codes

are syndrome decoding in nature. We have defined in Section 2.2

33.

the (n-k)-tuple syndrome vector s for a received n-tuple vector

r as:

s = Es1' s2, °°'" sn-0=0 i=[ri, r, rn=1 DT]

For a given r, we say that an error pattern em is a valid error

pattern if r - em is a codeword; this implies that there is some

codeword vm such that vm +em =r. However, there are qk distinct

s = (vm+em) [HT] = vM :HT] em DT] = em [HT] ; since v

EHTD = O.

The syndrome is independent of the codeword transmitted

but depends only on the channel error sequence. Thus every valid

error pattern is a solution of the syndrome equation;

s = e DJ.]

Since the set of qk valid error patterns is exactly

the set of solutions of the syndrome equation s = e BIT] / we see that a decoder which decodes r into r- e, where e is (one of)

the solution(s) of s=eHT that maximizes Pn(e), is a maximum

likelihood decoder for an algebraically additive channel*. This

follows from the fact that P(r/v)=Pn(e) so that v=y- e

maximizes P(r/v). The above can be summarized as follows: For

a given syndrome s, the qk solution of s=eH

T are the qk valid

error patterns. A maximum likelihood decoding rule for an

algebraically additive channel is to decode r into v=r- e

where e is the solution of s = eHT which maximizes Pn(e).

A channel is said to be algebraically additive if P(r/v)

depends only on e=r-v, i.e., P(r/v)=Pn(e) where Pn is

the probability distribution for error patterns.

codewords and hence there are exactly qk valid error patterns

for each possible r. Equation (2.4.1) can be rewritten as

follows:

• 34.

The syndrome decoding of linear block codes, therefore,

consists of three basic steps:

1) Calculation of syndrome of the received vector.

2) Identification of the correctable error pattern that

corresponds to the syndrome calculated in step 1.

3) Correction of the errors.

In general, step 2 is the most complex part of the

decoder to implement. This is because step 2 involves a one-to-

one mapping between the syndrome and the large number of qn-k - 1

correctable error patterns.

Most of the well known procedures of decoding employ

the algebraic structure of the code to simplify the implementation

of step 2. Some of the methods used are described briefly below.

2.3.2 Decoding of Binary Cyclic Codes

We have seen in Section 2.2 that a cyclic code is

defined in terms of a generator polynomial g(x) of degree n-k.

A polynomial of degree less than n is a code polynomial, if

and only if it is divisible by the generator polynomial g(x).

To encode a message polynomial m(x), we divide xn-k

m(x) by g(x)

and then add the remainder r(x) resulting from this division to

xn-km(x) to form the code polynomial F(x). That is,

xn-km(x) = g(x) q(x) + r(x).

where g(x) is the quotient and r(x) the remainder resulting from

dividing xn-km(x) by g(x). Since in modulo two arithmetic,

addition and subtraction are the same, we have,

F(x) = xn-km(x) + r(x) = q(x) g(x)

which is a multiple of g(x) and, hence, a code polynomial.

Furthermore, r(x) has degree less than n-k, and xn-km(x) has

35.

zero coefficients in the n-k low-order terms. Thus the k

highest-order coefficients of F(x) are the coefficients of r(x),

and these are the check symbols.

An encoded message containing errors can be represented

by:

H(x) = F(x) + e(x)

where F(x) is the correct encoded message and e(x) is a poly-

nomial which has a nonzero term in each erroneous position.

Because the addition is modulo two, F(x)+e(x) is the true

encoded message with the erroneous positions altered. If the

received message H(x) is not divisible by g(x), then clearly an

error has occurred. If, on the other hand, H(x) is divisible

by g(x), then H(x) is a code polynomial and we must accept it

as the one which was transmitted, even though errors may have

occurred. Therefore, an error pattern e(x) is detectable if and

only if it is not evenly divisible by g(x).

To detect errors, we divide the received, possibly

erroneous, message H(x) by g(x) and test the remainder s(x).

If the remainder is nonzero, an error has been detected. If

the remainder is zero, either no error or an undetectable error

has occurred. The syndrome of any linear block code is obtained

by taking the modulo-2 sum of the received parity check digits

with the parity check digits calculated from the received

information digits. Therefore the remainder's(x) resulting

from dividing the received vector H(x) by the generator poly-

nomial g(x) is the syndrome of H(x), i.e.:

H(x) = p(x) g(x) + s(x)

Therefore encoding and syndrome calculation can be accomplished

by a division circuit used to calculate the remainder of the

division. The hardware to implement this algorithm is a shift

36.

register and a collection of modulo two adders. Figure 1 below

shows an (n-k)-stage shift register to calculate the remainder

of the division r(x) = q(x) g(x) + xn-k

m(x)

F(x) + xn-km

(x)

•

denotes a connection if the coefficient of xi in

g(x) is 1, and no connection if the coefficient is zero.

The remainder calculating circuit that uses an (n-k)-

stage shift-register was proposed by Peterson(69), while the

circuit using a k-stage shift register was first reported by

Green and San Soucie(70), and Prange(71)

Once the syndrome is calculated, the second step is

to identify the correctable error pattern that corresponds to

that syndrome. A possible method is to form a logical

dictionary assigning for every possible syndrome a corresponding

error pattern. However, for long codes this becomes impractical

because of the large number of words in the dictionary. Meggit (72),(73) presented a general error-correcting decoder for

cyclic codes. This he achieved by noting that the error pattern

which corresponds to the calculated syndrome can be determined

by means of a combinational logic function. The procedure in

essence is as follows. The input of the combinational logic

circuit is the calculated syndrome polynomial s(x) from which

the output en-1xn-1 is produced: from this operation the error

polynomial e(x) is determined (i.e. a total of n-shifts is

required to estimate e(x)).

37.

Whether Meggitt decoder is a "practical" decoder for

a cyclic code depends entirely on the complexity of the logical

circuit, Generally, if only t or fewer errors are to be

corrected, where t is about 3 or less, the decoder is much

easier to implement than the corresponding BCH decoder. (The

BCH decoder will be discussed in a later section.)

2.3.3 Error-Trapping for Binary Cyclic Codes

This process is based on the simplification of the

following equation:

H(x) = p(x) g(x) + s(x)

which is simplified to:

s(x) = F(x) + e(x) + p(x) g(x)

We can see from the above that since the code polynomial F(x)

is divisible by the generator polynomial g(x) the remainder of

dividing s(x) by g(x) is entirely due to the error polynomial

e(x). We can also see that the syndrome polynomial is identical

to the error pattern polynomial by noting that the above

equation can be written as:

s(x) = e(x) + B(x) g(x)

where B(x) is the quotient of dividing e(x) by g(x). If the

error polynomial is of degree (n-k-1) or less, then B(x) = 0

due to the fact that g(x) is a polynomial of degree (n-k).

Hence it follows from the above equation that:

s(x) = e(x)

This implies that the syndrome is identical to the

error pattern if the errors are confined to the check digits

only. Note that the weight of the syndrome is t or less if

the correctable error pattern is confined to the check digits

38.

only, and the weight is greater than t if the correctable error

pattern is not confined to the check digits.

From the above results one is led to a simple decoding

algorithm for cyclic codes: for example, Mitchell(74),(75)

Rudolph(76) and MacWilliams(77) have described the operation of

the decoder, which can be summarised thus:-

•

(i) Calculate the syndrome s(x) by dividing the received

polynomial by the generator polynomial and retaining the remainder.

(ii) Test if the syndrome is of weight t or less in which

case the error pattern is identical to the syndrome polynomial.

(iii) If the weight of the syndrome is greater than t

shift the received polynomial H(x) by one position xfH(x)1 and

repeat the above steps. This procedure is repeated until the

weight of the new syndrome is t or less. If this cannot be

achieved having gone through the entire number of shifts, the

error cannot be corrected.

It can be seen, therefore, that the error-trapping

decoding procedure is very effective for decoding single error-

correcting codes. However, when it is applied to multiple

error-correcting codes, it becomes very ineffective and much

error-correcting capability is lost. This is because all

errors that occur at positions of (n-k) or greater digits

apart cannot possibly be corrected.

A modification to the above error-trapping procedure

was introduced by Kasami78) in which he made the error-trapping

technique effective for some multiple error-correcting codes.

The principle of the modification is as follows. A set of

polynomials tQi(x)I of degree k-1 or less is chosen such that

for any error pattern e(x) of weight t or less, there is one

polynornialQ.WinthesetforwhichxrQi(x) agrees with e(x)

39.-

or a cyclic shift of e(x) in its k information positions. Let

qi(x) be the parity sequence associated with the information

sequence. Ql(x). We recall that the syndrome is the sum of

the noise sequence in the information positions. Thus, adding

q.(x) to the syndrome s(x) gives the error pattern in the

parityplaceswhenxrQ.(x) coincides with xr+i

e(x) in the

information positions. Since the code has a minimum distance of

at least 2t+ 1, this coincidence can be detected uniquely by the

inequality:

wEsi(x)+qi(x)] t v&i(x)..]

Then the original error pattern e(x) is given by:

e(x) = )ck-i(s.(x) q.(x). Qi(x)) (Mod xn- 1)

Kasami shows that such a set of tQi(x) polynomials

can be found for some cyclic codes and gave the set of polynomials

required for the cyclic Golay code, the Bose-Chaudhuri (63,45),

(31,16), (31,11) codes and the (41,21) cyclic codes.

2.3,4 Majority Logic Decoding Algorithm

In this section we consider the majority principle of

decoding cyclic and non cyclic linear block codes, which often

enables a simple solution of the problem of correcting multiple

errors to be found. This type of decoding is based on the

possibility, that for certain cyclic and non cyclic codes, the

value of any one symbol of a codeword may be expressed by

several independent methods as a linear combination of other

symbols. Independence of the methods is understood in the

sense that no two expressions depend on the same symbols.

A single distortion can change only one expression, two

distortions can change not more than two expressions, etc.,

t distortions, therefore, can change not more than t

expressions. For correct decoding of the symbol it is then

40.

sufficient to have 2t + 1 independent expressions defining the

given symbol in terms of all the remainder and the decision as

to the definite symbol to be transmitted is taken from the

majority of the meanings given by each individual expression.

The possibility of using the principle of decision

by the majority (majority principle) for decoding cyclic codes

was first indicated by Green and San Soucie(79) using the

example of the (15,4) code with a Hanudng distance of 8.

Reed(26) used this principle for decoding noncyclic Killer codes(25)

Massey(9) unified the theory of majority logic decoding which is

outlined below.

Let e= e0' e1, •' en-1 be an error pattern, for

the syndrome of which is given by:

s = s s s 0' 1, 2' sn-k-1 = eH

T

where:

s = e0 0

S1 = e1

en-k +.7.:74-13k1 e

n-1

en-k 1-****4-Pk2 en-1

sn-k-1

= en-k-1 4V1,(n-k) en-k —4-21k,(n-k)en-1

Consider a check sum A of the syndrome bits defined as:

A = a0 s0

+ alsl + . . . . +

an-k-1 sn-k-1

= b0 e0

+ blel + . . . . + b

n-1 en-1

where d.1 =1 or 0 and b. = 1 or 0. A parity check A.1 will be said

to check an error digit ei if ei appears in the summation given

by A with a nonzero coefficient. A set tAl of parity checks is

said to be orthogonal on em if each Ai checks em but no other

error bit is checked by more than one Ai. Each parity check in

an orthogonal set thus provides an independent estimate of em

41.

when the noise bits are statistically independent. If, then,

there are 2t parity-check sums orthogonal on em, m=n.-k, n-k+l,..

.., n-1, the value of the error digit em is given as the value

assumed by a clear majority of the parity check sums orthogonal

on em. A code with minimum distance d is said to be completely

orthogonizable in one step if and only if it is possible to form J=d-1

parity-check sums orthogonal on every error digit. A set of J

parity-check sums Al, A2, ..., Aj is said to be orthogonal on

the setiEif and only if: (1) every error digit eit in E is

checkedbyeverychecksumA.for j= 1, 2, J, and (2) no

other error digit is checked by more than one check sum. The

process of estimating sums from sums of larger size is called

orthogonalization. The orthogonalization process continues

until a set of J or more parity-check sums, orthogonal on every

error digit, are obtained. A code is said to be L-step orthogon-

izable (or L-step majority-logic decodable) if L steps of

orthogonalization are required to make a decoding decision on

every error digit.

The majority method of decoding is appreciably simpler

and more reliable than the decoding scheme of Peterson(80)

or

the decoding algorithm of Berlekamp(3) for Bose-Chaudhuri codes.

The deficiency of this method is that, in all the cases

considered, cyclic codes with majority decoding schemes are no

better in correcting capability than the corresponding (n,k)

Bose-Chaudhuri codes. Nevertheless, because of the simplicity

of the decoding devices, codes with majority decoding schemes

are in many cases more effective than other schemes.

2.3.5 Decodin of BCH Codes

For the purpose of decoding BCH codes(69)

we note

that v(x) is a codeword polynomial of an (n,k,t) BCH code with

42.

ti

abeing a primitive nth root of unity in GF(2m) if and only if

al, i = 1, 2, ..., 2t are roots of:

v(x) = v0 +v

lx+v2x2 + +vn-1x

ri-1

Let the received polynomial be denoted by r(x), and

let it be corrupted by the noise polynomial e(x), i.e.:

r(x) = e(x) + v(x)

Ifwesubstitutetherootsa_of the code in r(x) we obtain:

s1 = e(0) + v(0) s2 = e(a2)+v(a2)

s3 = •e(CC3)+v(0C-3)

s 2t

e( a2t + v( ( t)

Since Ce", i = 1, 2, ..., 2t are roots of v(x), the v(4),

i = 1, 2, ..., 2t, are all equal to zero.

The vector s = s1, s2, s2t is defined(80) to be

the syndrome of the code.

If the received polynomial r(x) has errors in the

positions ji, j,`„ z5." <,t, the syndrome si, i = 1, 2, ...,2t

will be given by(6,9) :

si = (aik )1

Z=1 Any error correction procedure therefore will involve

the simultaneous solution of the set of the 2t equations given

in the above summation.

Peterson(80) presented an algorithm to facilitate the

above process of decoding. Several authors suggested various

improvements to this basic algorithm(81-85). We outline below

•

•

43.

one of the procedures which is a well known modification of

the Peterson algorithm, the simplest to date. We start by

defining the error location polynomial in the form(80)

a(x) = n (1 4- 0)2, x) 2.=1

where the roots of a(x) are the inverses of the error location

positions. The polynomial a(x) can be determined for the

received polynomial r(x) by completing the table below(3)'(8)&(5)

V

-% 2

o

a(11)(x)

1

1

dp 1

s 0

211414

-1

0

1 2

t

where the l)th row is given by:

(1) if dti = 0, the a(11-1)(x) = a(1)(x), or

(ii) if dµ # 0, find another two preceding the p, say.the pth, such that the number 2p - 2p in the last column is as large

as possible and dpi0. Then:

a(11+1) (x) = a( (x) dpdp-1 x2 (11-p) a( p ) (x)

In either case, 2 +1 is exactly the degree of a

("1)(x), and

11

d = s + al 2p (11+1)s + a2(1+1)s2

41+1 s 2 1-1 +3 - 12, µ+1

Therefore a(x) is given by a(t)(x), and hence the

roots of a(x) may be calculated to locate the errors.

At this point it might be well to point out that

finding a simple way to extend the BCH decoding algorithm to

+ a (1/41)

44.

•

- correct t:>

d1--f— errors is an important, but unsolved, problem.

The complexity of the decoder increases linearly

with the block length of the code and also with the square

of errors to be corrected. The implementation of the decoder

needs a special-purpose computer to carry out the computation

in GF(2m). Lin(5) estimates roughly that 127,000 [isec. of

computing time are needed to carry the required calculation for

each received block of (127,92) BCH code capable of correcting

5 errors, whereas 190,000 psec. are needed for (1023,923) BCH

code capable of correcting 10 errors.

Although BCH codes have in some sense very desirable

attributes (i.e. they encompass a very large class of codes of

various rates and error-correcting capabilities), they become

impracticable for large n due to the increasing complexity of

the decoder hardware and the long computation time needed

for the decoding of each block.

45.

2.4 MINIMUM-DISTANCE BOUNDS FOR BINARY LINEAR CODES

For a binary symmetric channel *(BSC), an (n,k) linear

block code V of minimum Hamming distance d can correct t random

errors if and only if d.3.2t+1; for a code that corrects t errors A t

and detects random errors, we have dt+Z+1(. This can be

justified as follows. Consider a received vector r corrupted

by an error vector of weight t or less. If t=',] d-71—, then the

received vector r is closer to the actual transmitted code

vector r than to any other code vector U. Thus, the decoder

will make a correct decoding and hence the error will be

corrected. However, the decoder cannot correct all the error

patterns of e errors, where e > t+1, for there is at least one case for which an error pattern of P. errors results in a

received vector which is closer to an incorrect code vector

than to the transmitted one. In this case the decoder can

detect Q < d-1 errors, due to the fact that no error pattern of

weight (d-1) or less will alter the transmitted code vector into

another [code vector. On the other hand, the decoder may not

detect an error pattern of Q errors fora > d since there is at

least one case of the random error pattern of Q errors that may

result in a received vector coincident with one of the codeword

vectors in the subspace V.

The maximum minimum-Hamming distance dmax of all

binary linear codes of fixed n and k, is therefore a critical

parameter in the evaluation of coding as a method of error

control. Thus to determine the ultimate capabilities and

limitations of error-correcting codes it is necessary to study

upper and lower bounds on d. The upper bound on d is defined

as the absolute theoretical maximum value of d for given values

BSC is that channel which has an equal, small probability

of changing a 0 to a 1• and a 1 to 0 (called the memory

less binary symmetric channel), and maximum likelihood

decoding . 46.

of n and k, and it is said to be "good" if it is the nearest to

the best lower bound.

s

The lower bound is defined as the upper

bound on d for those codes which can be shown to exist, and is

said to be "good" if it is nearest to the best upper bound. A

number of upper and lower bounds have been reported in the

literature. Most of these are based either on the well-known

sphere-packing argument introduced by Hamming(30),(27), the

"average distance" approach of Plotkin(63),(64)

, or a combination

of these(65)

. The Hamming upper bound(30) which is based on the

sphere-packing argument (given in Section 2.2 under the definition

of perfect codes) yields the following results. For any (n,k)

block code with minimum distance 2t+ 1 or greater, the number

of check digits (n-k) is such that: t

(q-1)1 (7)

i=0 For binary codes and large values of n the bound

approaches a simple asymptotic form that can be easily evaluated

numerically using the following inequalities(8),(83)

In\ = Tf n\ ■ii

i=0 i=n-t

n ( iia ) x- X n µ-µn

i=Xn where X2>1- and p.= 1-X.

Therefore the asymptotic form of Hamming's upper bound

is given by: -(n-t) -t

2n-k

/ ( IT. ) n

On the other hand, the Plotkifniper bound is based on the fact

that the minim= weight of a codeword in an (n,k) linear code

is at most as large as the average weight of the code. Thus the

47.

•

•

average codeword weight of an (n,k) linear block code can be

found by arranging the code vectors as rows of a matrix, where

each field element appears qk-1 times in each column. There-

fore, the number of non-zero elements in each column is

(q-1)qk-1, and since there are n columns,(will be)the sum of

the weights of all codewords in the code4q-1)qk-1

. It follows

therefore that the average weight of an (n,k) code over a field

of q elements is nqk-1

(q-1)/(qk- 1) bearing in mind that there

are qk - 1 non-zero code vectors, Hence the asymptotic form of

Plotkin's bound becomes:

d q

However, the maximum number of codewords possible,

B(n,d), in a linear code of length n with minimum weight, d,

if n>d,is such that B(n,d) -1::: qB(n-1,d). Then it can be shown()

that if n;“qd-1)/(q-1), the number of check symbols required to

achieve minimum weight d in an n-symbol linear block code is at

least equal to

Dqd-1)/(q-0] - 1 - logqd

which in the asymptotic form for binary codes becomes:

n k .3?2d-2 log2d

1 - k > 2d - - liogd n n n n 2

Hence the rate is given by R = n <1 -

Elias obtained another upper bound on the minimum

Hamming distance, by employing the concepts used in both the

Plotkin and Hamming bounds(3). In this case for large n the

bound is tighter than either the Hamming or Plotkin bounds.

For the binary case the Elias bound is given by:

-11.)(0T) d 2t(1 -

where t is any integer such that:

• 48.

T, (3) > 2n-k

j=0

and k is the smllest integer for which

k ( ())/2n-k

J-0

The Elias bound is an important upper bound since it

is the nearest known to the Varshamov-Gilbert lower bound. The

Varshamov-Gilbert bound was proposed by Varshamov(60) and is in

fact a refinement of a bound proposed by Gilbert(27). The same

bound was also found by Sacks(66)

. The Varshamov-Gilbert lower

bound on d is given by: d-2

(1) (q-1)1 qr i=0

For the binary case this bound can be simplified by

using the inequalities given in (2.2.2) and (2.2.3) and assuming

n to be very large. Hence the asymptotic form in this case

becomes:

2n-k tn-d+2\-(n-d+2)

d -(d-2) 2 (-) n I

For small values of n the numerical evaluations of-

the bound may require a large and tedious amount of computation.

However, the minimum distances of a number of well-known codes

such as the BCH, Hamaing and Srivastava codes also provide a

lower bound on the minimum Hamming distance. These difficulties,

the complexity of some of the bounds previously mentioned, and

the continued appearance of new codes and code construction

techniques have prompted Calabi and Myrvaagnes(67)

to compile

a table of upper and lower bounds on d. In May 1973 Helgert

and Stinaff(68) expanded the Calabi and Myrvaagnes table for

1<n<127, 1<k<n. This table is considered very important

and we shall refer to it throughout the thesis. This is due

to two main reasons, the first being that the table is based

49.

on all the information available to date, and the second reason

is that the table compiled the best results obtained from the

best known codes together with the bounds obtained by combining

codes with other codes according to certain rules of construction.

Or their generator and parity-check matrices can be increased

or decreased in size by adding or omitting certain rows and

columns, and the minimum distances of the codes thus obtained

can be related to the minimum distance of the original code.

The combination of these codes has the following properties:

(I) dmax(n,n) = 1; dmax(n,l) = n.

(ii) If dmax

(n,k) = 2t + 1, then

d x(n+1,k) = 2t + 2.

ma

(iii) d x(n+1,k)4C 1 + dmax

(n,k). ma

(iv) d x (n+1, k+1)< drrtax (n,k). ma

(v) dmax(n + n2k) dmax

(n1

k) + dmax(n2 ' k)

50.

3. NOVEL PROCEDURES FOR CONSTRUCTING

LINEAR BLOCK CODES

3.1 A Computerised Search for Linear Codes.

3.2 Application of Walsh Functions in

constructing Linear Block Codes.

3.3 Application of the Concept of Anticodes.

3.4 Two Procedures of Linear Block Code

Modification

(1) Code Puncturing.

(2) Code Lengthening.

51.

3.1 A COMPUTERISED SEARCH FOR LINEAR CODES

A linear block (n,k,t) error-correcting code(3W)

• ' comprises qk distinct codewords (q being the number of symbols

per sign and for binary codes q = 2), which form a subspace V

of the vector space Vn over the field F of q elements. The

basis vectors of the subspace V can be considered to be the

rows of a matrix ECC,called the generator matrix of V. The

basis vectors of the null space V' of the subspace V can be

considered as the rows of the parity check matrix Da. Since V'

is the row space of [HJ and the null space of V, a vector v is

in V if and only if it is orthogonal to every row of CH]. It

follows that for each codeword of Hamming weight W, there is a

linear dependence relation between W columns of DO, and

conversely, for each linear dependence relation involving W

columns of H, there is a codeword of weight U.)0).

Lemma 3.1:

If an (n,k) code V has a parity-check matrix Ildj,

then V will correct all errors of weight t or less if, and only

if, every 2t columns from H are linearly independent(7).

Theorem 3.1:

Let a subset U of n vectors in a vector space Vn-k be

formed over a field F of q elements, and let U contain at least

one set of the basis vectors of Vn_k. A parity-check matrix Eta

can then be produced such that its n columns are the vectors in

U. The (n,k) linear code corresponding to this matrix will

correct t random errors if, and only if, all linear combinations

of every t vectors in U give unique non-zero vectors in Vn_k.

Proof:

Since the set U contains at least one set of the basis

vectors of the vector space Vn_k, then the parity-check matrix

52.

•

Elij whose columns are the vectors of U, has a rank equal to its dimension. All the (n-k) rows of E11 are thus linearly

independent. Now consider 2t vectors of the set U. Let these

2t vectors form two sets U1 and U

2 of t vectors each. The

linear combination of all vectors of U1 form a set S

1 of qt

unique vectors. Similarly set S2 corresponds to the linear

combinations of all vectors of U2. Since S1 contains all the

vectors which are the linear combinations of the vectors (u1, u2,..

ut) EUl'

it follows that if a vector s is in S1, then all

vectors of the scalar product (as) are in S1 also, where a =

1, 2, ..., q-1. This implies that the modulo-q addition of any

vector in S1 with any vector in S2

is a non-zero vector, and

therefore every 2t vectors of U are linearly independent. As a

consequence of this, every 2t columns of the parity-check

matrix EH] are linearly independent and hence the corresponding (n,k) code can correct up to t random errors.

Using the above characteristics of the parity-check

matrix, a computer search is developed to find one code at a

time for a given number of parity-check digits and a given

error correcting capability t. The computer search follows

the following steps:

1) Read the given Da matrix. 2) Cross out the (n-k)-tuple vectors resulting from all

the linear combinations of t columns of the EH: matrix. 3) Take the vectors of the vector space Vn_k in turn,

starting with the all zero vector and ending with the

all one vector.

4) Test each vector for uniqueness. If not unique,

start again at step 3 with the next vector.

5) Test for uniqueness of all t linear combinations of

the vector with the columns of the DC matrix. If

53. •

any vector of the resultant linear combination is

not unique, start again at step 3 with the next vector.

6) Cross out the vector and all the t linear combinations

of step 5.

7) Increase the size of the DO matrix by one column by

adding the vector to theHmatrix.

8) Continue the search again starting from step 3 until

all vectors in Vn-k have been tested.

Appendix 2 contains a typical computer programme to

find a new binary code for error correcting capability t=3 and

the number of parity-check digits rr.:15. In this programme, the

parity-check matrix of the Karlin(47) (30,16,3) code, with a

dummy parity-check digit was used as a starting matrix. All

programmes are written in Fortran IV and were executed on the

London University computer CDC 7600.

Using the above search the following results have been

obtained as in table 3.1.1.

For t = 5, m = n-k = 19, and taking the 19th order

identity matrix as the starting matrix, a (26,7,5) code was

found. The best linear block code corresponding to the largest

previously known value of k for t = 5, and m = 19 was (25,6,5)(7).

For t = 4, m = 17, 18, 19 and 22, the mth order

identity matrix is taken as the starting [HJ matrix. The

following codes were found: (26,9,4), (30,12,4), (34,15,4) and

(47,25,4). For these values of t and m the previously best

known linear block codes were (23,6,4)(7), (25,7,4)(7), (30,11,4)(7)

and (46,24,4)(7).

For t = 3, m = 15, the parity-check matrix of the

Karlin(47) (30,16,3) code, with a dummy parity digit, was used

54.

as a starting matrix. A (34,19,3) code was found. The largest

previously known value of K for t = 3, and m = 15 is given by

the (32,17,3)(57) code.

For t = 3, and m = 19 and 20, starting with the mth

order identity matrix as the starting matrix, a (72,53,3) code,

and a (86,66,3) code were found. The best codes corresponding

to the largest known values of K for t = 3, and m = 19 and 20

respectively were previously (70,51,3) and (83,63,3)(84)

Finally, for t = 2, m = 11 and 13, and starting with

the mth order identity matrix as theKmatrix, the following

codes are found: (41,30,2), and (71,58,2). For these values

of t and m the previously best known linear block codes were

(39,28,2)(51), and (70,57,2)().

Using the above search 10 new good binary codes were

found plus the 10 even Hatmaing distance versions of the newly

found codes.

Owing to the limitations of computer memory and time,

the above results represent the limit of application of the

search procedure.

Table 3.1.2 gives the parameters (n,k,d) and the

first k columns of the NI matrix of twenty-six codes found by the above computerised search procedure. All codes listed in

this table are as good as the best previously known codes of

identical Hamming distance and the same number of parity-check

digits. Note that the columns of the EHJ matrix in both tables

(tables 3.1.1 and 3.1.2) are given in Octal, where the first

digit of the columns is taken to designate the least significant

digit of the Octal number.

55.

minimum

d : Helgert minimum

dL : Helgert

TABLE 3.1.1 TWENTY NEW GOOD CODES FOUND BY COMPUTERISED SEARCH

n : Code length.

k : Number of information digits.

d : Hamming distance.

and Russell(1) lower Hamming distance.

and Russell(1) upper Hamming distance.

bounds on

bounds on

d dL dU The first K columns of Do matrix in Octal. 26 7 11 11 11 1777 76037 316343 526554 653265 1132671 1255316

27 7 12 12 12 As above with addition of overall parity check digit

26 9 9 8 10 377 7417 31463 52525 65252 113152 213630 263723 306136

27 9 10 9 10 As above with addition of overall parity checkdigit

30 12 9 8 10 All columns of code (26,9) plus: 416246 520155 724616

31 12 10 9 10 As above with addition of overall parity check digit

34 15 9 8 10 All columns of code (30,12) plus: 1023305 1441516 1777651

35 15 10 9 11 As above with additionof overall parity check digit

47 25 9 8 12 All columns of code (30,12) plus: 1023305 1347214 2027151 2457261 3166444 4055666 4632577 5251417 7514712 10057307 11414574 12345175 17170103


34 19 7 7 8 23642 7504 7211 36422 35044 32111 24233 10447 21117 2236 4475 11172 22364 4750 11721 37777 40343 42507 56016


72 7 6 8 All columns of code (55,38) plus: 414510 425201 431744 612665 622311 657104 667425 1014517 1025230 1031772 1206630 1236343 1243167 1273441 1404303


86 66 7 6 8 All columns of code (72,38) plus: 2014523 225217 2031761 2317101 2327413 2352645 2362302 2407003 3106403 3400062 3431735 3733415 3776332

87 66 8 7 8 As above with additionof overall parity checkdigit

41 30 5 4 6 17 63 125 152 226 253 333 355 367 427 455 511 647 1031 1113 1214 1343 1562 1660 1710 1723 2034 2045 2203 2432 2563 3060 3102 3465 3611

42 30 6 5 6 As above with additionof overall parity checkdigit

56.

*

TABLE 3.1.1 Continued.

6 12543 15172 13115 16445 17131 17440 3220 1750 10347 14270 3066 4267 12220 5110 2444 521 14342 13353 15676 2633 4433 5073 12726 5353 12676 2573- 11166 12116 5047 12730 5354 10602 4301 15206 6503 12505 16747 17070 3626 1713 5122 2451 11137 11745 14471 16127 777 10064 2025 16012 7005 16510 7244 1661 10423 6071 13327 3374

6 As above with addition of overall parity check digit

I

n k d dL dU 71 58 5 5

72 58 6 5

57.

TABLE 3.1.2

TWENTY SIX NEW CODES AS GOOD AS THE CORRESPONDING BEST PREVIOUSLY KNOWN

LINEAR BLOCK CODES.

Code No.- of Minimum

Length Message Hamming The First K columns of [H] Matrix in Octal Digits Distance

n K d

8 2 5 17, 63

9 2 6 As above with addition of overall parity check digit

11 4 5 All columns of code (8,2) plus: 125 152


17 9 5 All columns of code (11,4) plus: 226 253 333 355 367


11 2 7 63 455


15 5 7 All columns of code (11,2) plus: 1331 1552 1664


23 12 7 All columns of code (15,5) plus: 2353 2561 2635 3174 3216 3447 3722


14 2 9 377 7417


17 3 9 All columns of code (14,2) plus: 31463





58.

TABLE 3.1.2 Continued.

Code Length

No. of Minimum Message Hamming The First K Columns of CH] Matrix in Octal Digits Distance

n K d


17 2 11 1777 76037

18 2 12 As above with addition of overall parity check digit.




24 5

12 As above with addition of overall parity check digit

59.

3.2 APPLICATION OF WALSH FUNCTIONS IN CONSTRUCTING LINEAR BLOCK CODES

Lemma 3.2:

If the result of multiplying together two Walsh

functions, Wal(h,9), where Wal(h,e) = Whl' Wh2"." WhL' Wij = amd Wal(k,9), is transformed by replacing all the positive

elements by 0's and the negative elements by l's, then the

subsequent product is identical to that produced by similarly

transforming the original functions and then performing a simple

modulo-2 addition.

Proof:

Since the Walsh functions 1Wal(i,9)1i = 1,2, ...,

(2j- 1), j = 1,2,3, ..., etc.1 form an Abelian group with

respect to multiplication, then the product of two Walsh

functions yields another Walsh function in the group

Wal(h,9) Wa14,9) = Wal(r,9)

where the value of r is equal to the modulo-2 sum of h and(85)

r = h + t

th The j element of Wal(r,9) is given by:

W = W x W . rj hj zj

The multiplication rules are given below:

X +1 -1

+1 +1 -1 -1 -1 +1

The above table is identical to modulo-2 addition, when +1's

are replaced by 0's and -1's are replaced by l's, as shown

below:

60.

Further details of the algebra of Walsh functions can be found

in Harmuth(85)

Theorem 3.2:

If the parity-check matrix EH] of an (n,k) systematic linear binary code, has the first k columns given by the subset

of the Walsh functions of length L, [Walsh, 9) I h = 2J-1, j = 1, 2,3,...,k J, then the code is capable of correcting t = L/4 random errors.

Proof:

Consider the subset V" of k vectors, of the vector

space Vk, such that each vector vEV" corresponds to the binary

representation of the number h = 2j -1, j = 1,2, thooyko This

subset is given below:

h = 1 , 0 0 0 0 0 0 1 i = 1 •

3 0 0 0 0 0 1 1 2

7 0 0 0 0 1 1 1 3

. • •

• . • .

. . . . • •

. • • • . • • . 2k-1 1 1 1 1111 k

It is apparent from the above array that the vectors vEV" are

linearly independent. Now consider the Abelian group of the

Walsh functions fWal(i,(41) 1 1=0,1,2,3,••,(2k_ 1)3 of length L = C2k where C = 1,2,...,etc. Since the vectors v in V" are

linearly independent, it follows from the one to one relationship

between the vectors v and the decimal numbers h that the subset

of tWal(h,9)i where fWal(h,0) (=IWal(1,0i has k linear

61.

independent members which form a basis for the group [Wal(i,9)3.

Since the weight of every Walsh function in the Abelian group

Wal(i,9) is L/2, it follows that every Walsh function resulting

from the linear combination of the members of the subset .XAla.1(h,9)1

are linearly independent of every (L/2)-1 of the remaining (n-k)

columns of the reduced echelon form of the Epli matrix. Hence every L/2 columns of the DC matrix are linearly independent

and according to lemma 3.2, t = (L/2)/2 = L/4 random errors can

be corrected. This completes the proof.

As an example, first consider an (n,k) linear binary

code of k = 2, where the first 2 columns of the [H] matrix are

given by Wal(1,9) and Wal(3,9). Let the length of the Walsh

functions be L = C.2k = 8, whence C = 2, and n-k = L = 8.

From theorem 3.3, the code will correct t = L/4 = 2

random errors. The parity check matrix is given by:

Wal(1,9)

1 0 1 0 0 0 0 0-

1 0 0 loo 0 f o 0

1 1 0 0 1 0 U n o 0

1 1 0 0 0 1 0 4 0 0

0 0 0 0 0 0

ol0000 0 0 1 0

Wal(3,9) Truncated columns

It can be seen that the fifth and sixth digit of Wal(1,9) and

Wal(3,9) are zero, and therefore they can be truncated with no

effect on the error correcting capabilities of the code. Thus

the code has a length n of 8 digits, k = 2, and the minimum

Hamming distance d is 5. The Helgert and Stinaff upper bound

=

Truncated rows

62.

for this code is 5. The truncation of some rows and consequently

all zero columns of the [H] matrix can be generalised by the

following corollary.

Corollary 3.2:

The truncation of the rows, 2 --F 1, 2, ... —+C, 2 ' 2 C = L/2k, and the consequent removal of the C all zero columns

of the parity-check matrix of any (n,k,t) linear binary code

derived by applying theorem 2.1, does not alter the error

correcting capabilities of the code.

Proof:

The Walsh functions of the subset tWal(h,9)1 h= 2' - 1,

j=1,2,...,k1 are all square waves of frequency equal to j Hz and

period L/2j. The minimum period therefore C=L/2k. Since the

Abelian group of Walsh functionstWal(i,9)1 i = 0,1,2,...,(2k-1)}

forms an: orthogonal square matrix of rows IWal(i,9)1 i = 0,1,2,

...,(2k-1)1 and columns of fWal(1,01, with Wal(0,9) appearing

at the mid columns, it follows that all 24- 1, 124 + 2, 2 + C

digits of Wal(1,9), Wal(1,9)EfWal(h,9)111 = 23-1, j = 1,2,...,k1

must be zero. Therefore the truncation of the corresponding C

rows and the consequent removal of the C all zero columns from

the parity-check matrix does not alter the error correcting

capabilities of the (n,k,t) code.

3.2.1 Code Construction and Decoding

Using theorem 3.2 and corollary 3.2, a class of linear

binary codes may be constructed by following steps given below:

i) fix the value of k.

ii) the length L of the Walsh functions to be used is

given by L = C.2k where C = 1,2,3,..., etc:, and must

always be a multiple of 4.

iii) The first k columns of the required parity-check

matrix are given by the subset of Walsh functions

1Wal(h,9)lh = 2j-1, j = 1,2,...,ki.

63.

iv) the error correcting capability of such a code is

given by: t = C.2k-2.

v) finally, truncate the appropriate C digits from each

function in the subset 1Wal(h,4

Table 3.2 shows some codes of this class for k'5-C10.

All (n,k) codes in this class can be decoded by a one

step majority logic algorithm. This can be seen as follows; the

parity-check matrix of the given (n,k) code can be arranged in

the following form:

[ 1

0 0 W1,1 W1,3 W1,(2k-1)

0 1 0 W2,1

W2,3 W2,(2k-1) • •

Da = . . . . . .

• .

• •

Let e = 0' 1, .' el, en-1] be an error pattern. The syndrome corresponding to e is given by:

S = e

=, 0, s 1 s2, sn-k-1]

where,

so = eo + W1,1

en-k + + W1,(2

k-1) en-1

s1 = el

+ W2!1 en-k + + W2'(2

k-1) en-1

8n-k-1= e

n-k-1 + W(L-C),1 en-k

+ + W(L-C1(2k-1) en-1

Consider a check sum A of the syndrome bits:

A = a0 s0 + alsl

+ + an-k-1 sn-k-1

= b0 e0 + blel

+ + bn-1 en-1

where ,a. = 1 or 0, and b1 = 1 or 0. ,A parity check Al will be

said to check an error digitej if e appears in the summation

• • •

•

0 0 1 w(L-C),1

W(L-C),3 w

(L-C),(2/j)

64.

3 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 7 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 l 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

15 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 l 0 0 1 1 0 0 1 1 311 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 00 1 0 1 0 1 0 1 0 1 0 1

given by Ai with a non-zero coefficient. A set 1.A1 of parity

checks is said to be orthogonal on em if each Ai checks em but

no other error bit is checked by more than one Ai. Each parity

check in an orthogonal set thus provides an independent estimate

of em when the noise bits are statistically independent. If then

there are 2t parity-check sums orthogonal on em, m = n-k, n-k+l,

..., n-1, the value of the error digit em is given as the value

assumed by a clear majority of the parity-check sums orthogonal

on em(7). It is apparent from the characteristics of the subset

tWal(h,G)) as given in tabular form below, that there is a

unique syndrome check digit for each information noise digit.

The functions of the subset tWal(h,9)lh = 2j-1, j = 1,

2, k, k = 5} h are given in decimal and binary notation

below: sib, checks en-1

h so, checks en—k o si ,7);/checks en-2 T7/r)111111111111111110000000000000000

Trunc'tedt sl, checks en-3 s 23 'checks en-4 Column s checks the first information noise digit en-k, and sr,

r0 = -2- +C.2k-m-1, checks the mth, m = 2,3, ..., k, information

noise digit, en-k+m-1. Since the subset {Wal(h,9)Ih = 2j-1,

j = 1,2, ..., kl has a minimum weight of 2t, 2t parity-check

sums orthogonal on each information digit can be formed, and

therefore all codes of this class are completely orthogonisable

in one step.

3.2.2 Remarks

From the above theory and results, we can make the

following remarks.

i) The rows corresponding to 7 + 1, 7 + 2, ..., Ti+ p of the parity-check matrix of any (n,k,d) code of the proposed

65.

•

class, may be deleted; L being the length of the Walsh functions used for the construction of the parity-check matrix. The

punctured code which is produced from such an operation may have

a minimum Hamming distance of d - if p is an even multiple of

the duration of Wal(2k - 1, 9) and if p<27.

The correctness of the procedure given in the above

remark can be shown by considering the Walsh functions matrix

fWal(i,9)1i = 0,1,2, (2k - 1)1 of length L. If the

columns 7 + 1, 7 + 2, ..., 7 + p of the matrix have been deleted the minimum weight of the matrix will be reduced at most by p/2,

providing p is an even multiple of the duration C of the last

row in the matrix Wal(2k-1, 9), and p<217. This can be seen

clearly if we state the fact that the deleted columns of the

matrix are Wal(1,9), Wal(2,9), ..., and Wal(P/C,9). Therefore

the resulting code corresponds to the punctured Walsh function

matrix has a minimum Hamming distance d - P/2. See the proof

of theorem 3.2.

ii) For the case corresponding to k = 1, the codes given

in this paper are repetitive and therefore perfect.

iii) Consider the (n,k,t) codes of the proposed class of

C = 1, where one of the columns of the parity-check matrix EH] is Wal(2k-1,9), which is a square wave of frequency kHz., and

duration one bit. If the complement of Wal(0,8) with the 2 —+ 1

bit is deleted, (Wal*(0,9)) is taken as the column of the EH] matrix of the code, then this column of weight (L-1) is

linearly independent of all the members of the truncated subset

of the Walsh functions fWal(h,O)In = 21-1, j = 1,2, ..., kl.

Moreover, the resulting vectors formed by this linear combination

are of weight 2t or 2t-1 and hence the required conditions of

Lemma 3.1 are satisfied. For example, the codes (n,k,d) of

Table 3.3, (5,2,3), (10,3,5), (19,4,9), (36,5,17), (69,6,3),

(134,7,65), (263,8,12), (520,9,257) and (1033,10,513), can

• 66.

therefore be modified as above to (6,3,3), (11,4,5), (20,5,9),

(37,6,17), (70,7,33), (135,8,65), (264,9,129), (521,10,257) and

(1034,11,513) respectively.

iv) The codes of the proposed class are of low rate and

as n increases to very large values the rate kin tends to zero

and the ratio d/2n tends to 0.25. The method of deriving this

class of codes may be extended to derive a new class of codes

of higher rate and useful characteristics for decoding. Work

in this direction is under way and it is hoped will appear in

future publications. (see section 4)

3.3 APPLICATION OF THE CONCEPT OF ANTICODES

A linear block (m,k,5) anticode over a Galois field

GF(q), forms an array of 2k rows and m columns in which each of

the m columns is some linear combination, (modulo-q), of the

first k, (1c-c.m), columns of the array(86). The anticode has a maximum distance, b, if the distance between any two rows of

the anticode array is less than or equal to 6, and since any

linear anticode forms an algebraic group over GF(q), then 6 is

equal to the maximum weight max of the rows of the array(3).

An anticode is good if it exhibits the least 6 for given m and

k; and said to be non-repetitive if no two columns of the anti-

code array are identical.

Farrell(14,86) was the first to introduce the concept

of anticodes. He suggested that for every good non-repetitive

(m,k,o) anticode,there exists an (n,k,d) good or nearly good

non-repetitive linear code which can be coistructed simply by

deleting the given anticode array from the appropriate m-sequence

code array.

The m-sequence code array is a qk by q

k-1 matrix in

which the first k columns are the information columns, and

each of the remaining qk-1-k columns is some distinct, linear

67.

combination (modulo-q) of the information columns(14)

. The

array can be partitioned into two parts, with one part

consisting of the codewords of a linear non-repetitive (n,k,d)

code, and the other the corresponding (m,k,6) anticode words.

This implies that:

m + n = qk - 1

Since the m-sequence codes are equi-distance codes(3)

of distance equal to qk-1, it follows that:

d + 6 = qk - 1

The object behind this section is the formulation of

a mathematical description of linear anticodes from which a

systematic procedure for the generation of anticodes is estab-

lished.

3.3.1 Matrix Description of Linear Anticodes*

Since an (m,k,15) linear anticode forms an algebraic

group under modulo-q addition", (q being the number of symbols

per sign), then an (m,k,o) linear anticode forms a subspace U

of the vector space Um over GF(q). If a set of the basis

vectors of the subspace U are considered as the rows of a

(k-by-m) matrix DJ, then matrix Di is called the generator matrix of the anticode U. The reduced echelon form of the

matrix [4..] has the form(3): - C Ik b 3 (1)

The matrix Elk] is the identity matrix of order k,

and E] is an arbitrary b-by-(m-k) matrix. The null space U'

of U is a subspace of the vector space Vm(3). A set of basis

vectors from the null space U' of the subspace U can be

considered as the rows of a (m-k)-by-m matrix [Q, the EL]

Since linear anticodes form algebraic groups, their matrix description is similar to that of group codes. The reader is referred to references (3) and (7).

68.

matrix is called the generator matrix of the null space U' or

the parity-check matrix of U. The reduced echelon form of L

has the form(3):

EL: = E-brr lin_k] (2)

where the matrix E-13T] is the transpose of the arbitrary matrix

[-b], and is the identity matrix of the order (m-k).

The parity-check matrix of an anticode can itself be a generator

matrix which generates the dual anticode of the anticode

generated by EC . For simplicity, let us introduce the term "quasi-

linear independence". The r-tuple vectors rl, r2, ri

over GF(q) are quasi-linearly independent if the modulo-q

addition of the scalar products, airi +a2r2+ airi does not

equal zero when the scalars (a., j= 1,2,...,1) may take any

values of the non-zero elements in GF(q). Since there are (q-1)

non-zero elements in GF(q) then the quasi-linear combinations

of the above vectors may have (q-1)1 combination sums, each sum

, resulting in an r-tuple vector over GF(q). If the vectors are

not quasi-linearly independent over GF(q), they are linearly.

dependent over GF(q).

From the above we have the following,

Theorem:

For any (m,k) linear anticode U over GF(q), which

has a parity-check matrix [(], U will have a maximum distance

6 or greater if and only if every combination of 6+i, i = 1,2,

..., m-6, columns of [1.] are quasi-linearly independent.

Proof:

Suppose that some m-tuple vector u, u(311, has a

weight of 6+i, 1=a4;m-O. Since U is a row-space of Ea, and U' is a row-space of [1] and a null space of U, an m-tuple vector such as u is in U if and only if it is orthogonal to

69.

every row of El]. That is to say:

uT

= °

Since u has a weight of 6+i, this specifies a

linearly dependent set of 6+i columns of [i] . Conversely the modulo-q addition of 5+i columns of EL] is equal to the zero vector which contradicts the conditions of the theorem. There-

fore, we conclude that for the condition of the theorem to be

fulfilled there must be no m-tuple vector in U of weight

greater than 6.

3.3.2 A S stematic Procedure of Generatin Linear Anticodes

Consideration of the above theorem suggests the

following systematic procedure for the construction of an

(m,k,6) anticode over GF(q) for given values of 6 and (m-k). The

columns of the [U matrix can be determined as follows: start

with the identity matrix of order (m-k). The (m-k+1)th column

is chosen arbitrarily from the vector space Vm_k over GF(q),

subject only to the condition that the chosen column must not

be the inverse of any of the vectors obtained from the quasi-

linear combinations of every 6+1, i = 0,1,...,(m-k) of the

previously chosen columns. The inverse of a vector r over

GF(q) is that vector which when added (modulo-q) to r results in th a zero vector. Similarly the j column is chosen so that it is

different from the inverse of any of the vectors obtained from

the quasi-linear combinations of every 6+i, i = 0,1,...,j-1 of

the previously chosen columns. This method of generation

guarantees the required independence of columns specified in

the theorem. As long as the set of all these inverse vectors

does not include all the (m-k)-tuple vectors, another column

can be added. This process of building up the L-matrix will,

therefore, continue until all the (m-k)-tuple vectors are

exhausted. The anticode is the null space of this (m-k)-by-m

matrix IX] .

70.

Using the properties of the parity-check matrix EL: for linear anticodes established above, a computerised search procedure

for new binary linear anticodes can be developed to generate

new anticodes. Such new anticodes may in turn lead to new

linear block codes.

3. TWO PROCEDURES OF LINEAR BLOCK CODE MODIFICATION

3.4.1 Introduction

In a communication system employing coding as a means

of error correction or detection, the usual constraint on the

choice of linear block code for use in such a system is that the

block length n or the number of information digits k must take

some specific value. Any arbitrary value of k or n may be

achieved in principle by some appropriate procedure whereby an

already existing code may be modified to have the desired value

of k or n.

The techniques by which codes may be modified to

generate new codes can be grouped into six basic categories.

These techniques in the past have been referred to in a variety

of ways: Berlekamp(3), however, adopted a unified approach and

classified them under the headings of extension, lengthening,

puncturing, shortening, augmentation and expurgation.

The first of these methods, code extension, involves

the annexation of extra check digits to the code. An example

of this technique is the construction of even Hamming distance

codes by annexing an overall parity-check digit to the approp-

riate linear block codes of odd Hamming distance. Another useful

example of code extension is given by Andryanov and Saskovets(87),

in which BCH codes are extended by annexing a number of parity

check digits in such a way that the minimum Hamming distance of

the given code is improved by at least a factor of 2.

71.

A second technique, puncturing, involves deleting

check digits from the code. In general, if one punctures p

digits from a linear binary code of length n and minimum

Hamming distance d, then the punctured code may have a minimum

distance as low as d-p(3). However, Solomon and Stiffler(88)

derived some good linear codes by puncturing subspaces of a

maximal-length shift-register code. If j subspaces, each with

corresponding dimensions 21, where i = 1,2,...,j, are punctured

from a maximal-length shift-register code of length n and

minimum Hamming distance d, then the newly formed code has a

length of En - (2Qi_1X] and a minimum Hamming distance

1=1 2..-1 given by [d - t (2 1 ).j. Farrell(86) generalised the Solomon

i-1 and Stiffler puncturing procedure by introducing the concept of

anticodes. Farrell establishes that for every good non-repetitive

anticode there exists a good or nearly good non-repetitive linear

code which can be constructed simply by deleting the given anti-

code array from the appropriate maximal-sequence code array.

The third method can be described as code augmentation(3)

If new codewords are added to a code, then the code is said to

be augmented. The effect of this operation is to increase the

rate, while maintaining the block length constant. Goldberg(90)

proposed an augmentation technique in which he improves the rates

of a few existing codes, while maintaining the minimum Hamming

distance constant:

A fourth procedure entails the expurgation of code-

words from the code. The expurgated codes thus formed always

have a rate lower than the original code but may have an increased

minimum distance. The most common expurgated code is the one

expurgated by g(x)(x-1). In the binary case, the codeword of

this expurgated code are the even-weight codewords of the

original code(3)

72.

Lengthening codes by the annexation of more message

digits constitutes a fifth procedure by which codes may be

modified. The resulting lengthened codes have higher rates

than the original code, but, as a consequence, the minimum

Hamming distance may be decreased. The technique of lengthening

has not yet been used in algebraic coding theory in any non-

trivial way(3).

The last technique of code modification is the short-

ening of codes by omitting message digits. These resulting

codes have a rate lower than the original code and the minimum

distance in general does not alter. In practice this technique is

often used for cyclic codes to achieve simplicity of implementation

Davida and Reddy(91) proposed a code modification

procedure which was based on a combination of puncturing columns

and rows from the parity-check matrix of an (n,k,d) linear block

code in such a way that the resulting punctured code has a 1+ minimum distance of dT —. The new modified t-error-detecting code,

- where t= d21-, is used for communication over a noisy channel. When

t or less errors are detected, a single request is made via a

feedback channel for transmission of the punctured parity check

digits to enable the decoder to correct t or less errors. This

method of puncturing codes is not applicable to all (n,k) linear

codes. However, it is shown that the following families of

codes may be punctured by the above suggested procedure:

Single-error-correcting Hamming codes(30), Reed-Solomon codes

(50)

the Golay (23,12) code(31), and a class of extended BCH codes

(Davida(93)).

The object of this section is to introduce two technique

of code modification which are extensions of the above discussed

procedures. These techniques lead to some new codes of high

rate.

• 73.

•

3.4.2 Linear Block Code Puncturing Procedures

A linear (n,k) error-correcting code is a set of qk

n-tuples that forms a subspace V of the vector space Vn over a

field of q elements. The code can be described in terms of a

k-by-n generator matrix a], whose rows are linearly independent

and form a basis for the code. For a systematic code, the

generator matrix [G] has the form given by:

. 1 bk,1 bk,2 .. .. .

.

.

.

. bk,(n-k)

•

.

[ .

.

.

. . 0 b1,1 b1,2 b1,(n-k)— •

. 0 b2,1

b2,2 . b2,(n-k)

0

.

.

•

. .

.

.

.

0 .

DJ = lo.

1 .

.

.0

.

where EIk is the identity matrix of order k and Chi is an arbitrary k-by-(n-k) matrix.

The null space of [CO, VI is a set of qn-k n-tuples

which is generated by an (n-k)-by-n parity check matrix pa,

whose rows are linearly independent and form a basis for v's

The null space V in its own right forms a linear (n,n-k) code

referred to as the dual of the (n,k) code V. The weight

distribution of the dual code is a function of the weight

distribution of the original code. The form of this function

is governed by the MacWilliams(94) identities.

The parity check matrix DJ of a systematic code is

always of the form:

-b1,1 -b2,1 . .

- -b1,2 -b2,2 . .

. .

. -bk,1 1 0 . .

• -bk,2 0 1. .

. • •

Eg3 = E:1::9Q . . • • .

-b1,(n-k) -b2,(n-k) -bk,(n-k 0 0 1

• .

74.

41

where C-1311 is the negative transpose of the matrix El)] , and

[In-kI is the identity matrix of the order (n-k).

A linear (n,k) code is saidtohaveaminimum Hamming

distance d if and only if every d-1 columns of its parity-check

matrix EH] are linearly independent(13)

It follows from the above description of linear block

codes that if one punctures any number of columns of the parity-

check matrix [H] of an (n,k) code, then the resulting punctured

matrix retains the same column independence. The row independ-

ence, however, may be altered unless the punctured columns are

carefully chosen.

ith Now consider the case of deleting the . row of the

parity-check matrix Ea and all the corresponding columns which contain a non-zero element in the i

th position. Since all the

columns of the original [0 matrix which have not been deleted

have a zero element in the ith position, the conditions for

column independence in the newly formed punctured matrix EH''.] are identical to those in the original Eta matrix. And since

only one column in the identity matrix EIn_kj of the NI matrix has a non-zero element in the ith position, then all n-k-1 rows

of the punctured matrix LW] are linearly independent.

ith Let the . row of the matrix have a weight y;

then the punctured (n-y, k-y+1) code, which is the null space

of the punctured (n-y)-by-(n-k-1) matrix [HI], has a minimum

distance identical to that of the original (n,k) code.

In general, if the J1, J2, ...., Jx rows (where J

J2, ...., Jx are the row location numbers) are deleted from the

parity-check matrix of an (n,k,d) code together with all the

corresponding y columns containing at least one non-zero

element at the position Jl, J2, ...., Jx, then the resulting

75. "

matrix DV] has (n-k-x) linearly independent rows and (n-y) columns having an independence identical to that of the

original Ell:1 matrix. The null space of the punctured parity matrix EH'] forms an (n-y, k+x-y, d) linear block code.

For example, consider the parity-check matrix al] of a three error correcting (15,5) binary BCH code, with a

generator polynomial:

g(x) 1 + x + x2 4. x4 .4. x5 x8 x 10

The parity-check matrix as is given by:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ELO =

1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1

1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 2

1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 3

0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 4

1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 5

1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 6

0 1 1 1 0 0 0 0 0 0) 0 1 0 0 0 7

0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 8

1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 9

0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 10

The punctured code may be generated by deleting

columns 1, 3, 5 and 6. This will leave row 1 with all zeroes,

which in turn can be deleted. The newly formed punctured matrix

0-111 is thus:

1 1 1 0 0 0 0 0 0 0 0

1 1 0 1 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0

0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0

1 1 0 0 0 0 0 1 0 0 0

0 1 0 0 0 0 0 0 1 0 0

0 1 0 0 0 0 0 0 0 1 0

1 1 0 0 0 0 0 0 0 0

76.

Cr] =

The null space of EH'j is a linear systematic (11,2)

code with a minimum Hamming distance of 7. Note that the Helgert

and Stinaff(68)

upper bound on d for an (11,2) code is 7.

The rate of the newly found punctured code will be

highest when the weight of the deleted row is lowest. Since

the rows of the parity-check matrix [1J are codewords of the

(n,n-k) dual of the original (n,k) code, the lowest weight y

of those rows cannot be less than the minimum Hamming distance

of the (n,n-k) dual code.

This provides a bound on the lowest number of columns

of the matrix DO that must be deleted to form a new punctured

code.

Binary BCH codes of length 255 and 511, and of

various error-correcting capabilities ranging from 6 to 14

random errors were punctured in accordance with the above

procedure and the best punctured codes were computed in each

case. From these codes listed in Table 3.5.1 seventeen were

found to be better than the best previously known codes of

identical Hamming distance and the same number of parity check

digits. For ease of comparison the parameters (n,k,d) of the

original and the resulting punctured codes are tabulated in=

Table 3.5.1 side by side with those of the best previously

known codes. These parameters are taken from a table of binary

codes of highest known rate published in 1972 by Sloane(15)


generate the [U] matrix of a given (n,k) BCH code and to

determine the row of the generated Da matrix which has the lowest weight. Then the punctured code parity check matrix

is printed out.

77.

Message Expander logic circuit

3.4.3 Encoding and Decoding of the Punctured Codes

The encoding and decoding of the proposed punctured

codes may be achieved by an algorithm based on the insertion

of additional puncture logic circuitry before and after the

circuitry used in the original encoding and decoding operations.

The structure of the puncture logic circuitry is similar in many

ways to that of the puncture logic suggested by Solomon and

Stiffler(96)

for their punctured maximal sequence codes.

Figure 1 shows the encoder for an (n',k',d) punctured code

Generator of location numbers of the punctured

columns

Original Code

encoder

`Compressor logic circuit

Binary, Channel

Fig. 1 The encoder for the punctured codes.

which has been derived from an (n,k,d) code. The k' message

digits (ml,m2,...,mk,) are fed to an expander logic circuit, the

function of which is to re-label the digits mi according to

their position in the original unpunctured message word and to

form a larger message word of length k by inserting zeroes in

the punctured positions. This new message word is fed into the

original encoder to form an output comprising a block of n

digits. Before transmission this codeword is passed through a

compressor logic circuit, the function of which is to delete

all the digits at the punctured positions.

At the receiver the n' digit codeword is expanded

to a length of n digits by means of an expander logic circuit

identical to that used at the transmitter. The expanded

received codeword can then be decoded normally by the decoder

of the original code and the appropriate k' message digits

78.

recovered by passing the output of the original code decoder

Generator of location numbers of the punctured

columns

Message

kl

Received

Expander logic

circuit -->

Original Code

decoder

Compressor logic

circuit Codeword

Fig. 2 The Decoder for the Punctured Codes.

through a compressor logic circuit identical to that used at

the transmitter. This operation is illustrated in Figure 2.

3.4.4 A Technique for Lengthening Linear Block Codes

The parity-check matrix [H] of an (n,k,t) linear

block code V consists of n columns, each column being an (n-k)-

tuple vector of the vector space Vn_k. The code can be

lengthened by one information digit while still retaining its

error-correcting capability, if and only if there exists a

vector v in Vn-k which is linearly independent of every (2t-1)

columns of D-]. This vector v can then be considered as a new

additional column of Da. Most of the best known codes, however,

cannot be lengthened in this manner because there is no vector v,

v€1.7n-k which satisfies the required conditions of independence.

The lengthening of linear block codes may be viewed in

a different way. Consider the D:1 matrix of an (n,k,t) linear code, where Di is given by:

aaT

I -111_0

Each column of the E-bT1 matrix is identical to the syndrome

vector which is obtained when an error occurs in the information

digits at that column position. Similarly each column of the

identity matrix Eln-k 3 is identical to the syndrome vector

79.

corresponding to an error occurring in the parity-check digit

at that column position. This implies that the code is capable

of correcting t random errors if and only if the linear

combinations of every t column vectors in Di] give unique

nonzero vectors in Vn-k

(89). Consequently it can be lengthened

by one information digit if there exists a vector v in Vn_k

such that all the linear combinations of v with (t-1) columns

of Eu] give unique nonzero vectors in Vn-k.

Let us consider a vector u1 in the vector space Vn-k such that all the linear combinations of u-1 with (t-1) columns

of the E-bril matrix give unique nonzero vectors in Vn-k.

Similarly all the linear combinations of ul with (s-1) of the

remaining columns of DJ yield additional unique nonzero

vectors in Vn-k'

where s is an integer such that 1<s<t. Now

let us construct a parity-check matrix Dqj such that:

3 uTIH] C T bT I I 1 E-1 - u I1 -

The null space of matrix [H1] is an (n+1, k+1) code.

Since alf the linear combinations of the t columns of [HI]

which contain at least one column of matrix E-brr] give a

unique nonzero vector in Vil-k and since all the linear

combinations of s columns of Di3 give unique vectors, it

follows that every 2t columns of [a1] containing at least one

column of E-hT] are linearly independent and similarly every

2s columns of cy1 3 are linearly independent(89). This implies

- that the (n+1, k+1) linear code is capable of correcting t

random errors or less if at least one of these occurs in the

block of k message digits and s random errors or less if none

of the'errors occur in any of the k message digits.

The (n+1, k+1) code can be lengthened further by

finding another vector u2 in the vector space Vn_k which is

subjected to the same independence conditions with respect to

80.

matrix EH1 as exist between vector u1 and matrix D]. This - procedure for building up the lengthened code may be continued

until the maximum possible number of vectors u , u fL 1 -2' le

are found. The newly constructed Dk':I matrix is therefore given by:

T D]

rie

T. T T 1 1121111 I I In-k]

And thus the null space of [pic,] is the lengthened

(n+k', k+k', t and s) code, where t and s indicate that the

code is capable of correcting t random errors or less if at

least one of these occurs in the first block of k information

digits and s random errors or less if none of the errors occur

in any of the k message digits.

For example, consider the lengthening of the two-

error correcting (15,7) BCH code. The Do matrix of this code is:

1 0 1 0 0 0 1 0 0 0 0 0 0 0

0 1 1 0 1 0 0 0 1 0 0 0 0 0 0

0 0 1 1 0 1 0 0 0 1 0 0 0 0 0

0 0 0 1 1 0 1 0 0 0 1 0 0 0 0

1 1 0 1 1 1 0 0 0 0 0 1 0 0 0

0 1 1 0 1 1 1 0 0 0 0 0 1 0 0

1 1 1 0 0 1 1 0 0 0 0 0 0 1 0

1 0 1 0 0 0 1 0 0 0 0 0 0 0 1

A computer search used to find all possible vectors

of the vector space V8 which satisfy the condition set by the

lengthening procedure; s was taken to be 1. Four vectors were

found. The Eti4 matrix of the lengthened code (15+4, 7+4, 2

and 1) has the form given below:

81.

k—Ds-TsJ * Ca3 )1

0 0 0 0 0 0 0 1

E11 4]

S

r

0 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 I 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 I 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0

Five linear codes with the highest rate for given

block length and error-correcting capability were lengthened

in accordance with the above procedure. The parameters (n,k,t)

of these codes are listed in Table 3.4.4 together with the

parameters (n+k', k+k', t and s) of the modified codes. The

first k columns of the parity check matrix of each original

code are also tabulated in Octal, where the first digit in

each column designates the least significant digits of the

Octal number. In the last column of the table the new columns

are given also in Octal.


determine all possible vectors of the vector space Vn_k which

satisfy the conditions set by the lengthening procedure for a

given (n,k) linear block code.

82.

TABLE 3.4.4 THE PARAMETERS OF THE NEWLY FOUND CODES BY THE PROPOSED PUNCTURING PROCEDURE

n is the code length, k is the number of message digits, r is the number of redundant digits, and d is the minimum Hamming distance of the code.

3riginal Codes

Punctured Codes

The Best previously

Known Codes the Original Codes

The Generator Polynomials of , The

• .uncturea Row

The Location Numbers of the Punctured Columns

n = 255 k - 207 r = 48 d = 13

n = 163 k = 116 r = 47 d = 13

n = 156 k - 109 r = 47 d = 13

1+x+x2+x4+x7+x10+x1141(12+x13+ 16 17 18 19 20 22 23 24 x +x +x +x +x +x +x +x

+x26 +x31 +x32 +x33 +x35 +x37 +x38+

39+x 40+x 41+x 42+x 46+x 47+x 48 x •

12

1,3,4,5,11,16,18,20,24,25,28,34,38,39,40,41,42,43,44,47,48,49, 55,57,58,60,62,65,68,71,73,74,80,81,83,84,86,87,88,89,91,93,97, 98,101,102,104,105,106,108,112,113,117,122,124,127,132,133,134, 135,138,141,145,146,147,148,153,155,157,160,163,164,167,170, 173,174,176,177,179,180,182,183,185,191,192,194,197,200,204, 205,207,219.

n = 255 k = 199 r = 56 d = 15

n = 167 k = 112 r = 55 d = 15

n = 140 k = 85 r = 55 d = 15

5 7 10 13 15 16 17 1+x +x +x +x +x +x +x +x

22 26 30 31 32 34 36 +x +x +x +x +x +x +x + 39 40 45 46 48 49 52 x +x +x +x +x +x +x +

x5354

+x55+x56

40

1,2,4,5,16,21,22,23,24,25,2S,32,34,39,40,41,44,46,48,50,51,53, 54,55,57,61,63,65,68,70,72,77,80,84,86,88,89,93,94,98,101,102, 103,105,107,111,112,114,117,120,122,123,124,126,130,131,132, ,134,135,143,144,145,146,150,151,154,160,161,162,163,164,165, 169,173,174,176,177,179,180,181,182,184,189,190,191,192,199,239

n = 511 k = 448 r = 63 d = 15

n = 311 k - 249 r = 62 d = 15

= n 269 k = 207 r = 62 d = 15

2+x + 11 13+x 16+x 19+x 21+x 22+

1+x x

23 24 25 29 31 32 33 36 x +x +x +x +x +x +x +x

+x 37+x 38+x 39+x 40+x 42+x 43+x 46+

x49+x

51+x56+x57+x58+x63

•

24

1,2,3,5,6,11,13,14,17,22,23,25,29,31,33,35,38,39,40,49,51,53, 55,57,60,62,63,65,76,78,80,82,83,84,86,87,89,91,93,94,95,97, 98,100,107,109,110,111,114,117,118,120,121,122J24,125,128,129, 132,134,136,145,146,148,151,153,156,157,159,160,161,162,163, 165,166,167,168,169,171,174,177,184,185,187,193,195,196,197, 198,199,202,204,205,206,207,209,211,215,216,217,218,221,222, 223,225,227,231,232,236,237,238,239,241,243,245,246,247,255, 256,264,265,272,274,275,279,282,285,286,291,292,294,297,299, 303,306,312,313,316,318,320,321,323,332,335,336,341,342,343, 345,346,349,351,353,358,361,365,367,369,371,372,373,374,377, 378,382,385,388,390,391,392,393,397,399,400,401,405,407,408, 413,414,416,417,418,419,422,423,424,425,430,433,434,437,441, 442,444,445,446,447,448,472.

n = 255 = k 191

r =464 d = 17

n = 169 k = 106

d r = 63 = 17

n = 141 k = 78 r = 63 d = 17

1+x+x2+x4+x5+x6+x8+x11+x12+x15+

16+x 17+x

18+x

19+x

21+x

22+x 24+x 25 x 27 29 30 33 37 38 39 +x +x +x +x +x +x +x + 40+x 41+x 42+x 48+x 49+x 50

+x 53+x 54 x

+x55+x58+x59+x61+x62+x64

2

1,2,3,9,10,11,12,17,21,22,23,26,31,34,35,36,42,43,45,47,49,53, 54,59,62,63,64,66,74,75,78,83,85,86,87,93,97,98,99,102,105,107, 109,110,113,114,115,116,118,119,120,123,124,126,129,132,135, 136,137,138,141,143,147,148,149,151,154,155,158,160,165,166,16: 169,175,176,181,183,184,185,186,188,189,190,191,193.

n = 511 k = 439 r = 72 d = 17

n = 315 k = 244 r = 71 = d 17

n = 270 k = 199

r = 71 d = 17

1+x2+x5+x6+x7+x9+x10+x13+x17+ 18 19 20 21 22 23 24 25 X +x +x +x +x +x +x +x 26+x 27+x

28+x 32+x 33+x 35+x 39+ +x 40 41 43 44 47 49 50 57 x +x +x +x +x +x +x +x

+x 59+x 60+x+ 61 63+x 67+x 68+x 69+ x

x71+x72

•

45

1,14,15,18,24,25,26,30,31,32,33,34,35,39,41,43,46,47,48,49,50, 51,52,53,55,56,58,59,60,61,62,66,70,73,76,77,78,79,80,81,84,86, 90,92,93,100,101,103,104,110,111,112,115,118,119,123,124,125, 129,134,135,138,140,146,147,149,150,152,154,156,157,159,161, 164,167,173,174,175,178,181,184,185,186,191,192,193,194,195, 197,201,202,204,205,207,209,215,217,218,219,220,224,225,226, 227,229,230,231,233,235,236,244,251,252,253,255,257,260,265, 270,271,274,275,276,278,283,285,287,290,292,293,297,301,302, 303,305,309,310,311,312,314,316,318,320,322,323,325,327,328, 330,331,334,335,337,339,343,345,346,349,351,352,359,366,369, 370,372,374,376,378,383,385,386,388,389,390,391,396,398,401, 403,405,406,408,410,411,412,413,414,424,425,427,428,430,431, 434,437,484.

w

83 .

TABLE 3.4.4 (continued)

Original Codes

Punctured Codes

e Best PreTh

Y viousl Known Codes

The Generator Polynomials of 1

the Original Codes

e -3unct

Thured

Row The Location Numbers of the Punctured Columns

n = 255 k = 187 r 68 d = 19

n = 173 k = 106 r = 67 d = 19

n = 145 k = 78 r = 67 d = 19

1+x3+x5+x6+x7+x10+x11+x12+x13+ 16 19 22 24 25 27 41 x +x +x +x +x +x +x + 42 44 45 46 48 51 52 x +x +x +x +x +x +x + 54+x 56+x 57+x 59+x 60+x 61+x62+ x

x64+x66+x68

57

1,6,7,9,11,12,14,16,19,23,25,26,27,29,31,37,38,40,41,43,45,48, 52,53,54,56,57,58,61,62,63,64,67,69,70,73,74,77,82,89,92,93,95, 96,97,99,100,101,102,106,113,114,116,118,119,120,125,126,128, 134,136,137,142,145,152,156,160,161,163,165,166,167,168,173, 176,177,181,182,183,185,187,244.

n = 511 k = 430 r = 81 d = 19

n = 319 k = 239 r = 80 d = 19

n = 272 k = 192 r = 80 d = 19

1+x+x2+x4+x7+x8+x9+x12+x13+x17

18 22 23 26 27 28 30 +x +x +x +x +x +x +x + 31 32 33 40 43 44 45 x +x +x +x +x +x +x +

x46

+x47

+x49

+x51

+x55

+x56

+x60

+ 61 71 72 73 74 76 78 x +x +x +x +x +x +x +

x79+x80+x81

8

1,2,4,6,8,9,12,15,18,19,22,29,32,37,38,40,41,43,44,47,50,52,53, 54,56,59,62,64,65,66,69,72,73,74,75,76,80,81,82,88,89,90,94, 101,104,109,112,113,119,124,125,126,129,130,132,134,135,136, 137,141,143,149,151,153,154,155,157,158,159,160,165,166,167, 169,170,173,176,179,180,181,183,186,188,189,192,199,201,203, 204,205,209,210,213,215,216,217,218,220,222,223,224,228,229, 232,233,235,236,238,241,244,246,247,249,254,255,257,262,264, 266,267,268,269,270,271,274,275,277,283,285,286,288,292,293, 296,297,299,303,304,305,207,308,314,315,318,320,323,324,325, 332,333,336,337,340,341,343,348,349,353,355,357,359,364,368, 371,372,376,379,380,382,384,387,388,393,396,399,401,402,405, 412,413,414,415,416,418,421,422,423,424,427,428,430,438.

n = 255 k = 179 r - 76 d = 21

n - 173 k = 98 r = 75 d = 21

n = 147 k = 72 r = 75 d = 21

1+x2+x3+x5+x8+x15+x16+x19+x20+

21+x

26+x

28+x 30+x 31+x

35+x 41+x 42 x

43 45 46 47 48 51 52 +x +x +x +x +x +x +x + 53+x 57+x 60+x

61+x 62+x 65

+x 67+x 70 x

+x71+x73+x76

27

1,4,7,8,9,10,16,18,19,20,22,24,27,33,34,3637,40,43,48,50,51 , , 52,53,55,56,57,58,60,61,62,64,68,70,73,74,75,76,77,79,85,90, 91,93,94,95,96,98,99,100,103,105,108,113,114,115,117,128,129, 130,132,133,134,137,138,140,141,147,148,149,150,151,154,159, 161,163,165,166,168,173,178,206.

•

n = 511 k = 421 r - 90 d = 21

n = 319 k = 230 r = 89 d = 21

n = 277 k = 188 r = 89 d = 21

1+x+x3+x6+x8+x9+x10+x11

-Ic15+x16

17 20 23 25 26 31 32 +x +x +x +x +x +x +x + 50 x 33+x 34+x 39+x 43+x 44+x 47+x 49+x

52 54 55 56 58 59 61 +x +x +x +x +x +x +x + 63+x 65+x 67+x 70+x 71+x 72+x 75+x

78 x 79 80 81 82 83 84 87 +x +x +x +x +x +x +x +

x88+x90

50

1,4,7,8,13,14,16,19,23,25,26,27,28,30,33,35,36,38,39,40,42,43, 44,48,49,55,57,59,60,66,67,69,70,71,72,74,77,79,80,83,84,86,90, 95,99,100,104,105,106,108,111,113,116,119,121,122,123,124,129, 133,135,139,140,141,143,147,149,150,151,152,153,154,155,158, 166,167,168,175,177,179,180,182,183,187,189,190,191,192,199, 205,207,208,209,210,215,216,226,227,229,231,232,238,239,240, 241,243,244,247,251,252,253,254,255,261,163,164,269,271,272, 273,274,275,276,277,281,290,292,294,295,296,297,298,300,301, 302,305,309,311,313,314,315,316,318,319,321,323,324,326,327, 336,340,343,344,348,351,352,354,357,358,359,362,367,371,372, 373,375,376,377,379,384,386,387,389,390,392,394,395,396,400, 402,404,408,409,410,413,414,415,417,418,420,421,471.

n = 255 k = 171 r = 84 d = 23

n = 183 k = 100 • r = 83 d = 23

n = 148 k = 65 r = 83 d = 23

1+x+x2+x3+x4+x7+x8+x12+x13+x16+

17 18 22 23 27 28 29 30 x +x +x +x +x +x +x +x 31 32 37 38 39 41 42 +x +x +x +x +x +x +x + 43 46 50+x 51+x +x55 +x57 +x

52 55 57 61 x43 +x46 +x 65 66 70 73 74 75 80 +x +x +x +x +x +x +x +

x81+x83+x84

28

1,2,3,5,7,9,12,16,17,18,19,20,23,24,25,27,29,30,38,40,45,46,49, 50,52,56,58,59,64,65,66,69,70,72,79,80,81,82,83,90,93,94,97,98, 102,104,106,108,111,114,115,119,120,122,123,128,130,131,133, 138,139,141,147,150,151,154,155,158,159,164,171,199.

84 •

• 0


Original Codes

Punctured Codes

e Best PreThviously Known Codes

The Generator Polynomials of the Original Codes

e ' nct

Thured

Row The Location Numbers of the Punctured Columns

n = 511 k = 412 r = 99 d = 23

n = 319 k = 221 r = 98 d = 23

n = 278 k = 180 r = 98 d = 23

1+x6+x9+x10+x

11+x14+x15+x18+

19+x 20+x 22+x

23+x 24

+x 25+x 26+ x 27 28 30 31 32 33 37 x +x +x +x +x +x +x + 38 39 42 43 46 48 51 x +x +x +x +x +x +x +

x52 +x53 +x54 +x56 +x60 +x61 +x66

+ 71+x 74+x 76+x 77+x 78+x 84+x 85+ x 87 89 90 91 92 93 94 x +x +x +c +x +x +x +

x96+x97+x99

19

1,3,7,8,10,11,12,13,15,17,18,19,21,22,25,27,29,31,32,34,35,36, 38,39,40,42,44,45,47,48,49,51,54,56,59,60,61,62,65,69,72,75,76, 77,78,82,84,85,86,87,92,96,97,98,105,106,108,110,111,115,117, 119,120,123,124,125,128,130,134,140,141,144,145,148,149,150, 157,162,165,168,171,175,177,178,184,185,186,188,189,191,192, 195,197,199,200,207,213,215,219,220,221,224,226,230,231,235, 238,239,240,242,243,244,245,248,250,253,255,257,262,263,267, 268,270,272,278,282,284,288,289,291,293,294,301,302,304,308, 309,311,317,318,320,323,324,325,326,328,329,333,334,335,336, 337,341,345,350,352,355,357,358,359,361,362,363,364,366,367, 369,370,373,374,375,378,380,382,385,390,391,393,394,396,397, 399,400,402,403,404,407,408,409,411,412,431.

n = 255 k = 163 r = 92 d = 25

n = 181 k = 90 r = 91 d = 25

n = 149 k = 58 r = 91 d - 25

1+x7 +x

8 +x11 +x12 +x15 +x17 +x

20 +x22 24 25 26 29 30 31 32

+x +x +x +x +x +x +x +

x33 +x35 +x36 +x39 +x41

+x42

+x44 +x46

52 53 54 56 S7 58 59+ 6

+x +x +x +x +x +x +x x 69+x +x72 +x74 +x75 +x80 +x 71 72 74 75 80 87+x 89 x

+x90+x91+x92

26

1,3,4,7,9,13,14,16,19,22,27,28,32,33,34,36,37,38,40,42,44,46, 47,48,49,52,53,54,55,57,59,61,64,65,72,73,74,75,82,83,85,86,88, 95,101,102,103,107,110,116,118,120,123,125,126,127,130,133,137, 138,140,147,148,150,151,152,154,155,157,158,159,160,163,189.

n = 511 k = 403 r = 108 d = 25

n = 327 k = 220 r = 107 d = 25

n = 279 k = 172 r = 107 d = 25.:

1+x3+x5+x7+x8+x

9+x

10+x16+x18+

21+x

23+x

25+x

26+x

29+x 31

+x 33+x 34 x 35 36 37 38 39 41 43

+x +x +x +x +x +x +x +

x 44+x 46+x 47+x 49+x 51+x 52+x 53+x 55

58 59 60 61 62 63 66 +x +x +x +x +x +x +x +

x 69+x 70+x 72+x 73+x 74+x 75+x 77+x 79

80 81 82 87 88 91 92 +x +x +x +x +x +x +x + 93+x 94+x 98+x 99+x 100+x 102+x 103 x

x104+x105+x

107+x108

10

1,3,7,10,13,15,16,17,19,25,28,29,31,35,38,40,43,44,45,50,52, 57,58,59,60,67,69,70,71,73,80,83,86,87,89,91,92,93,94,95,96,7,98 99,100,102,104,105,110',112,117,119,121,123,124,125,126,129,130, 131,132,133,134,137,138,139,140,145,148,149,152,155,157,260, 163,166,168,171,173,174,177,179,182,183,185,187,192,194,196, 199,201,202,203,204,209,213, 214,215,216,217,223,225,227, 230,232,243,246,247,248,249,252,256,257,261,262,268,269,273, 274,275,276,277,282,289,290,291,295,298,303,304,305,308,310, 311,313,315,317,319,321,322,326,327,328,331,336,1340,343,345, 346,347,348,349,350,354,355,356,357,362,364,365,366,367,371, 373,374,375,377,378,380,381,382,383,384,386,387,388,389,390, 394,396,398,400,403,413.

n = 255 k = 155 r = 100 d = 27

n = 187 k = 88 r = 99 d = 27

n = 165 k = 66 r = 99 d = 27

1+x+x3+x4+x7+x8+x

9+x10+x11+x12

13 14 16 17, 20 22 23 26 x +x +x +x +x +x +x +x

28 29 33 35 37 40 42 +x +x +x +x +x +x +x + 43 44 45 47 48 54 56 58 x +x +x +x +x +x +x +x

59 61 62 63 64 65 71 +x +x +x +x +x +x +x + 72 74 81 82+x 84+x

87+x 89+x 90 x72 +x74 +x +x 91 92 93 95 96 97 98 +x +x +x +x +x +x +x +

x99+x100

8

1,2,4,6,10,11,12,16,20,22,24,28,33,36,37,39,40,41,45,47,49,50, 52,53,56,58,59,63,66,68,69,74,79M,81,83,86,87,89,92,94,100, 104,108,109,112,113,119,123,125,127,131,134,135,139,141,143, 144,145,146,147,148,150,151,152,153,155,163.

85 .


Original Codes

Punctured Codes

e Best PreThviously Known Codes

The Generator Polynomials of the Original Codes

e Punct

Thured

Row The Location numbers of the Punctured Columns

n = 511 k = 394 r = 117 d = 27

n = 331 k = 215 r = 116 = d 27

n = 280 k = 164 r = 116 d = 27

1+x+x3+x4+x

11+x12+x13+x16+x18+

23+x 24+x 25+x 27+x 29+x 33+x 34+x 38 x 39 40 41 44 45 46 47 +x +x +x +x +x +x +x + 53+x 57+x 58+x

59+x 61+x 65+x 66+x iK

68 69 x 70 72 73 75 76 78 79 +x +x +x +x +x +x +x + 84+x 85+x 87+x 92+x 95+x 96+x 97+ x

x102

+x105

+x107+x109

+x114

+x117

62

1,3,5,6,8,9,11,12,13,16,17,19,24,26,30,34,38,39,44,45,47,49,50, 54,55,60,62,63,67,70,71,72,82,84,86,89,91,95,97,98,100,101,102, 104,106,107,109,112,113,114,118,119,120,123,126,127,130,132p4, 135,144,145,146,147,156,157,159,161,164,166,169,170,171,174, 176,180,181,182,184,187,188,189,192,193,195,204,205,209,211, 213,215,216,218,220,225,226,227,228,231,239,241,244,245,247, 250,251,259,260,261,262,265,267,270,271,273,274,275,276,277, 280,282,283,287,289,290,291,292,293,296,297,298,300,302,303,306 308,309,311,313,314,319,320,321,322,323,324,325,326,327,328, 329,330,331,332,335,336,340,342,343,345,350,356,357,358,359, 360,362,363,365,367,369,370,375,376,378,379,385,386,391,456.

n = 255 k = 147 r = 108 d = 29

n = 193 k = 86 r = 107 d = 29

n = 160 k = 53 r = 107 d = 29

1+x3+x6+x7+x8+x9+x12+x18+x23+

8 x26 +x27 +x29 +x33 +x34 +x36 +x3 +x40

+x 41+x 42+x 47+x 51+x 52+x 54+x 56+ 58 60 62 63 65 67 68 69 x +x +x +x +x +x +x +x 72 73 74 75 76 78 80

+x +x +x +x +x +x +x + 81+x +x 82 84+x 85+x 86+x 87+x 93+x 94 x

+x96+x100+x104+x106+x107+x108

19

1,2,4,7,14,15,21,26,30,32,34,38,40,41,42,43,44,48,49,52,53,55, 57,58,61,62,65,66,67,72,75,80,81,82,84,85,87,88,91,92,98,100, 101,102,103,104,105,109,114,116,119,122,123,125,127,131,132, 134,135,139,143,166.

.

n = 511 k = 385 r = 126 d = 29

n = 343 k = 218 r = 125 d = 29

n = 281 k - 156 r = 125 d - 29

3 4 11 14 17 18 21 22 1+x +x +x +x +x +x +x +x 24+x

25+x 30+x 31+x 33

+x 35+x 42+ +x 43 45 46 47 51 55 57 59 x +x +x +x +x +x +x +x 62+x 63+x 65+x 66+x 68+x 71+x 73+ +x

75 76 77 78 79 82 84 87 x +x +x +x +x +x +x +x 88+x 89+x 90+x 91+x 92+x 93+x94+ +x 95 96 97 98 99 101 103 x +x +x +x +x +x +x +

x109+x110

+x113+x114

+x116+x117

+ 119 120 121 122 123 124

x125-1-x126

x +x 4tt +X +x +x

+

26

411.

1,3,4,6,8,9,11,13,15,16,17,18,20,21,23,24,26,32,3137,39,40,42,43, 44,45,46,49,50,51,56,57,59,60,62,63,66,67,68,69,70,74,75,76,78, 81,82,83,87,90,93,104,106,107,109,112,113,117,124,125,126,127, 129,130,131,132,134,135,136,137,141,143,151,152,154,156,167, 169,171,176,177,178,179,181,184,185,187,190,192,196,199,200, 202,204,205,206,207,214,217,218,228,229,232,234,236,239,240, 241,242,243,246,250,257,258,261,262,265,266,268,273,275,276, 278,280,281,283,286,288,289,292,294,296,297,298,302,303,307, 311,313,314,318,319,320,326,327,329,331,333,334,335,336,338, 340,348,350,351,353,354,357,361,363,364,366,368,370,380,381,

86.

•

TABLE 3.4.4 THE PARAMETERS OF THE NEWLY FOUND CODES GENERATED BY THE PROPOSED LENGTHENING PROCEDURE

n is the code length, kr is the number of message digits, t is the error-correcting capability of the first k message digits, and s is the error-correcting capability of the annexed k message digits.

Original Codes

(n,k,t) (n+k',k+k',t

Lengthened Codes

&- s) The first k columns of CA in Octal The new k' columns in Octal

7, 1, 2 17, 1 + 10, 2 & 1 17. 23,25,26,43,45,46,61,62,64,70.

12, 4, 2 20, 4 + 8, 2 & 1 17,63,25,152. 26,203,205,206,221,222,224,230.

14, 6, 2 20, 6 + 6, 2 & 1 17,63,125,152,226,253. 114,214,314,310,320,340.

15, 7, 2 19, 7 + 4, 2 & 1 321,163,346,35,72,164,350. 13,203,211,212.

18, 9, 2 25, 9 + 7, 2 & 1 17,63,125,152,226,253,333,355,367. 403,405,406,421,422,424,430.

4. A NEW CLASS OF NESTED LINEAR BLOCK CODES AND THEIR NESTED DECODING ALGORITHM

4.1 Introduction.

4.2 The Construction of Nested Codes.

4.3 The Nested Decoding Algorithm.

4.4 Features and Merits of the Nested

Decoding Algorithm.

88.

4.1

INTRODUCTION

A large volume of research in coding theory has

concentrated on searching for codes that have sufficiently well

defined mathematical structures so as to facilitate their encoding

and decoding with equipment of moderate complexity and cost. In

general the overall objective of the effort is to incorporate

the ideas of Shannon(103-4) into the design of engineering

systems in such a way that the improvement in performance

justifies the extra cost and complexity(1). Since Berlekamps(3)

decoding algorithm for BCH codes, (the complexity of which

increases as the square of the number of errors to be corrected),

no major work has been published on the subject of simplifying

the complexity of the necessary decoding procedures for linear

block codes.

In this section a new class of multiple error-correcting

linear block codes is introduced. The codes of this class,

referred to as nested codes, cover a wide range of code lengths

and rates. An (n,k) linear block code is said to be a nested

code if and only if its syndrome, corresponding to errors in the

k information digits, are codewords of either a nested or a

Hamming code.

The decoding algorithm of a nested code, referred to as

the nested decoding algorithm, is shown to be very simple, its

complexity increasing only linearly with code length and error-

correcting capability. The essence of this algorithm is as

follows:-

Consider an (n1, k1, t) nested code. Its syndromes,

corresponding to errors in the kl information digits, are code-

words of a second(n1-k1, k2, t-1) nested code. In addition, the

syndromes of this second nested code, corresponding to errors in

89.

the k2 information digits, are codewords of a third (n1-k1-k2'k3' t-2) nested code. This process of building up a nest of codes

may be continued until the (t-1)th nested code. Its syndromes,

corresponding to errors in the kt-1 information digits, are

codewords of an (ni-ki-k2-....-kt_i, kt, t-(t-1)) single error-

correcting Hamming code. It follows that there is a nest of t

syndromes for every codeword of an (n1, kl, t) nested code

corrupted by noise. And for every (n,k,t) nested code there is

a corresponding nest, referred to as the (n;k;t) nest, of t

codes. The (n,k,t) code forms the outer code of the nest, whilst

the Hamming code forms the inner code. The other (t-2) codes are

obviously nested codes.

ith syndrome of the nest is calculated according to the

i th

parity-

check equations of the (n1-k1-...-k.1-1' k.,t+l-i) nested

code of the(nl' .k1 ;0 nest.

. A codeword vector of an (n k t) l' J' nested code corrupted by noise

nl-tuple I - 1 nf 1 k1 I k vector I 1

I Syndrome Corresponding nested code n2-tuple I I k0 1st (n1, k1, t) vector 1'

I

vector I n-tuple

I I i ka] ' 2nd (n1-k1, k2, t) v

1 .

I nt-tuple 7-6 I vector (n

t-1' kt-1'2)

nt+1-tuple tth vector LI (nt, kt)

Figure 4.1. THE SYNDROMES NEST OF AN n1 -TUPLE VECTOR.

Figure 4.1. shows a nest of an nctuple vestor. The

90.

The (t-1)th syndrome of the above nest may be an

nt-tuple vector which forms a codeword, corrupted by noise,

of an (nt' kt) double error-detecting and single error-correcting

cyclic Hamming code (or shortened version of it) which is

very easily decodable particularly by error trapping techniques.

This implies that all the first nt digits of the original code

can be easily cleared of errors. It is also shown that the

remaining (n1 -nt) digits of the corrupted codeword v can be

cleared of errors as easily as the first nt digits by using

the nested characteristics of the other codes of the (n •k • t)

nest.

91.

4.2 THE CONSTRUCTION OF NESTED CODES

An (n,k) linear block code is by definition a sub-

space V of the vector space Vn over a field of q elements.

The code V is a rowspace of its generator matrix Ed], whose k rows are the basis vectors of V. For a systematic code, the

generator matrix EGD has the form given by:

CG] = [Ikib] where DIJ is the identity matrix of order k and Lb] is an arbitrary k-by-(n-k) matrix.

The null space of DI VI , is a set of qn-k n-tuples,

which is generated by an (n-k)-by-n parity check matrix 1.7A, whose rows are linearly independent and form a basis for V'.

The reduced echelon form of [H] is given by:

CH] = E-bT 11.11_0 The reduced echelon form of the parity-check matrix

[H] of an (n,k) nested code may be constructed by using two

known error-correcting codes (see Figure 4.2.1), the first being

an (ni, kl) linear block code or nested code, and the second an

(n2, k2) nested code. The procedure of construction can be

described in the following steps:-

- ki

Da k2=n1-k1

The parity-check matrix of the first (n1,1(1) linear block code

EIn_kj CHID = E-bT, I ini__1(i] Parity-check digits generated byl the second (n2,k2) nested code.

k

2=11-

.NC

Figure 4.2.1 THE CONSTRUCTION or THE PARITY-CHECK MATRIX OF AN (n, k) NESTED CODE

k=n1

>n-

92.

•

(i) The last (n-k) columns of the DI:1 matrix form an identity matrix of order (n-k), [In_k] •

(ii) The first k columns of the first (nl-k1) rows of the

parity check matrix [H], form the parity-check matrix [H1] of the first (ni,ki) linear block code (i.e. ni = k).

(iii) Take each of the columns of the [H1] matrix as a block of k2 information digits of the second (n

2,k2) nested

code, (i.e. k2 = n1-k1), and then generate the corresponding

n2-k2 parity check digits of the second (n2,k2) code to form

the remaining (n-k)-(nl-k1) digits of each of the first k

columns of the pci matrix (i.e. n2 = n-k).

The null space of the constructed parity-check matrix

Da is an (n,k) nested code. The validity of this statement

can be shown by the following argument.

Since each of the k columns of the Da matrix is a codeword of the second (n2,k2) nested code, and since (n2,k2)

is a linear code (i.e. the modulo-q addition of any two code-

words of the code is also a codeword of the code), the (n-k)-.

tuple vectors resulting from the linear combination of the

first k columns of the EH] matrix are therefore codewords of the (n2, k2

) nested-code. Now consider the syndromes which

correspond to errors in the k information digits of an (n,k)

Linear block code. These syndromes are (n-k)-tuple vectors

resulting from the linear combination of the first k columns

of the parity-check matrix or conversely they are codewords

of the (n2, k2) nested code. The resulting (n,k) linear block

code is therefore, by definition, a nested code.

The Hamming distance of the resulting (n,k) nested

code is determined by the following:

93.

Lemitia 4.2.1

A block code that is the null space of a matrix [H]

has a minimum Hamming distance, d, if and only if the linear

combination of every (d-1) columns of the [H] matrix is linearly

independent ()

Theorem 4.2.1

An (n,k) nested code derived from two codes, the

first being a (k, k1) block code and the second a (n-k, k-k1)

nested code, has an odd minimum Hamming distance, d, if the

minimum Hamming distances of the first and second codes are d

and d-1 respectively.

Proof:

Consider the parity-check matrix Era of the (n, k)

nested code which is given by:

[H] = E.-13TI/n_k3

The first (k-k1) digits of each column of E-1311 matrix

is a column of the parity-check matrix [H1] of the (k,k1) block

code. Since the (k,k1) block code has a maximum Hamming distance,

d, it follows that the linear combination of every d-f column of

[-bill is linearly independent.

The vectors resulting from these linear combinations

are codewords of the (n-k, k-k1) nested code and they are all

non-zero vectors. Since the minimum Hamming distance of the

(n-k, k-k1) nested code is d-1, the minimum weight of these

vectors is d-1. It follows that the vectors resulting from

the linear combination of every (d-1) columns of the E.-01

matrix are linearly independent of every (d-2) columns of the

identity matrix Hence every d-1 columns of the [H]

• 94.

1 1 1 0 1

1 1 0 1 0

1 0 1 0 0

1 0 0 1 0

0 1 1 0 0

0 1 0 1 0

0 0 1 1 0

DO

matrix are linearly independent and according to Lemma 4.2.1

the minimum Hamming distance of the (n,k) nested code is d.

This completes the proof.

4.2.1 Double error-correcting nested codes

An (n,k) nested code, of minimum Hamming distance

equal to five, may be constructed by using for the first code

the best known block error-correcting code for a given number

of parity-check digits and with a minimum Hamming distance

equal to five and an appropriate double error-correcting and

single error-correcting Hamming code or its shortened version

as the second code.

For example, consider the construction of a nested

binary code from the (11,4) Slepian(105) block code of minimum

Hamming distance equal to five, and the shortened double-error

detecting and single error-correcting (12,7) Hamming code.

The parity-check matrix of the derived (23,11) nested code may

be constructed as follows:-

The [H1:1 matrix of the Slepian (11,4) double error-

correcting code is given by:-

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

The generator polynomial of the (12,7) shortened

Hamming code () is given by:-

95.

g(x) = x5 + x4 + x2 + 1.

The codeword of the (12,7) code corresponding_to the

information given by the seven digits of the first column of the

Ey1:1 matrix of Slepian code is:- x11

+ x10 + x

9 +x8 + x

2 + x

The eleven codewords corresponding to the information

pattern given by the eleven columns of the Slepian code EHJj matrix may be determined in a similar manner. The parity-check

matrix [H] of the (23,11) nested code is given by the eleven

columns which are identical to the calculated codewords of the

(12,7) code and the 12th order identity matrix EI12D. The EH3 matrix therefore has the form:

MN/

1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1101010 0 0 0 0 010 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0110 0 0 0 010 0 0 0 0 010 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 010100110110 0 0 0 0 0 010 0 0 0 0 1 1 1 1 0 1 0 1 1 0. 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0

1 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0

2 1 1 1 o 1 1 o 1 1 1 o 0 o o o o o o o o o

Table4.2.1provides a list of double error-correcting

nested codes derived from the best previously known binary

(n,k) block codes of parity-check digits from five to twenty

digits. The parameters and the generator polynomial of the

required double error-deteCting and single error-correcting

Hamming code are also tabulated.

96.

n=n1+n2 k=n1 k1 Ref. n2=n-k k2=n1

13 5 1 Repetitive (7) 9 4

19 8 2 Slepian(105) 11 6

23 11 4 Slepian(105) 12 7

30 17 9 Prange(7) 13 8

37 23 14 Wagner(51) 14 9

46 31 21 BCH(7) 15 10

57 41 30 (89) 16 11

81 63 51 BCH 18 12

90 71 58 (89) 19 13

147 127 113 BCH 20 14

164 143 128 (106) 21 15

277 255 239 BCH 22 16

294 271 254 (106) 23 17

535 511 493 BCH 24 18

1049 1023 1003 BCH 26 20

Notes.

(8,4) Hamming Shortened -from (16,11)

ditto

ditto

ditto Shortened from (32,26)

ditto

ditto

ditto

ditto

ditto

ditto

ditto

ditto

ditto

TABLE 4.2.1

A family of double error-correcting nested codes.

n: The length of the nested code (n = n1+n2).

k: Number of information digits of the nested code (k=n).

n1: The length of the best previously known double error-

correcting block code for a given number of parity

check digits.

k1: The number of information digits of the best known

linear block code.

The length and the number of information digits of the

double error-detecting and single error-correcting

Hamming code or the shortened version of it. With a

generator polynomial g(x), (n2=n-k).

97.

4. 2 . 2 Multi le error-correcting nestecicacies

An (n,k) nested code capable of correcting t random

errors or less (where t>2) may be constructed by using the

best previously known block codes of the required number of

parity check digits and also capable of correcting t random

• errors. The second code is the appropriate t-detecting and

(t-1)-correcting nested code.

Table 4.2.2 provides a list of three-error-correcting

nested codes. The first group of codes contains the best

previously known block codes capable of correcting three errors,

and the second group os codes are the nested codes of Table 4.2.1

or their shortened version with the addition of an overall parity-

check digit to each of them.

Similarly, the parameters of the nested codes capable

of correcting four random errors are calculated and then

tabulated in Table 4.2.3. The parameters of those capable of

correcting five errors are tabulated in Table 4.2.4.

This procedure can be applied to generate nested

codes for various values of rates and error-correcting

capabilities.

98.

TABLE 4.2.2

A family of three-error correcting nested codes.

Notations are similar to those of Table 4.2.1.

(n1k1)

(n2k2)

The best previously known three-error correcting block codes.

Three-error detecting and double-error correcting nested codes of Table 4.2.1 with the addition of an overall parity-check digit.

n=n1+n2 k=n1 k1 Ref. _ n2 k2=n1-k1 Notes

24 7 1 Repetitive () 18 6 From (19,8)°

33 11 2 Slepian (105) 22 9)

38 15 5 BCH 23 10) Short (24,11)

47 23 12 Golay(7) 24 11 From (23,11)

58 30 16 Karlin(47) 28 14) Short (31,17)

63 34 19 (89) 29 15)

96 63 45 BCH 33 18) 106 121

72 86

53 66

(89) (89)

34 35

19) 20)

Short (38,23)

163 127 106 BCH 36 21)

181 143 120 (106) 38 23 From (37,23)

298 255 231 BCH 43 24) 331 287 261 (106) 44 26) Short (47,31) 556 511 484 BCH 45 27) From (46,31) 1069 1023 993 BCH 46 30)

99.

TABLE 4.2.3

A family of four-error correcting nested codes.


(n1, k1) The best previously known four-error correcting block codes.

(n2, k2) Four-error detecting and three-error correcting nested codes of Table 4.2.2 with the addition of an overall parity-check digit.

n=n1-f-n.2

40

50 55

59

63 68 73 78 86 89 94

116 139 184

212 317

353 578

1094

k=n1

9

14 17

20

22 26 30 34 41 43 47

63 83 127

151 255

287 511

1023

k1

1

2 3

5

6 9 12 15 21 22 25

39 56 99

120 223

252 475

983

Ref._

Repetitive

Fontaine(56) (56) Fontaine

BCH

Wagner(51) (89) (89)

Chen (89) (89)

BCH (106) BCH

(106) BCH

(106) BCH

BCH

n2

31

36 38

39

41 42 43 44

46 47

53 56 57

61 62

66 67

71

45 45

k2=n1-k1

8

12 ) 14 )

15

16 ) 17 ) 18 ) 19 ) 20 ) 21 ) 22 )

24 ) 27 ) 28 )

31 ) 32 )

35 ) 36 )

40

Notes

Short (34,11) From (33,11)

Short(39,15)

From(38,15)

Short(48,23) From(47,23)

Short(59,30) From (58,30)

Short(64,34) From(63,34)

Short(71,40)

From(70,40)

100.

Short(51,14) from (50,14)

15 Short(56,17)

17 From(55,17)

Short(60,20)

20 From(59,20)

26 From(68,26) Short(74,30) from(73,30)

34 From(78,34)

35 ) 39 ) 40 )

45

10

18) 19)

27

Short(91,43) from(90,43)

Short(96,47) from (95,47)

50 Short(117,63) from (116,61) ,

•

TABLE 4.2.4

A family of five-error correcting nested codes.


(nk1) The best previously known five-error correcting block codes.

(n2,k2) Five-error detecting and four-error correcting nested codes of Table 4.2.3 with the addition of an overall parity-check digit.

n=n +n 1 2

k=n1 k1 Ref. n

2 k2=n1-k1 Notes

58 11 1 Repetitive(7) 47

71 17 2 Fontaine(56) 54

76 20

3 BCH 56

81 23

5 MacDonald(107) 58 85 26

7 (89) 59

91 31 11

BCH

60

114 45 19

(106)

69

134 63 36

BCH

71

170 91 57

(106)

79

208 127 92

BCH

' 81 238 153 114 (106)

85

340 255 215

BCH

85

604 511 466

BCH

93

1127 1023 973

BCH

104

101.

4.3 THE NESTED DECODING ALGORITHM

We shall now examine the decoding procedure for the

previously proposed nested codes. For this purpose consider an

(n,k,d) nested code over a field of q elements derived from an

(k, k1, d) block code and (n-k,

k-k1, d-1) nested code.

The parity-check matrix [H1] of the (k, kl, d) block

code can be arranged in the following form:

1 0 0 0 . . . 0 al,r+1 a1,r+2 a1,r+3 • . . al,k 0 1 0 0 . . . 0 a2,r+1 a2,r+2 ,a2,r+3 • • ' a2,k 0 0 1 0 . . . 0 a3,r+1 a3,r+2 a3,r+3 • • . a3,k • .

[Hi] • • •

• . •

• • • •

0 0 0 0 . . . 1 ar,r+1 ar,r+2 ar,r+3 • . . ar,k

wherer=k-kl anda..ij may have any value of the q-elements of

the field.

Let the parity-check digits of a codeword in the

(n-k, k-k1, d-1) nested code corresponding to the information

digits given by the ith column of [Hi] matrix, al,i a2,1 a3,1

.. ar,i, be given by the digits Cr+1,i Cr+2,i Cr+3, i Cn-k,i'

Thus the parity-check matrix of the (n,k,d) nested code can be

arranged in the following form:-

1 0 0 1 0 0 . • . 0 0 0 0 0 0

• 0 0

0 0 1

0 0 0

0

0 . . 0 . . 0 . .

0 . . 0 . 0 .

0 .

. 0 1 0 0 0 . . .

. 0 0 _ 1 0 0 . . .

. 0 0 0 1 0 . . . . .

• •

• •

. 0 0 0 0 0 • • • . . 0 .

Cr+1,1Cr+1,2 Cr+1,3 cr+1,4 "

. . 0 . Cr+Z1Cr+2,2 Cr+2,3 Cr+2,4 ' ' . .

• •

. . . . 1 . . .

Cn-k,1%-k,2cn-k,3Cn-k,4 •

0 a a1,r+1 a1,r+2 1,r+3 ' • a a 0 2,r+1 a2,r+2 2,r+3 • •

0 a3,r+1 a3,r+2 a3,r+3 ' •

• 1 ar,r+1 ar,r+2 ar,r+3 • • C • Cr+1,r Cr+1,r+1 cr+1,r+2 Cr+1,r+3 • . Cr+2,r Cr+2,r+1 r+2,r+2 Cr+2,r+3 •

. Cn -k,r cn-k,r+1 cn-k,r+2 Cn-k,r+3 •

• al,k • a2,k • a3,k

• ' • ar,k • r+1 k . C ' r+2,k .

. Cn-k,k

=141 11 ] n- C . Irk]

2 [..1 ---- n-k C

102 .

•

1

Let e = [el e3 e ] be an error pattern. The digits (e 1 2 3 .•. n 1, e2 ... en-k) e •.. e are parity error digits and en-k+1' en-k+2' n are information error digits. The syndrome corresponding to

error pattern e is:

ss 1 2 s3 sn-kJ

= [eJ COI The above equation yields the following syndrome bits:

s1=e1

+dn-k+1 +a1,r+1en-k+r+1+...+a1,k en

s2= e2 en-k+2 +a2,r+1en-k+r+1+...+a2,k en s3= e3 en-k+3 -F - 4...+a a3,r+1en-k+r+1 3,k en

sr- er

en-k+r+ar,r+1en-k+r+1+...+ar,k en

e ...+ Sr+1= r+1 +Cr+1,1en-k+1+ Cr+1,ken

e Sr+2 r+2 ... +Cr+2,1en-k+1+ 1-Cr+2,ken

• •

en -k,1e+ ...+ Sn-k= n-k+1 n-k Cn-k,ken

Each bit of the syndrome is a modulo-q addition of certain error

digits given by one of the above (n-k)-syndrome equations.

However, if the information digits are free of errors (i.e.

= en-k+1 en-k+2 = = en = 0) then the (n-k)-tuple vector s is

given by:

s = Eel e2 e3 . . . en-k] This implies that the syndrome vector s is identical to the error

pattern e or conversely, for a t-error correcting code, the

syndrome vector which corresponds to.a pattern of t errors or

103.

less in the parity check digits is of weight t or less. Then,

any other syndrome vector corresponding to other correctable

errors pattern is of weight greater than t.

Since the (n-k) rows of the Da matrix are linearly

independent, the (n-k) syndrome-equations are also linearly

independent. This suggests that if the code rate is 50% or •

less (i.e. the number of information error digits is less than

or equal to the total number of syndrome equations), and if

the parity check digits are free of errors (i.e. el = e2 = e3 = .

= en-k = 0), the numerical values of the information error

digits, en-k+1 en-k+2

en, correspond to the solution of k

of the (n-k) syndrome equations.

Theorem 4.3.1

In an (n,k) nested code over a field of q elements,

which is capable of correcting t-random errors and derived from

two codes, the first a (k, kl, d) block code, and the second an

(n-k, k-k1, d-1) nested code, no syndrome which corresponds to

t errors or less and involves at least one of the parity check

digits is a codeword of the (n-k, k-kl, d-1) nested code.

Proof

Consider a syndrome vector, s, of the (n,k) nested

code, which corresponds to j errors (where j<t), so that (j-i)

errors are in the k information digits (where 1<i<j), and the

remaining i errors are in the parity check digits.

Let the syndrome vector corresponding to the (j-i)

errors in the information digits be sl and that corresponding to

the i-errors in the parity check digits be s2. Thus s is given

by:

s = s1

+ s2 (modulo-q addition)

104.

The above equality indicates that the Hamming

distance between s and s1 (d(s,s1)) is equal to the Hamming

weight of s2, w(s„,L) , i.e.

d(s,s1) = w(s2)

However, the vector s2 is identical to the vector corresponding

to errors in the parity check digits. This implies that:

w(s2) = i

and thus:

w(s2 ) C t

and therefor

d(s,s1) t 4; d21

Since the vector s is by definition a codeword of the

(n-k, k-k1, d-1) nested code and since the minimum Hamming

distance between any two codewords in the (n-k, k-kl, d-1) nested

code is d-1, the vector s is therefore not a codeword of the

(n-k, k-k1, d-1) nested code. This completes the proof.

Corollary 4.3.1

In an (n,k,d) nested code over a field of q-elements,

which is capable of-correcting t-random errors and is derived

from two codes, the first being a (k,kl,d) block code and the

second a (n-k, k-k1,

d-1) nested code, the syndrome of the (n-k,

k-k1,

d-1) nested code for an (n-k)-tuple vector, given by a

syndrome corresponding to t errors or less in the (n,k,d)

nested code, will correspond to errors in the parity check digits.

Proof

Since the syndromes of the (n,k,d) nested code

corresponding to errors in the k information digits are codewords

105.

of the (n-k, k-k1, d-1) nested code, the proof of the corollary

becomes self evident.

From the above argument the principles of the nested

decoding algorithm follow immediately for codes of rate less

than or equal to 50%. The decoder generates a nest of t syndromes

for the received vector (see Figure 4.1) which leads to a simple

operation for clearing the errors from the parity check digits.

As a result of this, the errors in the information digits can be

corrected by a solution of the syndrome equations.

Consider the nested decoding algorithm of an (ni,ki,t)

nested code. If all nested codes of the (n • k • t) nest are of

rate less than or equal to 50%, then the operation of this

decoder can be described in the following steps:

(1) Calculate the syndrome, s, corresponding to the

received vector, u, according to the syndrome-equations of the

first nested-code of the (n1, k1, t) nest:

s = 11[1-11] = V DT] e EHT3 e DiT3

where v is the transmitted vector and e is the error pattern

vector.

If the syndrome vector is zero, the decoder assumes

that no error has occurred and no correction is necessary.

If the syndrome is not zero, the decoder performs

the following tests:-

(a) If the weight, w(s), of the syndrome vector is less

than or equal to t, the decoder assumes that all errors

are confined to the parity-check digits and no correction

is necessary.

(b) If w(s)>t, the decoder proceeds to the second step.

106.

(2) The syndrome of the previous step is considered as a

received vector for the next code down the nest (see Figure 4.1.1)

and its corresponding syndrome is calculated according to the

syndrome-equations of this code.

If the calculated syndrome is zero, the decoder

proceeds to step (3); otherwise the following operations are

performed:-

(a) If the weight of the calculated syndrome vector is

less than or equal to the error-correcting capability of

this code, the decoder assumes that the error pattern in

the first L digits of the received vector is identical to

the calculated syndrome, the dimension of the calculated

syndrome vector being L.

The inverse of the calculated syndrome is added to itself

to the first L digits of the received vector u, and to all

the previously calculated syndromes. The decoder then

proceeds to step (3).

(b) If the weight of this syndrome is greater than the

error-correcting capability of this code, and if the code

is not an (nt, kt) Hamming code, the above procedure is

repeated starting with step (2).

(c) If the code IA the (nt' kt) code, the errors in the

first nt digits of the received vector are determined by

using an appropriate decoding algorithm of Hamming codes.

The inverse of the determined error vector is added

to the first nt digits of vector u and to the first nt

digits of all previously calculated syndromes. The decoder

then proceeds to step (3).

(3)

Assume that the code undergoing processing in the

previous operation is the (ni, ki, t-i+l) nested code, i.e. the

107.

.th code of the nest (see Figure 4.1. ). Since the calculated

syndrome of the ni-tuple vector of the previous operation is

found to be zero or made to be zero, the decoder assumes that

the first n. digits of the resulting processed received vector

u are free of errors. This implies that the errors in the

ni-l -tuple vector are now confined to the Ri-1 information

digits. These errors are assumed to be a solution of the

syndrome equations of the (i-1)th code of the nest (ni_i, ki_1,

t-i).

The inverse of the error pattern in the ni_1-tuple

vector is then added to the-tuple vector, to the first ni-1 ni-1

digits of the received vector, and to all the previously found

syndromes of the nest.

If the (i-1)th

code is not the outer code of the nest,

the decoding procedure, starting with step (3), is repeated.

Figure 4.3.1 gives a flow chart of the above basic

nested decoder.

However, to decode an (n,k) nested code of rate'

greater than 50%, some particular code properties are required.

The characteristics of these properties are as follows:-

Consider the syndrome equations of an (n,k) nested

code constructed from the codes (k,ki) and (n-k, k-k1) respect-

ively. If the first r parity check digits are free of errors,

(i.e. el = e2 = = er = 0), the first r bits of the syndrome

vector, s, are therefore given by:-

s1 =en-k+1

s2= en-k+2

S3= e

n-k+3

+ . . . + a1,r+1 en-k+r+1 + a1,k en

+ a2,r+1

en-k+r+1 • • • a2,k

en

. . . + a3,k

en

+ a3,r+1'

en-k+r+1 +

sr= . . . en-k+r + ar r+1 en-k+r+1 +

+ ar,k en

1.08.

GO TO PAG E NEXT 109.

Calculate the syndrome sl, of u, according to the syndrome-equations

of (n1, k1,

t) code.

Is w(s1) equal to

zero

Assume no errors in u

Assume no errors in the information

di:its of u

Yes Calculate the nest syndrome down the nest s., of si_i, according to tke syndrome-equations of (ni, ki, t-i+l)

code

Is w(s) equal to

zero

Add the inverse.of s. to the first L. 1 digits of uls.1,s . 2 ..s., where -L. is the

1 dimensionlof s.

I.,k.,t-i+1) the 1 inner

1 code of t

Yes

Calculate the corresponding errors pattern e, in si-1 using an appro-

priate decoding algorithm of Hamming codes

FIGURE 4.3.1 The Flow Chart of a basic Nested Decoder of an appropriate (n1.11, t) Nested Code

Read and store the received vector u

Calculate e' as a solution to the syndrome-equations of (n.

1 ,k.

1, t-i+l) code

-

Add the inverse of e' to the first digits of u'ss2'

..., and si 2

Consider the nest code up the nest (ie reduce i by 1)

Yes

Is n.1 ' k. t-i+l) th outer i-1 , code of the

nest

Add the inverse of e to the first L

1-1 digits of us

1,s2'

...., and s . i-1

Assume the errors pattern e is confined to the last (

- )digits of s -Li-1 i-2

Assume the decoded received vector is identical to the

transmitted vector

110.

where r = k-k1.

These equations are in fact the syndrome equations of

the (k,k1) block code. A possible approach for decoding is

therefore: if the (k,k1) block code is itself a decodable

nested code, the numerical evaluation of the information error

digits (en_k+i, en-k+2,

..., en) can be determined by decoding

the (k,k1) nested code after clearing all errors from the first

r parity check digits of the (n,k) code.

In general an (n,k) nested code is decodable, if it

exhibits one of the following properties:-

(1) The rates of all the codes of the (n;k) nest, other

than the inner code, are less than or equal to 50%.

In this case, the basic decoder is as explained above.

(2) The rate of the (n,k) nested code is greater than 50%.

However, all other codes of the (n,k) nest are of rate less than

or equal to 50% except for that of the inner code, and the (k,k1)

code is itself a decodable nested code.

The decoder first clears the errors in the parity check

digits of the (n,k) code and then proceeds to decode the (k,k1)

nested code.,

(3) The second code of the (n,k) nest and the (k,k1) code

are both decodable nested codes.

These properties are the generalised form of (2).

(4) The first r parity check digits of the (n,k) nested

code are decodable (i.e. the jth code of the (n,k) nest, (ni,kj),

whose length n. is a decodable nested code) and the 3

(k,k1) is a decodable nested code.

(5) Both second codes of the (n,k) nest and the (k,k1) nest

are decodable nested codes, where k is less than or equal to

(n k1).

The decoder in this case first clears the errors from

the parity check digits of the (n,k) code (e e2' ...,

en-k)

and then proceeds to clear the errors in the parity check digits

of (k,k1) code (en-k+1' en-k+2'' en-k+r)• The remaining

information error digits are, therefore, a solution of the (n-k)

syndrome equations.

An (n,k) decodable nested code can, therefore, be

constructed in a variety of ways, so as to exhibit any one of

the five properties. These many possible ways of construction

lead to a wide range of code rates and error-correcting

capabilities.

Table 4.3.1 provides a list of examples of decodable

nested codes with rates greater than 50% and of various error-

correcting capabilities.

112.

TABLE 4.3.1 Some examples of decodable binary nested codes of rate greater than 50%

Decodable nested code.

Rate of the above code.

The first nested code for constructing the (n, k, d) nested code.

(n-k,k-ki,d-1)

(n, k, d) R in %

The second code of constructing the (n, k, d) nested code.

(k, k1, d) (n-k,k-ki,d-1) Notes

26, 13,5 50.0 13, 5,5 13, 8,4 Shortened from (16,11,4) Hamming code.

35, 19,5 54.2 19, 8,5 16, 11,4 Hamming code. 41, 23,5 56.0 23, 11,5 18, 12,4 Short Hamming (32,26,4) 49, 30,5 61.2 30, 17,5 19, 13,4 ditto 74, 49,5 66.2 49, 30,5 25, 19,4 ditto 105, 74,5 70.4 74, 49,5 31, 25,4 ditto 142, 105,5 73.9 105, 74,5 37, 31,4 ditto (64,57,4) 186, 142,5 76.3 142, 105,5 44, 37,4 ditto 237, 186,5 78.4 186, 142,5 51, 44,4 ditto 295, 237,5 80.3 237, 186,5 58, 51,4 ditto 361, 295,5 81.7 295, 237,5 66, 58,4 ditto (128,120,4) 435, 361,5 82.9 361, 295,5 74, 66,4 ditto 517, 435,5 84.1 435, 361,5 82, 74,4 ditto 607, 517,5 85.1 517, 435,5 90, 82,4 ditto 705, 607,5 86.0 607, 517,5 98, 90,4 ditto 811, 705,5. '86.9 705, 607,5 106, 98,4 ditto 925, 811,5 87.6 811, 705,5 114,106,4 ditto 1047, 925,5 88.3 925, 811,5 122,114,4 ditto 1178,1047,5 88.8 1047, 925,5 131,122,4 ditto (256,247,4) 1318,1178,5 89.3 1178,1047,5 140,131,4 ditto 1467,1318,5 89.8 1318,1178,5 149,140,4 ditto 1625,1467,5 90.2 1467,1318,5 158,149,4 ditto 1792,1625,5 90.6 1625,1467,5 167,158,4 ditto 1968,1792,5 91.0 1792,1625,5 176,167,4 ditto 2153,1968,5 91.4 1968,1792,5 185,176,4 ditto 2347,2153,5 91.7 2153,1968,5 194,185,4 ditto 2550,2347,5 92.0 2347,2153,5 203,194,4 ditto 2762,2550,5 92.3 2550,2347,5 212,203,4 ditto 2987,2762,5 92.5 2762,2550,5 221,212,4 ditto 3213,2983,5 92.8 2983,2762,5 230,221,4 ditto 3452,3213,5 93.0 3213,2983,5 239,230,4 ditto

113.

(n, k, d) R in % (k, k1 (n-k k-k ,d-1)

76, 38,7 50.0 38, 15,7 38, 23,6 91, 47,7 51.6 47, 23,7 44, 24,6

118, 61,7 51.6 61, 30,7* 57, 31,6 124, 66,7 53.2 66, 34,7* 58, 32,6 161, 91,7 56.5 91, 47,7 70, 44,6 214, 124,7 57.9 124, 66,7 90, 58,6 342, 214,7 62.7 214, 124,7 128, 90,6 515, 342,7 66.5 342, 214,7 173,128,6 739, 514,7 69.5 514, 342,7 225,173,6

1023, 739,7 72.2 739, 514,7 284,225,6 1374,1023,7 74.4 1023, 739,7 351,284,6 1800,1374,7 76.3 1374,1023,7 426,351,6 2309,1800,7 77.9 1800,1374,7 509,426,6 2909,2309,7 79.3 2309,1800,7 600,509,6 3608,2909,7 80.6 2909,2309,7 699,600,6 4414,3608,7 81.7 3608,2909,7 806,699,6 5335,4414,7 82.7 4414,3608,7 921,806,6

94, 47,9 50.0 47, 25,9 47, 22,8 186, 94,9 50.5 94, 47,9 92, 47,8 227, 116,9 51.1 116, 63,9** 111, 53,8 253, 139,9 54.9 139, 83,9** 114, 56,8 458, 253,9 55.2 253, 139,9 205,114,8 792, 458,9 57.8 458, 253,9 334,205,8

1300, 792,9 60.9 792, 458,9 508,334,8 2034,1300,9 63.9 1300, 792,9 734,508,8 3053,2034,9 66.6 2034,1300,9 1019,734,8 4424,3053,9 69.0 3053,2034,9 1371,1019,8 6222,4424,9 71.1 4424,3053,9 1798,1371,8 8530,6222,9 72.9 6222,4424,9 2308,1798,8

Notes

From (37,23,5) nested code. Shor.from (49,30,5) nested code. ditto (74,49,5) ditto ditto ditto (105,74,5) ditto (142,105,5) ditto (186,142,5) ditto (237,186,5) ditto (295,237,5) ditto (361,295,5) ditto (435,361,5) ditto (517,435,5) ditto (607,517,5) ditto (705,607,5) ditto (811,705,5) ditto (925,811,5)

Shor.from (47,23,7) ditto (91,47,7) ditto (118,61,7) ditto ditto (214,124,7) ditto (342,214,7) ditto (515,342,7) ditto (739,514,7) ditto (1023,739,7) ditto ditto (1800,1374,7) ditto (2309,1800,7)

j. Some supporting nested codes in which the decoder is capable of

correcting the errors in parity check digits of the code only:- (n,k,c1) (k,k1211 (n-k k-k d-1) it

61,30,7 30,16,7 Karlin( 31,14,6 47) From (35,19,5) 66,34,7 34,19,7 (89) 32,15,6 From (35,19,5) 98,63,7 63,45,7 BCH 35,18,6 From (35,19,5)

** o% Aft

116,63,9 63,39,9 BCH 53,24,8 From (58,30,7) 139,83,9 83,56,9 (106) 56,27,8 From (58,30,7)

114.

4.4 FEATURES AND MERITS OF THE NESTED DECODING ALGORITHM

The main two features of the nested decoding algorithm

are as follows:-

4.4.1 The capability of correcting pattern of errors of weight greater than t

Theorem 4.4.1

The decoder of an (n,k,t) nested code which is

constructed to fulfil the conditions of decodable nested codes

given in (1) is capable of correcting any pattern of errors of

weight greater than t if the pattern of errors exhibits the

following properties:-

(1) The pattern of errors is a coset leader of the

• standard array of the (n,k,t) code.

(2) The number of errors in the parity check digits is

less than t.

(3) The pattern of errors in the remaining k information

digits is also a coset leader of the standard array of the code.

Proof

Consider a pattern of errors e (where e = Ce/e2e3...enj)

which fulfils the conditions of the above theorem. Let the

n-tuple vector em where em ... en be = E0 0 0 .•• en-k+1en-k+2 n

J, the pattern of errors in the information digits and let the

n-tuple vector ep where ep = [ele2 en-k 0 0 ... 0:1 be the

pattern of errors in the parity check digits.

Since e is of weight less than t, it is therefore a

coset leader of the (n,k,t) nested code and since all the errors

of e are in the parity check digits, it follows that the

syndrome, s , corresponding to e , is identical to the first

(n-k)-digits of e , i.e.

• 115.

•

s = Ee le2e3 en-kJ

Since the n-tuple vector e is a coset leader and of

weight greater than t, the syndrome s corresponding to e is a

unique nonzero (n-k)-tuple vector of weight greater than t and

given by:

s = e [HT]

where EHT3 is the transpose of the parity-check matrix of the (n,k,t) nested code.

Let the syndrome vector corresponding to errors in

the information digits em be S. Thus the syndrome vector s

is given by:

s = sp + sm ep CHT3 + ern CHT3 The first operation of the nested decoder is the

calculation of s. Since s is of weight greater than t, the

decoder then proceeds to calculate the syndrome vector s1 corresponding to the vector s according to the syndrome

equations of the (n-k, k-kl, t-1) nested code. This code is

obviously the second code of the (n; k; t) nest. Let the

parity-check matrix of the (n-k, k-kl, t-1) nested code be

. Then sl is given by:

si =sEHT2j=(sp +sm) 2 EHTJ= sp 2 DIT3+ sm 2 DT]

Since sm is a codeword of the (n-k, k-k1,

t-1) nested

code we have:

s DITJ m 2 = 0, and therefore

s1 = sp CHTJ = Ee1 e2 e3 ... e n- 0 [H2]

However, since the weight of the error vector Eele2e3 ... en_k]

is less than or equal to the error correcting capability (t-1)

116.

of the (n-k, k-k1, t-1) nested code, the error pattern, e ,

can be corrected by the nested decoder*.

Since em is a coset leader, the remaining information

error digits Cell _k4.1 en-k+2 ej can be determined as a

solution of the syndrome equations of the (n,k,t) nested code.


Corollary 4.4.1

The decoder of an (n,k,t) nested code, which has the

properties given in one of the conditions of (2), (3) or (4) of

constructing decodable nested codes, is capable of correcting

any pattern of errors of weight greater than t, if the pattern

of errors exhibits the following properties:-

(1) The pattern of errors is a coset leader of the standard

array of the (n,k,t) code.

(2) The pattern of errors in the parity-check digits is

a coset leader of the (n,k,t) code and is decodable by the

decoder of the second code of the (n;k;t) nest.

(3) The pattern of errors is in the k information digits,

is a coset leader of the standard array of the (n,k,t) code and

is decodable by the.(k, kl, t) nested code.

It is interesting to note that since (n-k, t-1)

is itself a nested code, it is capable of correcting some

patterns of errors of weight greater than (t-1). This

suggests that the second condition of Theorem 4.4.1 may be

restated in a more general form as follows:-

"(2) The pattern of errors in the parity-check

digits is a coset leader of the (n,k,t) code and

is decodable by the decoder of the second code

of the (n;k;t) nest."

117.

S

Corollary 4.4.2

The decoder of an (n,k,t) nested code which is constr-

ucted to fulfil the conditions of decodable nested codes given

in (5) is capable of correcting some pattern of errors of

weight greater than t, if the pattern of errors exhibits the

following properties:-

(1) The pattern of errors fulfils the first two conditions

set by Corollary 4.4.1, and

(2) The pattern of errors in the information digits is a

coset leader of the (n,k,t) code and the first (k-k1) digits

(i.e. en-k+1 en-k+2 en_ki) are decodabie by the second code

of the (k; kl; t) nest.

It has been established by the above theorem and two

corollaries that the nested decoding algorithm is capable of

correcting some of the correctable pattern of errors* of weight -

greater than t (where t = d21). The importance of this feature

is realized from the fact that most known decoding algorithms

of known classes of codes are not capable of correcting errors

of weight greater than t. It is also of interest to note that

finding a simple way to extending Berlekamp's decoding

algorithm for BCH codes, so as to enable the decoder to

correct the correctable errors of weight greater than t, is an

important but as yet unsolved problem.

4.4.2 The complexity of the nested decoder

The complexity of decoding is generally measured by

the number of operations required for decoding a single

*

A correctable pattern of errors is one that is a coset

leader of the standard array of the given code. Since

there are 2n-k

-1 nonzero coset leaders, the number of

. correctable patterns of errors if therefore equal to

2n-k

-1.

118.

information digit.

The number of operations required for decoding a

sequence of n digits at the output of a channel using an

(n,k,t) nested code may be tabulated as follows:-

(1) The calculation of a nest of t syndrome vectors:-

The dimension of the first vector is (n-k) whilst

the dimension of other vectors of the nest approximately

decreases exponentially with their positions down the nest. The

calculation of such syndromes requires the evaluation of each

of the individual syndrome bits. Each bit of a syndrome in the

nest is a modulo-q addition of certain digits of the syndrome

directly above it in the nest and is determined by the syndrome

equations of the corresponding code of the (n;k;t) nest.

Syndrome calculations therefore involve a number of modulo-q

additions (for binary codes modulo-q adders are simply

exclusive-OR gates). The number is equal to the total

dimension of the t syndromes, which increases linearly with

code length and t.

(2) The syndromes weighting:-

Each of the calculated syndromes in the above operation

is tested as to whether its weight is greater than, equal to or

less than the error-correcting capability of the corresponding

code of the (n;k;t) nest. The number of operations obviously

depends only on the number of the syndromes under test.

(3) The decoding of the inner code, (Hamming code), of the (n; k; t) nest:-

The number of operations required in this decoding

process depends entirely on the code length and increases

linearly with it.

119.

(4) The decoding of the corresponding information

error d* its of all codes other than the outer code of the

(n; k; t) nest:-

Since each information error digit of these codes is

a modulo-q addition of certain parity check digits of the

individual code, the operation required in this process

involves a number of modulo-q adders. The number of these

adders is equal to the total number of information digits of

these codes, which is a linear function of the (n,k,t) code

length and t.

(5) Decoding the information error digits of the

(n,k,t) nested codes:-

This stage involves, for most decodable nested codes,

the repetition of the above operation for the corresponding

first code of the construction, (n,k,t) nested code.

From the above it may be concluded that the average

number of operations required to decode one symbol of the

received sequence of a digits, is a linear function of the code

length, n, and the error correcting capability t.

Finally, it is worth noting that the decoding procedure

does not require any operation which consumes computation time,

such as filling a table of information or the mathematical

operation in GF(p°). The time required for nested decoding of

an (n,k,t) nested code is therefore approximately equal to 2t

multiplied by the transit time of the modulo-q adders.

120.

• 5. LINEAR BLOCK CODES FOR NON-INDEPENDENT ERRORS

5.1 Introduction.

5.2 Block Codes for Non-independent errors,

Definition and Construction.

5.3 Single-Burst-Error-Correcting Codes.

5.4 Burst-and-Random-Error-Correcting Codes.

5.5 Multiple-Burst-Error-Correcting Codes.

121.

5.1 INTRODUCTION

Previous sections of this thesis have been mainly

concerned with the construction and decoding of block codes

capable of correcting independent (random) errors. However,

in many communication channels, noise disturbances tend to

occur in a (single) burst and in many instances these noise

disturbances cause the errors to cluster in the form of low-

density bursts. For example, on telephone lines, lightning or

man-made electrical disturbances frequently affect several

adjacent transmitted digits. Moreover, in many digital systems

for data transmission (and particularly data storage), distur-

bances of a burst nature and those of random nature are both

present. The burst errors are mainly due.to impulsive distur-

bances or spot imperfections in the storage media, whilst the

random noise may be largely due to circuit noise or noise from

the environment. Under these circumstances, it is not efficient

to use either random error-correcting codes or codes for

correcting (single) burst errors. A need therefore exists for

classes of codes that are capable of correcting either multiple-

bursts of errors or random-and-burst errors. In the literature

on error-correcting codes, there has been a good deal of attention

given to the correction of a (single) burst of errors(3,5,7,108-118)

However, the literature concerned with the correction of either

random-and-burst errors or multiple bursts of errors is

comparatively more limited.

The most effective method of constructing either random-

and-burst or multiple-burst-error-correcting codes is the inter-

lacing of a t random error-correcting (n,k) code with itself,

times; an (n9, kZ) code, capable of correcting any combination

of t bursts of length Q or less, is then obtained. Other attempts (

to construct such codes are given in the literature-5'7,119-130)

122.

However, in many of these cases, the derived codes for correcting

short length of bursts are of extremely large length; as was the

case with the codes found by Bahl and Chien(119), Wolf(120), and

Chien and Spencer(121)

In this section an investigation of other methods of

generating codes (capable of correcting either random-and-burst

errors or multiburst errors) is given in detail. The ideas and

techniques proposed in the previous sections of this thesis are

modified to construct such codes. A considerable number of such

codes have been found by these modified techniques.

123.

5.2 BLOCK CODES FOR NON-INDEPENDENT ERRORS DEFINITION AND CONSTRUCTION

A burst of length t is defined as a vector whose non-

zero components are confined to t consecutive digit positions,

the first and last of which are non-zero(5).

An (n,k) linear block code is said to be an i-burst

error-correcting code, if it is capable of correcting all n-tuple

error vectors of a set [Ph which contain all burst errors of

length t or less (not all bursts of length t+1).

Definition 5.2.1

An (n,k) linear block code V, over a field of q

elements, is capable of correcting a set of errors [P], if and

only if all elements of {P1 are coset leaders of a standard

' array of V(131)

Lemma 5.2.1

An (n,k) linear block code V, over a field of q

elements, is capable of correcting a set of errors tP) if and

only if (e1-e2) in V and (e1 and e2 etpi ) implies el equals e2.

Proof

Consider two n-tuple vectors el and e2 in the set [Ps.

Since the (n,k) code is capable of correcting all vectors in the

set .1211, the n-tuple vectors el and e2 are, according to

definition 5.2.1, coset leaders.

Suppose that (el-e2) is a code vector (say V). Thus

el-e2 equals V or conversely el equals VA-e2. That is, el is

in the coset whose leader is e2, which contradicts the constr-

uction rule of a standard array of the code V, that the coset

leader e1 should be previously unused. Therefore it may be

124.

concluded that for e1 not equal to e2, e1 - e2 does not equal v,

otherwise e1 equals e2. This completes the proof.

From Lemma 5.2.1 it follows that for an (n,k) block

code (that is, the null space of a matrix Ho to correct a set

of E error vectors [131, it is necessary and sufficient that no

code vector consists of the sum of two error vectors el and e2,

where e1and e

2 elpi; consequently, it is necessary and

sufficient that for any two vectors in fP3 such as e1 and e2,

the following inequality holds:

Eel + e2 [HT] 0

The generalised form of the Hamming upper bound on the

number of information digits for a linear code that is capable

of correcting a set of E errors €P1, may be obtained by using

Definition 5.2.1 as follows:

Lemma 5.2.2

For an (n,k) block code V over a field of q elements

(capable of correcting a set of E errors [Ps), the largest value

of k is bounded by:

qn-k > E.

Proof

Since the code is capable of correcting all E error

vectors of set fP), all error vectors in tP1 by definition must

be coset leaders. Thus the number of error vectors of {PI must

be no greater than the total number of cosets. That is:

qn-k > E.


The generalised form of the Varshamov-Gilbert bound,

for an (n,k) block code capable of correcting a set of E

125.

error vectors 1:1, follows from the fact that the parity check matrix [H] for the code may be constructed in such a way that

the chosen columns of [H] satisfy the inequality:

Ee. e .J DTI 0 for all values of i, i = 1, 2, 3, E, and j, j = 1, 2, 3, .

E, where e. and ej G[P'. Thus the worst conceivable case 1

would be for each Lei ej [HT] to result in a unique non-zero vector of the vector space The total number of these

vectors is therefore bounded by qn-k

, including all the zero

vectors for the no error case.

The above properties of the parity-check matrix of

an (n,k) block code capable of correcting non-independent

errors, may be viewed in a slightly different, but nevertheless

useful way, as follows:-

Theorem 5.2.1

An (n,k) linear block code that is the null space of

a matrix [HD is capable of correcting a set of errors (P1 if and only if for every e in f131, the matrix product e[HT3results in a unique non-zero vector in the vector space Vn-k.

Proof

Let e be an error vector in [Ps. The syndrome

corresponding to e is given by:

s = e [HT].

Since the syndrome of any error vector e, eCiP1, is

unique, and since all the vectors of a coset have the same

syndrome, no two error vectors (el, e2) in {Pi are therefore

in the same coset. This implies that every error vector

e, e (Pi, is a coset leader and according to Definition 5.2.1,

126.

the code is capable of correcting all error vectors in {P.I.


An (n,k) block code may also be constructed by using

the construction procedure for nested codes, described in

Section 4.2, in such a way that the first and second codes of

construction are linear block codes. The error-correcting

capability of the newly derived code is determined by the

following theorem:-

Theorem 5.2.2

An (n,k) block code constructed in accordance with

the construction rules for nested codes (using two block codes

for construction) is capable of correcting a set of errors tP1,

if the first code of construction is capable of correcting all

patterns of errors in the k information digits corresponding

to the error vectors of fP1 and the second code is capable of

correcting all patterns of errors in the (n-k) parity check

digits corresponding to the error vectors of (Pi.*

Proof

The parity check matrix [H] of the (n,k) code is given

In-k = E.-13 lin_k]

where [H1J is the parity check matrix of the first (k,ki) code of

Since the (n,k) code is constructed in accordance with the construction rules of nested codes, it follows (see Section 4 of this thesis) that the syndromes corresponding to the correctable error patterns in the information digits are codewords of the second code of construction (i.e. the (n-k, k-k1 ) code), and moreover all other syndromes corresp-onding tocorrectable error patterns involving at least one error in the parity check digits are not codewords of the (n-k, k-k

1) code. The correctness of these statements can also be seen as a consequence of the proof of the theorem.

•

*

by:

=

127.

construction, Dn_kj is an identity matrix of, dimension (n-k)

and Ed.] is that matrix in which each of its (k) columns are

formed by the parity check digits of a codeword (in the second

(n-k, k-k1) code of construction) corresponding to the inform-

ation digits given by the appropriate column of DJ] matrix.

Let e = E.e1 e2 en] be an error vector in P .

Let ei = Eel e2 ek 0 0 ... 0] be the error pattern of e

that is confined to the first k digits of the received vector

and let ep = 0 ... 0 ek+l ek+2 • enJ be the error pattern

of e that is confined to the remaining (n-k) digits.

The syndrome vector s corresponding to e is given by:

TT

e = e DT 11 I C

] = (e. +e p) [.1

In-kT T T

=E e2 ek o o [1111C + CO 0 ... 0 ek+1

T

Ti c T In-k

ek+2 en.„ n-k

= 1 e2 ... D 1 ek] T I CT] Eek+1 ek+2 . . . e] n-j]

= e.' T lcT 3 + e' 1

wheree.'= [el e2 ekJ and e = E.ek+1 ek+2 '" er]

Since the first k columns of CH] matrix are code vectors of the (n-k, k-k1) block code, the matrix product e., [HTicT] therefore results in a code vector of the (n-k,

k-k1) code, say v. Thus s = v + ep

Since the first (k-k1) digits of each column of [-bT]

matrix is a column of the parity check matrix [H1] of the (k,k1)

128.

block code, and since the block code is capable of correcting

all the patterns of errors in the k information digits corr-

esponding to e, e 0)1, it follows that for every pattern of

errors such as e., there is a corresponding unique nonzero

syndrome vector (v) in the (n-k, k-k1) code. Thus s equals v

fore' = 0.

Suppose that the syndrome s (where s = v + e e 0)

is a code vector u in the (n-k, k-k1) code; thus u equals v+e 1.

That is, e ' is also a code vector of the (n-k, k-k1); ). this

implies e ' is not a correctable error pattern, contradicting

the condition of the theorem that the (n-k, k-k1) code is

capable of correcting all patterns of errors in the parity

check digits corresponding to e, eel.pi. Therefore it may be

concluded that s is not a code vector of (n-k, k-k1) code unless

e = 0.

Now suppose that the syndrome s (s = v+e ') is equal

to some other syndrome vector si, corresponding to an error

pattern ,

e.3 where e,(3tPl. Let si be given by v.

3 + e.', where

v.3 issomecodevectorofthe(n-k,k-kdcodeande.'is that

errorpatternofe.3 confined to the last (n-k) digits. Thus

v + e ' equals v. +e.'. That is, e.' and e ' are in the same J 3

coset of a standard array of the (n-k, k-k1) code contradicting

the condition of the theorem that the (n-k, k-k1) code is

capable of correcting all patterns of errors in the parity check

digits,suchase'ande.3'. It may be concluded therefore that

s does not equalsj . This implies that for every error pattern

e, where e G [Ps, there is a corresponding unique syndrome and according to Theorem 5.2.1, the (n,k) code is capable of

correcting all patterns of errors in the set {P3. This completes

the proof.

129.

From the above arguments, two methods for constructing

block codes to correct the required set of errors tPi, approp-

riate for the given channel characteristics are suggested. The

first makes use of Theorem 5.2.1 to establish a computerised

search procedure for new codes. However, this method is limited

by the computer core size, computation time or both these

factors. The second method is based on the modification of the

principles used for constructing a nested code for random errors

as given in Theorem 5.2.2.

130.

5.3 SINGLE-BURST-ERROR-CORRECTING CODES

From Theorem 5.2.1 the following corollary follows

immediately:-

Corollary 5.2.1.1

An (n,k) block code V, that is the null space of a

matrix E], is capable of correcting a burst of length 2, or

less, if and only if the linear combinations of every g

adjacent columns of the Efij matrix result in unique nonzero

(n-k)-tuple vectors.

This corollary suggests a computerised search procedure

for single-burst error-correcting codes. A computer search

progrannue is developed to find such codes for a given number of

parity check digits and given burst length 2. . The search

follows the following steps:-

(1) Read the identity matrix Ein_k] of dimension (n-k).

(2) Cross out the (n-k)-tuple vector of the linear

combinations of every adjacent columns of the [H] matrix

from the set of qn-k-1 non-zero vectors of the vector space

Vn-le

(3) Take.the vectors of the vector space Vn_k in turn,

starting with the all-zero vector and ending with the all

q vector,

(4) Test the vector(chosen in step (3))for uniqueness,

if it is not unique, start again at step (3).

(5) Perform the linear combination of the vector(under

test)with the last - 1) columns of the Da matrix and

test the resulting vectors from this linear combination for

uniqueness - if any of these vectors is not unique, start

again at step (3).

131.

(6) Cross out the vector resulting from the last

performed linear combination.

(7) Increase the size of the Da matrix by one column,

by adding the vector to the CH: matrix. (8) Continue the search again, starting from step (3),

until all vectors in Via-k have been tested.

Table 5.3.1 gives some examples of the single-burst-

error-correcting codes that were found by the above search

procedure. However, the search was not carried out extensively

due to the fact that there is a large range of cyclic codes with

a minimum or near-minimum redundancy; moreover, the codes are

capable of correcting burst errors of various lengths

Furthermore, there exists a simply implemnted decoding procedure

(error trapping techniques for cyclic codes) that is applicable

to all of these codes. Thus Peterson(7) suggests, from an

engineering viewpoint, that the problem of designing burst-

correcting codes and decoders appears to be solved. However,

the general decoder for these codes is capable of correcting

only the specified 2 burst errors or less and moreover that for

certain range of code parameters the choice of codes is limited.

To decode some of the remaining correctable error patterns and

to widen the range of choice of single burst error correcting

codes, the following corollary (which follows from Theorem 5.2.1)

provides a basis for constructing single burst codes, from other

best known codes. The decoder complexity for such codes is

slightly greater than that of the decoder of the original codes.

Corollary 5.2.2.1

An (n,k) block code constructed in accordance with

the construction rules of nested codes, using two block codes

for construction, is capable of correcting any burst of length

132.

2 or less, if both codes of the construction are .t-burst-error-

correcting codes.

For example, consider the construction of (40,27)

single-burst-error-correcting block code, capable of correcting

a burst of three errors or less from two other block codes. The

first is a (27,20) shortened cyclic code capable of correcting

a burst of three errors or less and the second a (13,7) shortened

cyclic code also capable of correcting a burst of three errors or

less. The generator polynomial, g1(x), of the (27,20) code is

given by(7) :

gi(x) = x7 + x6 + x3 + 1.

Thus the parity check matrix, [HI], is given by:-

EH 3. 011001111010010101110001000 010110001100101111000000100 101011000110010111100000010 110101100011001011110000001

The generator polynomial, g2(x), of the (13,7) code is

given by(7):

g2(x) = x6 + x5 + x4 + x3 + 1

Determine the 6-by-27 matrix DO, whose columns are defined as the parity check digits of a set of 27 codewords in

the (13,7) code. Each of these codewords corresponds to those

information digits given by the appropriate column of EHij matrix. Thus matrix [CD is given by:

1 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 0 I 1 1 0 0 0 1 0 0 0 0

Cc]=

Nam. ••■•■

1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0.0 0 1 0 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 1_

133.

The parity check matrix Eta of the newly derived

(40,27) code is therefore given by:-

Da Dll I I13

A wide range of Z-burst-error-correcting codes may be

derived (by the above procedure) from various lengths and rates.

If both codes of construction (k,k1) and (n-k, k-k1)

have a simply implemented decoding procedure, the derived (n,k)

P-burst-error-correcting code may, therefore, be decoded simply

by the following steps:

1) Calculate the symdrome (s) which corresponds to the

received vector (u), according to the syndrome equations of the

(n,k) code. If the calculated syndrome s is zero, the decoder

assumes that no errors occurred in u, otherwise it proceeds to

step (2).

2) Assume the syndrome vector s, as a received vector for

the (n-k, k-k1) code and the corresponding error pattern e

(where e ' =[ek+l ek+2 ... en.] ), may be determined by using the

appropriate decoding procedure for the (n-k, k-k1) code. Add

the inverse of e ' to s and to the parity check digits of u.

3) Assume that the remaining errors e1' (where e1' =

[e1 e2 ... %Dare confined to the k information digits. The

syndrome vector of the (k,k1) code is assumed to be identical

tothefirst(k-kddigitsofs.Thene.'may, therefore, be

determined by using the appropriate decoding procedure for the

(k,k1) code.

The complexity of this decoder is approximately equal

to the sum of the complexities of both decoders of the (k,k1)

and the (n-k,k-k1) codes. However, this decoder exhibits some

of the characteristics of the nested decoders; that is, the

decoder is capable of correcting some of the correctable random

errors and burst-errors of length greater than Q. This charac-

teristic for some applications of error-correcting codes may be

considered as the major factor in the choice of codes. 134.

Add the inverse of e.' to the first (k) digits of u.

(\

Assume the decoded received vector is identical to the transmitted vector

A

•

FIGURE 5.3.1 The flow chart of the decoder of an (n,k) block code (capable of correcting a set of errors {PP) constructed in accordance with the construction rules of the nested codes. Both codes of the construction have simply implemented decoding procedure and are capable of correcting the set of errors (Pi.

\\\\\

Receive and store the received vector, u.

Calculate the syndrome s, of u, according to the syndrome equations of the (n,k) code

Consider s as a received vector to the (n-k,k-k

1) second code of the construction

Determine the error vector e by using the appropriate decoder

of the (n-k,k-k1) code

Add the inverse of e to s and to the last (n-k) dgits of u

Consider the first (k-k1) digit of s as a syndrome vector to the

(k,ki) code

Determine the error vector e.' by using the appropriate decoder

of the (k,k1) second code of the construction.

135.

TABLE 5.3.1

Some Single-Burst- -Error-Correcting-

Codes obtained by the described Computer Search.

• NUMBER OF

CODE INFORMATION THE FIRST K COLUMNS OF THE PARITY-CHECK LENGTH DIGITS (k) MATRIX EH] IN OCTAL

CODES CAPABLE OF CORRECTING SINGLE BURST OF LENGTH TWO OR LESS

7 3 13,5,17.

15 10 26,13,22,11,32,16,21,15,12,5.

30 24 56,26,53,36,74,51,23,64,43,24,62,35,44, 22,77,61,33,50,41,32,21,15,12,5.

62 55 144,36,130,51,145,72,110,45,150,104,71,44, 143,135,160,47,173,103,74,43,117,33,56,167, 142,125,102,120,133,64,136,42,24,163,75, 123,141,53,122,52,101,77,166,105,61,50,16, 41,32,113,21,15,12,5,26.

126 118 330,217,162,354,272,224,46,372,301,27,331, 250,225,207,147,114,103,71,55,23,16,52,201, 156,244,54,216,131,204,163,72,141,202,66, 122,175,305,67,256,120,303,154,117,34,302, 101,212,77,132,61,227,322,346,125,11,344,170, 211,65,374,50,166,373,205,124,314,151,42, 104,371,226,41,160,32,102,206,116,62,213,21, 51,356,214,155,222,173,107,235,350,15,12, 243,266,255,360,5,110,276,233,171,112,333, 26,247,167,220,363,137,31,337,47,364,36,244 13,273,164,237.

250

241

CODES CAPABLE OF CORRECTING SINGLE BURST OF LENGTH THREE ERRORS OR LESS

7 3 13,5,17.

15 10 26,13,22,11,32,16,21,15,12,5.

30 • 24 56,26,53,36,74,51,23,64,43,24,62,35,44,22, 77,61,33,50,41,32,21,15,12,5.

62 55 144,36,130,51,145,72,110,45,150,104,71,44, 143,135,160,47,173,103,74,43,117,33,56, 167,142,125,102,120,133,64,136,42,24,163, 75,123,141,53,122,52,101,77,166,105,61,50, 16,41,32,113,21,15,12,5,26.

136.

126 118 *

253 144 *

507 497 *

Codes obtained using the described computer search

procedure but not included here due to their being

inordinately long.

137. /

5.4 BURST-OR-RANDOM-ERROR-CORRECTING CODES

From Theorem 5.2.1 the following corollary follows

immediately: -

Corollary 5.2.1.2

An (n,k) block code, V, that is the null space of a

matrix EHJ, is capable of correcting t random errors or less,

or a burst of length Q or less, if and only if the linear

combinations of every t columns and every Q adjacent columns of

the D-1:1 matrix result in unique nonzero vectors of the vector space

This corollary suggests a computerised search proced-

ure for burst-or.-random error-correcting codes. A computer

search programme is developed to find such codes for given

numbers of parity-check digits and given values of t and P.

This programme is simply the synthesis of the previously developed

programmes, the first being the search programme developed in

Section 3 of this thesis for finding some new random error-

correcting codes and the second the search programme developed

in the last chapter for finding single-burst-error-correcting

codes.

Table 5.4.1 provides a list of the codes obtained by

the above computerised search; the codes length (n) and the

number of information digits (k) are tabulated together with the

columnS (in Octal) of the determined EH] matrix. The techniques employed in constructing and decoding

random-error-correcting nested codes can be modified slightly

to construct a wide class of nested burst- or -random error-

correcting codes. Codes of this class are simply decodable by

the nested decoding algorithm described in Section 4 of this

thesis, and cover a wide range of code rates and error-

correcting capabilities. Buv.st - ar ) - damcJo,m . _ C ov.ve c 6'7 Cale, = 9.vit -

)

138.

An (n,k) linear block code is said to be a nested

burst-.or:-random (B C)R) error-correcting code if and only if

its syndromes, corresponding to errors in the information

digits, are codewords of either a nested B UR code or a single

burst-error-correcting code. The inner code of the nest of

such an (n,k) nested code is usually a cyclic (or shortened

cyclic) single-burst-error correcting code.

The reduced echelon form of the parity-check matrix Dij

ofan(n,k)nestedBOR code may be constructed by using the

construction procedure of random error-correcting codes

described in Section 4.

The error-correcting capability of a nested BO R

code is given by the following corollary which follows from

Theorem 5.2.2:-

Corollary 5.2.2.2

An (n,k) nested B C)R code is capable of correcting

t random errors or less, or any burst of length for less, if

both codes of the construction are 2-burst error-correcting

codes, the first having a minimum Hamming distance equal to

(2t+ 1) and the second a minimum Hamming distance equal to 2t.

Proof

In accordance with Corollary 5.2.2.1, the (n,k) nested

B"OR code is capable of correcting a burst of length 2,, due to

the fact that both codes of the construction are capable of

constructing any burst of length 2, or less.

Since the first and second codes of the construction

are of minimum Hamming distance, equal to (2t+ 1) and 2t

respectively, the (n,k) code is therefore (according to Theorem

4.2.1 of Section 4) capable of correcting any random errors of

weight t or less. This completes the proof.

139.

An (n,k) nested double random-or-single 2--burst

error-correcting code may be constructed by using, for the

first code, the best known double random-_on-single 2.-burst

error-correcting block code for the given number of parity-

check digits and an appropriate cyclic (or shortened cyclic)

single 2.-burst error-correcting code. An overall parity check

digit may be added to the single 2--burst error correcting

cyclic code to increase its minimum Hamming distance to four.

For example, consider the construction of a nested

binary double-random- or-single Q.-burst error-correcting code

from two other block codes. Let the first code of the

construction be a (21,12) block code of Table 5.4.1 capable of

correcting two random errors or less or a burst of length

three or less. The parity check matrix EH1] of this code is

given by:-

111110 0 0 0 0 0 010 0 0 0 0 0 0 0 110 0 01110 0 0 0 010 0 0 0 0 0 0 00100100110 0 0 010 0 0 0 0 0 110100101010 0 0 010 0 0 0 0

EH13= 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 010001100111000001000 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0

_0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1_

The number of information digits of the second code

must be equal to the number of redundant digits of the second

code (i.e. k-k1=9). The second code must be capable of

correcting a burst of three errors or less and of minimum

Hamming distance equal to four. The (15,9) cyclic single 3-

burst-error-correcting code of the generator polynomial(7)

(g(x) = x6 + x5 + x4 + x3 + 1) can therefore be used as the

second code of the construction by adding an overall parity

check digit to increase its minimum Hamming distance to four.

140.

The columns of matrix Ec:1 may therefore be determined by forming the parity check digits of those codewords of the

even Hamming distance (16,9) code, whose information digits are

given by the columns of the [I1] matrix. Thus the matrix [C] is given by:

-I 0 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1- 1 0 0 010 0 0 1 11111 010 010 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1

[C]=011000000001111001100 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 0 1 1 0 0 0 01010 0 0 0 01001110011 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0 1

The parity check matrix Da of the derived (37,21)

nested B 0 R code is therefore given by:-

1.111

DO[ I 1 6] In a similar manner, t-random errors-or -single c--burst

error-correcting nested codes may be constructed by using the

best known t-random errors-or_-single t-burst error-correcting

block codes for the first code of the construction and the

derived (t-1)-random errors- or-single-burst error-correcting

nested codes for the second code of the construction.

The basic principles and techniques of the nested

decoding algorithm for the nested B C) R codes are identical to

the nested decoding algorithm for random error-correcting nested

codes described in Sub-section 4.3. However, the two nested

decoders differ only in that the decocer of the inner code of

the latter decoder (as previously explained) is the error-trapping

decoder for the single-error-correcting Hatmaing code, whilst in

the case of the B 0 R nested decoder, the decoder of the inner

code is obviously the error-trapping decoder of the single-

burst-error-correcting cyclic codes.

ARM 1111•1116

141.

All nested B C) R codes which exhibit one of the

properties of decodable nested codes given in Chapter 4.3,

are decodable by the nested B C)R decoder. Table 5.4.2

gives the parameters of some decodable B C) R nested codes.

It may be useful to emphasize at this stage that the

features and merits of the B C)R nested decoder are identical

to those of the random-error nested decoder.

142.

TABLE 5.4.1

Some burst-asi-random-error-correcting codes found by computer search.

CODE NUMBER OF LENGTH INFORMATION

(n) DIGITS (k) THE COLUMNS OF THE PARITY CHECK MATRIX [HD IN OCTAL

Codes capable of correcting two random errors or less,or a burst of length three or less.

8 2 55,33,40,20,10,4,2,1.

11 4 165,117,55,33,100,40,20,10,4,2,1.

18 8 252,343,321,235,165,117,55,33,200,100,40,20,10,4,2,1.

21 12 664,651,506,447,425,314,252,226,165,117,55,33,400,200,100,40,20,10,4,2,1.

29 19 1637,1504,1363,1324,1205,1043,1035,736,611,577,446,425,314,252,226,165,117,55,33,1000, 400,200,100,40,20,10,4,2,1.

37 26 3474,3353,2640,2217,2144,2135,2036,1637,1504,1363,3124,1205,1043,1035,736,611,577,446, 425,314,252,226,165,117,55,33,2000,1000,400,200,100,40,20,10,4,2,1.

51 39 7744,7414,6517,6343,6213,5634,5471,4462,4237,4170,4112,4053,3771,3552,3017,2461,2267, 2162,2103,2036,1637,1504,1363,1324,1205,1043,1035,736,611,577,446,425,314,252,226,165, 117,55,33,4000,2000,1000,400,200,100,40,20,10,4,2,1.

68 55 17305,17073,15042,14745,13715,13125,12743,12675,12432,11203,10624,10506,10277,10174, 10105,10056,7744,7414,6517,6343,6213,5634,5471,4462,4237,4170,4112,4053,3771,3552, 3017,2461,2267,2162,2103,2036,1637,1504,1363,1324,1205,1043,1035,736,611,577,446,425, 314,252,226,165,117,55,33,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

89 75 36147,35251,34773,34030,31531,30747,26362,25004,23214,23124,22722,22152,21656,21632, 21507,20753,20454,20260,20126,20077,17305,17073,15042,14745,13715,13125,12743,12675, 12432,11203,10624,10506,10277,10174,10105,10056,7744,7414,6517,6343,6213,5634,5471, 4462,4237,4170,4112,4053,3771,3552,3017,2461,2267,2162,2103,2036,1637,1504,1363,1324, 1205,1043,1035,736,611,577,446,425,314,252,226,165,117,55,33,20000,10000,4000,2000, 1000,400,200,100,40,20,10,4,2,1.

119 104 77701,76555,75720,73635,70564,67670,65310,63534,62337,60726,57361,54202,53441,53046, 52100,50455,47735,47241,45055,44443,44027,42714,42013,41531,41011,40604,40516,40126, 40077,36550,35463,35335,35037,31402,30330,26251,24641,24072,23554,23207,22447,22355, 21301,21116,21052,20627,20415,20306,20064,17305,17073,15042,14745,13715,13125,12743, 12675,12432,11203,10624,10506,10277;10174,10105,10056,7744,7414,6517,5343,6213,5634, 5471,4462,4237,4170,4112,4053,3771,3552,3017,2461,2267,2162,2103,2036,1637,1504,1363, 1324,1205,1043,1035,736,611,577,446,425,314,252,226,165,117,55,33,40000,20000,10000, 4000,2000,1000,400,200,100,40,20,10,4,2,1.

153 137

201 184

261 243

341 322

444 424

572 551

Codes capable of correcting two random errors or less, or a burst of length four or less.

12 4 255,177,125,63,200,100,40,20,10,4,2,1.

17 8 603,571,507,445,227,177025,63,400,200,100,40,20,4,2,1.

25 15 1735,1664,1421,1215,1153,1045,1037,603.507,455,433,227,177,125,63,1000,400,200,100, 40,20,10,4,2,1.

34 23 3251,3742,3065,2735,2554,2416,2213,2145,2037,1661,1515,1222,1113,1047,1035,603,507, 455,433,227,177,125,63,2000,1000,400,200,100,40,20,10,4,2,1.

47 35 7763,7430,7253,6710,6202,6075,5236,4715,4233,4174,4123,4045,3521,3303,3052,2416, 2270,2153,2122,2037,1772,1631,1566,1217,1106,1047,1035,603,507,455,433,227,177,125, 63,4000,2000,1000,400,200,100,40,20,10,4,2,1.

62 49 16656,16120,15172,14425,13701,13205,13162,12107,13607,11027,10677,10443,10310,10233, 10051,7641,7512,7060,6413,6243,5467,5151,4611,4462,4206,4113,4045,3367,2524,2346,2270, 2153,2122,2037,1772,1631,1566,1217,1106,1047,1035,603,507,455,433,227,177,125,63, 10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

143.


CODE NUMBER OF LENGTH INFORMATION (n) DIGITS (k)

THE COLUMNS OF THE PARITY CHECK MATRIX [NJ IN OCTAL

Codes capable of correcting two random errors or less, or a burst of length four or less. (Continued).

85 71 34472,25746,37612,35360,34617,32074,31630,3]342,30241,26454,26174,24325,23423,22477, 22324,22013,21734,21556,21064,20425,20205,20112,20057,17130,16067,15271,14762,14214, 13616,12443,12375,11432,11222,10712,10272,10162,10117,10053,7307,6754,6620,6043,5467, 5151,4611,4462,4206,4113,4045,3367,2524,2346,2270,2153,2122,2037,1772,1631,1566,1217, 1106,1047,1035,603,507,455,433,227,177,125,63,20000,10000,4000,2000,1000,400,200,100, 40,20,10,4,2,1.

111

96 65663,76211,75617,74470,72351, 70005,64302,63077,61651,61361,56301,55112,54637,52716, 51750,50424,47777,47013,44335, 44166,42700,42416,41375,41214,41041,40306,40205,40057, 36734,32436,31071,30721,27621, 27135,25744,25460,25276,24073,23033,22417,22330,22222, 21150,20412,20242,20234,20051, 17333,17242,16424,15777,14320,14023,13616,12443,12375, 11432,11222,10712,10272,10162, 10117,10053,7307,6754,6620,6043,5467,5151,4611,4406,4206, 4113,4045,3367,2524,2346,2270, 2153,2122,2037,1772,1631,1566,1217,1106,1047,1035,603, 507,455,433,227,177,125,40000, 20000,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

148

132

193

176

255

237

334

315

430

410

558

537

729

707

Codes capable of correcting two random errors or less, or a burst of length five or less.

15 5 1077,467,357,245,143,1000,400,200,100,40,20,10,4,2,1.

25 14 3533,1666,3232,3017,2576,2246,2211,2107,2065,1061,447,357,245,143,2000,1000,400,200, 100,40,20,10,4,2,1.

35 23 1127,7232,6621,6033,5650,5031,4413,4351,4207,4063,3724,3322,3017,2414,2246,2211,2107, 2065,1061,447,357,245,143,4000,2000,1000,400,200,100,40,20,10,4,2,1.

50 37 17511,16437,16234,16130,15565,14517,14362,12727,12564,11326,11074,11013,10424,10242, 10133,10065,7471,6174,6041,5714,5031,4415,4226,4105,4057,3763,3014,2403,2207,2162, 2115,2053,1061,447,357,245,143,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

69 55 37566,33156,31441,31334,30136,26755,26266,25766,24121,23705,22017,21250,21045,20465, 20232,20111,20063,17141,16406,15475,14661,14546,14423,13062,12733,12427,12124,11007, 10473,10221,10122,10057,7005,6660,6031,5320,5104,5033,4437,4222,4176,4107,4055,3763, 3014,2403,2202,2162,2115,2053,1061,447,357,245,143,20000,10000,4000,2000,1000,400, 200,40,20,10,4,2,1.

96 81 74751,75532,71274,66607,65425,63662,61556,60130,56155,55535,55410,51214,51177,50601,50064 46133,45441,445U,44012,43204,43066,40550,40327,40276,40065,37226,35641,34042,33510, 33261,32503,30250,24211,23521,23044,22610,22017,21703,21567,21255,20405,20232,20111, 20063,17762,16347,15627,14474,13367,12544,12220,11307,11154,11011,10432,10215,10122, 10057,7005,6660,6031,5320,5104,5033,4437,4222,4176,4107,4055,3763,3014,2403,2207, 2162,2115,2053,1061,447,357,245,143,40000,20000,10000,4000,2000,1000,400,200,100,40, 20,10,4,2,1.

132

116

175

158

237

219

314

295

414

394

538

517

144 .

TABLE 5.4.1 (continued

CODE NUMBER OF LENGTH INFORMATION

THE COLUMNS OF THE PARITY CHECK MATRIX [HJ IN OCTAL.

(n) DIGITS (k)

Codes capable of correcting two random errors or less, or a burst of length six or less.

18 6 4141,2137,1127,717,505,303,4000,2000,1000,400,200,100,40,20,10,4,2,1.

31 18 17346,16017,15165,14722,14003,12245,12027,11011,10423,10352,10207,10125,4141,2121, 1107,717,505,303,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

49 35 26757,35304,33732,31522,30573,27140,26470,25032,24376,24227,24031,22016,21053,20467, 20306,20251,20123,16006,15033,14647,14270,14166,14003,12013,11011,10423,10352,10207, 10125,4141,2121,1107,717,505,303,20000,10000,4000,02000,1000,400,200,100,40,20,10,4,2,1.

72 57 35756,77165,75211,70120,67525,65042,64700,62147,61217,60044,56675,55364,54351,52510, 50526,45414,45370,44272,44035,42552,41665,41161,41030,40251,40123,34743,34255,30065, 26416,25066,24422,24333,24150,24015,22053,21031,20421,20342,20117,16337,16005,15061, 14460,14011,12230,12003,11006,10423,10352,10207,10125,4141,2121,1107,717,505,303, 40000,20000,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

104 88 70512,174412,155512,124476,176470,170732,167317,164533,153043,151731,144247,142436, 142205,141350,140132,130547,126267,125515,123006,122275,121033,117256,111154,110603, 110425,110321,104017,102047,101034,100567,100253,100125,71667,65252,62620,61062,60714, 60375,57736,55131,50541,50073,47562,45706,44457,44316,44031,42126,41153,41046,40435, 40241,40123,37142,34721,32716,32671,30143,27244,26112,24462,24255,24030,23445,22063, 21067,20366,20225,20117,16506,16021,15026,14670,14212,14134,14005,12042,11003,10407, 10322,10215,10113,4141,2121,1107,717,505,303,100000,40000,20000,10000,4000,2000,1000, 400,200,100,40,20,10,4,2,1.

147 130

203 185

275 256

377 357

Codes capable of correcting two random errors or less, or a burst of length seven or less.

21 7 2031,10241,4237,2227,1617,1205,603,20000,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

43 28 47266,33653,24647,22066,21135,70636,70324,70061,64042,62121,61117,60514,60276,60013, 50102,44027,42011,41023,40652,40407,40225,20301,10241,4221,2207,1617,1205,603,40000, 20000,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

67 51 146075,132575,37036,36272,172240,170154,165516,155714,150222,143341,140654,140045, 135144,134512,130165,122035,120521,120030,115056,114132,110152;106234,105273,104702 104431,104017,100227,72003,70077,65663,65522,64005,60021,52120,51564,50770,50515,50006, 44013,42011,41023,40652,40407,40225,20301,10241,4221,2207,1617,1205,603,100000,400001 20000,10000,4000,2000,1000,400,200,100,40,20,10,4,2,1.

101 84

153 135

220 201

307 287

422 401

Codes capable of correcting three random errors or less, or a burst of length four or less.

9 1 267,200,100,40,20,10,4,2,1.

11 2 533,400,**.

14 4 1706,1455,1000,**.

16 5 2476,2000,**.

22 10 7262,6730,6071,5143,4315,4000,**.

25 12 11073,10336,10000,**.

28 14 32027,21534,20000,**.

33 18 74506,62702,53035,40652,40000,**.

39 23 152601,135555,123230,115352,102255,100000,**.

145.


CODE NUMBER OF LENGTH INFORMATION (n) DIGITS (k)

THE COLUMNS OF THE PARITY CHECK MATRIX Da IN OCTAL.

Codes capable of correcting three random errors or less, or a burst of length four or less. (continued)

45 28 340521,323426,235563,221157,202356,200000,**.

53 35 716576,651257,544232,450417,431624,415321,403547,400000,**.

62 43 151040201361322,1350046,1222051,1127734,1041665,1015607,1003715,1000000,**.

Ilk 73 53 3657453,3231643,3074272,2747671,2435207,2263136,2240516,2140262,2030352,2006341, 2000000,**.

86 65 7776040,7154337,6126444,5422412,5307523,4664522,4530504,4470606,4240323,4057207, 4022122,4010711,4000000,**.

Codes capable of correcting four random errors or less or a burst of length five or less.

11 1 1357,1000,400,200,100,40,20,10,4,2,1.

14 2 6467,4000,2000,**.

17 3 30533,20000,10000,**.

20 5 65542,52235,40000,**.

22 6 112750,1000000,**.

26 9 363666,304572,213524,200000,**.

• 29 11 524747,415436,400000,**.

di 33 14 1623671,1167061,1017641,1000000,**.

37 17 3444515,2251767,2024774,2000000,**.

42 21 7377137,4662710,4510463,4046345,4000000,**.

Codes capable of correcting five random errors or less, or a burst of length six or less.

13 1 5737,4000,2000,1000,400,200,100,40,20,10,4,2,1.

17 2 72157,40000,20000,10000,**.

20 3 322663,200000,100000,**.

23 5 654465,513274,400000,**.

26 7 1274703,1131565,1000000,**.

28 8 2427161,2000000,**.

32 11 7614452,6036427,4470632,4000000,**.

Codes obtained using the described computer search procedure but not included here due to

their being inordinately long.

* *

Plus the columns of the parity check matrix of the above code.

146 .

TABLE 5.4.2

Some Examples of Decodable Binary Nested B OR Codes.

(n,k) Decodable B C)R nested code.

R Rate of the above code.

(k,k1) The first code of construction.

(n-k,k-k1)The second code of construction.

(n,k)

(k, k1) Notes ( -k,k-k1 Notes

Codes capable of correcting two random errors or less or a burst of

errors of length three or less.

Shortened from the (16,9) single 3-burst-error-correcting cyclic code (7).

ditto

21, 8 0.380 8 2(From Table 5.4.1) 13, 6

25, 11 0.440 11, 4 ditto

14, 7

54, 11 0.574 31, 16

61, 37 0.606 37, 21

94, 61 0.648 61, 37(From Table 5.4.1)

136, 94 0.691 94, 61 ditto

187,136 0.727 136, 94 ditto

247,187 0.757 187,136 ditto

23,15 Shortened from the (28,20) single 3-burs& error-correcting cyclic code (7).

24,16

ditto

33,24 Shortened from the (64,55) single 3-burst error-correcting cyclic code (7).

42,33 ditto

51,42 ditto

60,51 ditto

Some supporting nested B C)R codes in which the

decoder is capable of correcting the errors in the parity-

check digits only:

31, 16 16, 8 (From Table 15, 8 Shortened from (16,9) 5.4.1)

37,21 21,12 (From Table 16,' 9 Shortened from (16,'9) 5.4.1)

147.

Shortened from the (20, 11) single 4-burst-error-correcting cyclic code (7).

18, 9 ditto Shortened from the (39, 29) single 4-burst-error-correcting cyclic code (7).

ditto

Shortened from the (86, 75) single 4-burst-erF9N correcting cyclic code. '

ditto

ditto

ditto

ditto Shortened from the (165, 153) single 4-burst error correcting cyclic code (7).

17, 8

29, 19

30,20

41, 30

52, 41

63,r52

74, 63

85, 74

97, 85

R (k,ki) Notes n-k,k-k1) Notes

317,247 0.779 247,187 (from Table 5.4.1) 70, 60 Shortened from the

(122,112) single 3-; burst-error-correcting cyclic code (7).

397,317 0.798 317,247 ditto 80, 70 ditto

487,398 0.815 397,317 ditto 90, 80 ditto

587,487 0.829 487,397 ditto 100, 90 ditto

697,587 0.842 587,487 ditto 110,100 ditto

817,697 0.853 697,587 ditto

•

120,110 ditto

.

• • • •

Codes capable of correcting two random errors or less or a

burst of errors of length four or less.

29, 12 0.413 12, 4 (From Table 5.4.1)

35, 17 0.485 17, 8 ditto

73, 44 0.602 44, 25

80, 50 0.625 50, 30 Shortened*

121, 80 0.661 80, 50 (From Table 5.4.1)

173,121 0.699 121, 80 ditto

236,173 0.733 173,121 ditto

310,236 0.761 236,173 ditto

395,310 0.784 310,236 ditto

492,395 0.794 395,310 ditto

Some supporting nested B C)R codes in which the decoder is capable of correcting the errors in the parity check digits only.

44,25 25,15 Table 5.4.1 19,10 Shortened from the (20,11) single 4-burst-error-

148. correcting cyclic code

54,34 34,23 ditto 20,11 ditto

(n,k) R (k,k1) Notes

601, 492 0.818 492, 395 From Table 5.4.1

712, 601 0.844

835, 712 0.852

970, 835 0.860

1117, 970 0.868

1276,1117 0.875

•

n-k, kl) Notes

109, 97 Shortened from the (165,153) single 4-burst error corre-cting cyclic code (7).

601, 492 ditto 111,109 ditto

712, 601 ditto 123,111 ditto

835, 712 ditto 135,123 ditto

971, 835 ditto 147,135 ditto

1117, 970 ditto 159,147 ditto

•

• •

Codes capable of correcting 2-random errors or less or a burst

of errors of length 5 or less.

36, 15

81, 47

93, 58

200,140

273,200

359,273

458,359

570,458

695,570

833,695

0.416

0.580

0.623

0.700

0.732

0.760

0.783

0.803

0.820

0.834

15, 5 From Table 5.4.1

47, 25

58, 35

140, 93 From Table 5.4.1

200,140 ditto

273,200 ditto

359,273 ditto

458,359 ditto

570,458 ditto

695,570 ditto

21, 10

34, 22

35, 23

60, 47

73, 60

86, 73

99, 86

112, 99

125,112

138,125

Shortened from the (28,17) 5-burst error-correcting cyclic code (7).

Shortened from the (49,37) single 5-burst error-correct-ing code (7).

ditto Shortened from the (132,131) single 5-burst error correct-ing code (7).

ditto

ditto

ditto

ditto

ditto

ditto

Some supporting nested B C)R codes in which the decoder is capable of correcting the errors in the parity check digits only.

47,

58,

25

35

25,

35,

14

23

From Table 5.4.1

ditto

' 11

23,12

22, 11 from the

(28,17) single 5-burst error correct-ing cyclic code (7).

ditto 149.

(n,k) "R (k k ) ---2--±- Notes (n-k,k-k1) Notes

Codes capable of correcting 2-random errors or less or a burst of

errors of length six or less.

43, 18

97, 51

109 68 ,

0.418

0.525

0.623

18, 6 From Table 25, 12 5.4.1

57, 31 40, 26

68,41 Shortened* 41 27 from (76,49) '

Shortened from the (35, 22) single 6-burst-erTo\-- correcting cyclic cod ?)

Shortened from the (68, 54) single 6-burst-ero;..- correcting cyclic codI

ditto

192,109 0.567 109, 68 From Table 5.4.1 83, 68 Shortened from the (170,

155) single 6-burst-erra correcting cyclic code.

290,192 0.662 192,109 ditto 98, 83 ditto

403,290 0.719 290,192 ditto 113, 98 ditto

531,403 0.758 403,290 ditto 128,113 ditto

674,531 0.787 531,403 ditto 143,128 ditto

832,674 0.810 674,531 ditto 158,143 ditto

Codes capable of correcting 2-random errors or less or a burst

of errors of length seven or less.

50, 21

119, 73

181,119

259,181

353,259

0.420

0.613

0.657

0.698

0.733

21, 7 From Table

5.4.1

73, 43

119, 73 From Table 5.4.1

181,119 ditto

259,181 ditto

29,

46,

62,

78,

94,

14

30

46

62

78

Shortened from the (39,24) single 7-burst-error- (7) correcting cyclic code":.

Shortened from the (104, 88) single 7-burst-error-correcting cyclic code.

ditto

ditto

ditto

* Some supporting nested B C) R codes in which the decoder is

capable of correcting the errors in the parity check digits only.

57, 31 31 18 From Table 26 11 Shortened from the (35,22) , ,

5.4.1 single 6-burst error- (7) . correcting cyclic code

49, 35 Ditto 27,14 ditto

43, 28 Ditto 30,15 Shortened from the (39,24)

single 7-burst error- (7) correcting cyclic code 1.

76, 49

73, 43

150.

1

(n,k) R (k,k1) Notes (n-k,k-k1) Notes

Codes capa ble of correcting 3-random errors or less or a burst

of errors of length four or less.

35, 9

38, 11

42, 14

45, 16

52, 22

57, 25

61, 28

67, 33

139, 74

147, 81

180, 101

191, 111

204, 123

230, 137

376, 230

586, 376

871, 586

1242, 871

1711,1242

0.257

0.289

0.333

0.355

0.423

0.438

0.459

0.492

0.532

0.551

0.561

0.581

0.602

0.595

0.611

0.641

0.672

0.7012

0.725

• •

9 ,

11,

14,

16,

22,

25,

28,

33,

74,

81,

101,

111,

123,

137,

230,

376,

586,

871,

1242,

•

1 From Table 5.4.1

2 ditto

4 ditto

5 ditto

10 ditto

12 ditto

14 ditto

18 ditto

39

45

53

62

73

86 137 From Table

5.4.1

230 ditto

376 ditto

586 ditto

871 ditto

26, 8

27, 9

28, 10

29, 11

30, 12

32, 13

33, 14

34, 15

65, 35

66, 36

79, 48

80, 49

81, 50

93, 51

146, 93

210,146

285,210

371,285

469,371

•

Shortened from (30,12) of this table.

ditto

ditto

ditto

ditto Shortened from (36,17) of this table.

ditto


ditto

Shortened from (81,50)

ditto

ditto

Shortened from (122,80)

Shortened from(174,120

Shortened from( 237,171

Shortened from(311,23)

Shortened fr om( 396,31.0)

Shortened from(493,395)

Some supporting nested B 0 R codes in which the decoder is

capable of correcting the errors in the parity check digits only.

74,39

81,45

101,53

111,62 123,73 137,86

From Table 39,23 5.4.1

45,28 ditto

53,35 ditto

62,43 ditto

73,53 ditto

86,65 ditto

35,16

36,17

48,18

49,19 50,20 51,21

Shortened from (36,17) of this table.


ditto ditto ditto

151.

5.5 MULTIPLE-BURST-ERROR-CORRECTING CODES

The corollaries of Chapter 5.4 may be modified for

multiple-burst-error-correcting codes as follows:

Corollary 5.2.1.3

An (n,k) block code V, that is the null space of a

matrix [H], is capable of correcting b bursts of errors or less

of length t or less, if and only if the linear combinations of

every combination of b or less sets of 2, adjacent columns of the

Da matrix result in unique nonzero vectors of the vector space

Vn-k.

A computer programme is developed, in accordance with

the above corollary, for finding multiple-burst-error-correcting

codes for a given number of parity check digits and given values

of b and Q. The programme follows the same steps as the

programme given in Chapter 5.3 for finding single burst error

correcting codes. However, it differs in the type of linear

combinations of steps (4) and (5); in this case the computerised

search performed the linear combinations of every combination

of b or less sets of 2 adjacent columns of the [H] matrix and

the resulting vector is tested for uniqueness. Table 5.5.1

supplies a list of some of these codes. The code lengths and

number of information digits are given, together with the

Octal equivalent of the columns of the determined parity check

matrix.

The conditions of constructing multiple-burst-error-

correcting codes (exhibiting some of the characteristics of

nested codes) are given by the following corollary which

follows from Theorem 5.2.2:-

• 152.

Corollary 5.2.2.3

An (n,k) block code, constructed in accordance with

the construction rules of nested codes using two block codes

for construction, is capable of correcting b bursts of error

or less of length 2 or less, if both codes of construction

are capable of correcting b bursts of errors or less of length

2, or less.

For example, consider the construction of a block

code capable of correcting double bursts of length two or less

from two other block codes of Table 5.5.1 ; the first code of

construction being the (41,27) code and the second code of

construction the (27,14) shortened code from the (30,17) code

of Table 5.5.1.

The parity check matrix EHij of the (41,27) is given by:

-110110 0 111110 0 0 0 0 0 0 0 0 0 0 0 00 010 0 0 0 0 0 0 0 0 0 0 0 0 1011010110 0 011110 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0'0 0 0 00 1010 0 0110 10 0110 011110 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 10110 010 0 01010 10110 0110 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 O1010100001000001010001100000001000000000 10100111000001000000101010000000100000000

Eig. 0 0 0110 0110 101010 010 010 0 0 010 0 0 0 0 0 0 10 0 0 0 0 0 0 11110 011110 01 0 0 0 0 0 101010 0 01 0 0 0 0 0 0 010 0 0 0 0 0 O0100000110001110101000101000000000100000 O11101010101010 0 0110 010 01010 0 0 0 0 0 0 0 010 0 0 0 11101101010110 01110 0 1101010 0 0 0 0 0 0 0 0 0 010 0 0 O 0101110 0111110 01011111 010 10 0 0 0 0 0 0 0 0 0 010 0 110 010 0 110 011111111010 10 110 0 0 0 00 0 0 0 0 0 0 010

_0 11 0 0 01011 010 0 01110111111010 0 0 0 0 0 0 0 0 0 0 0 01

The parity check matrix EH23 of the (27,14) code is

given by:

1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0

[112j= 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 i 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 01111101010110 0 0 0 0 0 0 0 0 0 0 010 0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1

153.

Using the above [-12:1 matrix, the parity-check equations

of the (27,14) code may be determined. Thus the 41 columns

of the matrix EC] are formed by the parity check digits of

those 41 codewords of the (27,14) code whose information

digits given by the columns of the [HI] matrix.

If 2: is the generator matrix of the (27,14) code,

the [C] matrix is therefore given by:

Ec3 = [Hi] EG2] The parity check matrix [H] of the derived (68,41)

code is therefore given by:

[1.11

= n-j

It has been established in Chapter 5.3 that if both

codes used for constructing (41,27) and (27,14) have a simply

implemented decoding logic, the derived (68,41) code can be

decoded simply by following the steps of the decoder suggested

for single-burst-error-correcting codes (see Chapter 5.3).

154.

TABLE 5.5.1

Some multiple-burst-error-correcting-codes found by

computer search capable of correcting double bursts

of length two or less

CODE NUMBER OF THE FIRST K COLUMNS OF THE PARITY CHECK LENGTH INFORMATION MATRIX [H] IN OCTAL (n) DIGITS (k)

10 2 252,125.

15 5 1507,1051,427,252,125.

19 8 1244,3155,2037,1507,1051,427,252,125.

23 11 7021,6265,5130,4053,2717,2035,1507,1051,427, 252,125.

30 17 12513,11147,4477,14705,12242,10053,7017,6273, 5126,4045,2717,2035,1507,1051,427,252,125.

41 27 36512,21133,16575,33320,20216,11434,6505, 34732,30343,24175,23204,20037,16316,14466, 12242,10053,7017,6273,5126,4045,2717,2035, 1507,1051,427,252,125.

50 35 66246,43451,24404,62557,53143,41157,33547, 21132,10734,70464,60117,57423,50005,45325, 40047,34732,30343,24175,23204,20037,16316, 14466,12242,10053,7017,6273,5126,4045,2717, 2035,1507,1051,427,252,125.

66 50 121714,57324,143625,112721,42165,145534, 102066,75130,46223,142232„125402,100346, 61603,40731,24404,16022,175540,160363,151652, 140151,131536,120043,111234,100172,70420, 60117,57423,50005,45325,40047,34732,30343, 24175,23204,20037,16316,14466,12242,10053, 7017,6273,5126,4045,2717,2035,1507,1051,427, 252,125.

86 69 206343,72374,342016,216424,143421,313247, 246146,207204,125035,303027,225250,201747, 106533,50515,346454,301432,271535,242372, 200631,165712,124701, 101171,65453,42605,23442, 361242,340141,321715,300153,260550,240070, 237763,22000,216050,200172,173502,1602471, 152614,140217,131512,120067,111210,100055, 70420,60117,57423,50005,45325,40047,34732, 30343,24175,23204,20037,16316,14466,12242, 10053,7017,6273,5126,4045,2717,2035,1507,1051, 427,252,125.

155.

(n) (k) THE FIRST K COLUMNS OF THE PARITY CHECK MATRIX EH] IN OCTAL

110 92 764103,615115,427116,215004,646732,412015, 247634,613171,526221,410717,205735,106311, 724642,602515,511221,403775,315540,271204, 204050,126130,73334,756026,704273,600474, 552071,501202,444444,401150,353612,305112, 242641,201013,144474,102401,42164,741036, 700335,640266,635753,600133,573051,540407, 525147,500037,463524,440315,422345,400056, 360702,341171,320523,300245,260355,240123, 236504,220426,217603,200071,173502,160271, 152614,140217,131512,120067,111210,100055; 70420,60117,57423,50005,45325,40047,34732,30343, 24175,23204,20037,16316,14466,12242,10053, 7017,6273,5126,4045,2717,2035,1507,1051, 427,252,125.

140 121 1116163,555373,1622251,1265300,1034564, 550104,1426121,1010434,530357,1455417,1325333, 1017251,503140,235012,1750660,1401217,1232241, 1005566,647331,411761,240554,1621660,1403747, 1376227,1216705,1003043,604766,401542,264231, 155065,1733253,1602472,1511165,1401074,1317223, 1201273,1115723,1000616,761375,710062,603666, 504524,400443,312143,275252,201016,114654, 53103,1767006,1700432,1654740,1600233,1546256, 1500353,1444523,1400136,1344544,1300210,1243755, 1200066 , 1142712,1100113,1042044,1000172,741204, 700106,675107,640767,600174,573046,540347, 525142,500032,463653,440062,422345,400056, 360702,341171,320523,300245,260355,240123, 236504,220426,217603,200071,173502,160271, 152614,140217,131512,120067,11210,100055,70420, 60117,57423,50005,45325,40047,34732,30343, 24175,23204,20037,16316,14466,12242,10053, 7017,6273,5126,4045,2717,2035,1507,1051,427, 252,125.

156.

6. BOUNDS ON THE HAMMING DISTANCE OF LINEAR CODES AND ANTICODES

6.1 Introduction.

6.2 An Improvement on the Varshamov-Gilbert Lower

Bound on the minimum Hamming Distance of

Linear Block Codes.

6.3 The Weight Distribution of Linear Binary Codes.

6.4 Maximum Hamming Distance Bounds for Linear

Anticodes.

•

157.

6.1 INTRODUCTION

An upper bound on the minimum Hamming distance d (of

an (n,k) linear block code V over the Galois field of q elements

GF(q)) is defined as the absolute theoretical maximum value d

can take for any arbitrary code length (n) and number of message

digits (k). The upper bound is said to be "good" if it is close

to the best lower bound. A lower bound on d is defined, for

arbitrary values of n and k, as the largest value of d

associated with any code which can be shown to exist, having

the value of n and k. The lower bound is said to be "good" if it

is close to the best upper bound.

A number of upper and lower bounds have been reported

in the literature. Most of these are based either on the well-

known sphere-packing argument introduced by Hamming(30,27), the

average distance" approach of Plotkin(63' 64) , or a combination

of these(65)

. The values of d for a number of well-known codes,

such as the Hamming(30)

, BCH(3,7), Goppa(57,58,82) and 84)

Srivastara (3, codes, also provide'a lower bound on d(67,68)

The Varshamov-Gilbert bound was proposed by Varshamov(60)

and is in fact a refinement of a bound proposed by Gilbert(27)

The same bound was also found by Sacks(66)

from a consideration

of the characteristics of the parity-check matrix EH] of the code. Sacks shows that since the minimum Hamming distance of an (n,k)

linear code is d, if and only if every (d-1) columns of matrix

Da are linearly independent(3), the following systematic

procedure for constructing an (n,k) code with r-parity check

symbols and minimum distance (d) follows immediately:-

1) The first column of the EH] matrix is chosen arbitrarily

from the vector space Vr, subject only to the condition that it

is not a null vector.

158.

2) The second column is chosen so that it is different

from all multiples of the first column and the null vector.

3) The third column is chosen to be different from the

null vectors, and from the linear combination of the first two

columns.

4) The remaining columns are then chosen such that they

are different from the null vector and from all possible linear

combinations of the previously chosen columns.

This method of generation guarantees the required

independence of the columns. The method cannot be successfully

completed unless r is sufficiently large. To find the minimum

value of r, the worst possible outcome of making n arbitrary

choices for the columns of [lj] is considered. The outcome is such

that all the entries in the nth stage are distinct and then to

ensure success in the face of this worst outcome, qr need only be

greater than the total number of entries in the nth stage (q

being equal to 2 for the binary case). The smallest integer (r),

that satisfies the following equation, is thus the Varshamov-

Gilbert lower bound on d.

d-2

(q-1)1 = qr i=d

(6.1)

The object of this section is to introduce a possible

improvement on the above bound by showing that the number of

distinct vectors, resulting from the linear combination of all

d-2 columns of the Da matrix, are much less than the summation

in equation (6.1).

An expression for the largest possible number of these

distinct vectors (for any (n,k) group codes) is proposed and a

tighter bound is therefore obtained.

159.

6.2 AN IMPROVEMENT ON THE VARSHAMOV-GILBERT LOWER BOUND ON THE MINIMUM HAMMING DISTANCE OF LINEAR BLOCK CODES

Let an (n,k) group code V over GF(q) have a parity

check matrix Dd.]; then a vector v of the vector space Vn is a

codeword (i.e. v EV) if and only if it is orthogonal to every

row of Da. That is(7)

[HTJ = 0 (6.2)

This implies that any vectory (ve_V) of weight w, specifies a

linearly dependent set of w columns of the Efa matrix, or

conversely a linear combination of w columns of DI resulting

in the zero vector, establishes the existence of a codeword v

of weight w(7).

It is convenient here to employ the term "quasi-linear

independence" introduced in Section 3.4.1. The r-tuple vectors

rl, r2, ri over GF(q) are quasi-linearly independent if the

vectors, formed by modulo-q addition of the scalar products

(a1r1 + a2r2 + + air.) are all nonzero vectors, where a.

may be any one of the nonzero elements of GF(q). Since there are

(q-1) nonzero elements in GF(q), the quasi-linear combinations

(QLC) of the above vectors may have (q-1)1 combinational sums,

each sum resulting in a nonzero r-tuple vector over GF(q). It

follows that if the vectors are not quasi-linearly independent

over GF(q) they must be linearly dependent over the field.

The numerical value of the Varshamov-Gilbert bound,

given by equation (6.1), may be rewritten as follows:

1 + (ni)(q.4)+ (2)(q-1)2+ ...+(dn3)(q...i)d-3

(d1.12)(q-1)d-2 qn-k (6.3)

We may say therefore that since elLrepresents the total number

of the quasi-linear combinations of one, two, ..., and (d-2)

columns of the EH: matrix, the bound assumes that all the vectors resulting from these combinations are distinct.

160.

Now let us consider a codeword vector v of weight d,

which has nonzero elements at the positions p1, p2, Pd* It follows from equation (6.2) that the modulo-q addition of the

scalar products of these nonzero elements with the corresponding

pi, p2, ..., and pd columns of the Ea matrix, results in a zero vector. Let the set of these d scalar products, say [C), be

divided into two sub-sets tA} and -Pl. Moreover, let the

modulo-q addition of the members of sub-set [Al result in an

r-tuple vector a and similarly the members of a sub-set

result in a vectot b. Then

a + b = 0 (6.4)

This equality indicates that a is the inverse vector of b and

vice versa. If the sub-set has two members, then the vector

a must be a vector in the set of vectors resulting from the

quasi-linear combination of every two columns of the DO matrix (denoted by OT IV GQLC(2)(q-1)2 ). And, since the inverse of a vector a is equal to the scalar product ((q-1).a) it follows

that the vector b is one of the vectors resulting from the

quasi-linear combinations of every two columns of DJ (i.e.

b I v eQLC(ni)(q-1)2 } ). However, the vector b is in the set

[v y +v GQLC( q-2 )(q-1)ad-1- 21 5, which suggests that vector b is not distinct. Similarly all the multiples of vector b, g.b, (whetrie

g is any nonzero element of GF(q)) are in the set iviveQu(d_ 2 ). (q-1)d-21 5 and since g.a + g.b = 0, according to the above

argument, the (q-1) multiples of vector b are not distinct.

Since set [C has d members, it can be divided, in

two sub-sets 1A1 and [BI of two members and (d-2) members

respectively, in many ways. The number of these combinations is

equal to (E21). Consequently, the corresponding number of non-

distinct vectors in the set tvivGQLC(dn...2)(q-1)d-21, for every

codeword of weight d, is given by:

161.

((2)(q-1).

Moreover, the sub-set tAl may also have 3, 4, ... or j members

and consequently the sub-set f13.1 may have (d-3), (d-4) .... or

d-j members. If j is less than half the number of members of

the set (C1, then all possible combinations of 3, 4,..., j

members out of d members of set IC) (used to form the sub-set {Aj)

are different from all the corresponding combinations forming

the sub-set 1131. Each of these ways of dividing set {C1

corresponds to a non-distinct vector in one of the sets

[Nr v- QLC(dn2)(q-1)d-2 tv I v GQLC(d n3 )(q-1)d-31

non-distinct vectors, for every codeword of weight d, is given (d+7)/2)(q-1)

(d+1)/21 [v I v G QLC( The total number of these

by:-- rd-

21

(6.5)

In general every codeword of weight w, where d<w<d-2-Ft, t =

r-21-1 (q-1) (1) (6.6)

i=w-d+2

n d-22 non distinct vectors in the sets [viveQLC(d2)(q-1) 3,

{v I v GQLC(dn3)(q-1)(1-31,..., fv i v GQI,c((d÷i)/2)(q-1)(d+1)/21

Since the linear combinations of every t columns of the EH]

matrix give unique nonzero vectors in the vector space Vr(89)

no two codewords of weight w (where d<w<t+d-2) suggest the

same non distinct vectors.

means the largest integer, smaller or equal to 2 d-1

(d-1)/2, i.e. 1-- 2 — t.the number of random errors the

code can correct.

, suggests the existence of: Fdi 2

162.

The total number of these vectors may be computed as

follows:-

Let the minimum number of codewords in an (n,k,d)

linear code of weight w be fw; then the whole computation of

these non-distinct vectors may be tabulated in the following

table:-

- The minimum CODEWORD number of WEIGHT codewords of

weight w.

Number of non-distinct vectors in the linear combinations of (d-2) columns of the DJ, matrix, corresponding to sub-sets [Al and tBi of "i" members and (w-i) members respectively.

i = 2 3 4

fd(q -1)

d+1

fd+1

fd-Ei (q-1) [ d+2

fd+2 fd+2(c1-1) [

• •

d-2 d-1 2 fd-24e1 &1-5(c1-1) [I 2

The total summation is then given by:

d-2fd-1 2 2

(q-1)y--; (7) w=d i=w-d+2

(6.7)

The subtraction of the total number of non-distinct

vectors of equation (6.7), from the summation of equation (6.1),

w fw

d

fd

163.

results in an improved Varshamov-Gilbert lower bound on the

minimum Hamming distance d. The improved bound is thus given

by:

d-2 d-1 Ed-11 2

2

T()(1) -(q-1) n i (7) = qr (6.8)

i=0 w=d i=w-d+2

The numerical evaluation for this improved bound

requires the determination of the lowest possible values of

fw (where w = d, d+1, 3d-5) in terms of the code parameters

n, k and d. In the following section an attempt to evaluate

these values for binary codes is outlined.

d-2

164.

6.3 THE WEIGHT DISTRIBUTION OF LINEAR BINARY CODES

The 2k codewords of an (n,k) binary block code can

be tabulated as the rows of an nk xn matrix; this matrix being

referred to as the code array. The first k columns of the

binary array are the k information columns, and each of the

remaining (n-k) check (redundant) columns is some linear

(modulo-2) combination of the k information columns.

Consider the code array of an (n,k) binary block code

arranged as follows:-

The codeworcb of the upper half of the array are

tabulated in a sequence that corresponds to the decimal value

(from 2k-1 up to 2k - 1) of the k binary information digits of

each codeword. The codewords of the lower half of the array

are tabulated in a sequence that corresponds to the decimal

value (in the range 0 to (2k-1 - 1) of the k information digits

(as shown on the next page).

If all zero elements of the first k columns of this

array are replaced by +1's and the 1 elements by -1's, then the

resulting columns form the k functions of Rademacher(85). Walsh

functions are usually defined by products of Rademacher functions (86)

. This definition, however, does not yield the Walsh

functions ordered by the number of sign changes as does the

difference equation*. Rademacher functions correspond to the

The Walsh functions Wal(j,9) may be defined by the

following difference equation(85).

.

Wal(2j+p,9)= (-1)1-3/71+P(walE),2(94fl+(-1)-14-1Walp,2(9 44

p=0 or 1; j= 0, 1, 2, ...; Wal(0,9)=1 for 4.-5Z9<-2;

`Wal(0,9)= 0 for 9

r)/21 means the largest integer smaller than or equal to Tie

165.

Redundant Digits

0 0

1 • • • . . • 1

1 1

0 0

0 1

1 0

• •

• •

• •

• '2k

Decimal Value of Information Digits Information Digits

2kT1 + 0 1 0 0 0 •

2k-1 + 1 1 0 0 1

2k-1 + 2 1 0 1 0

2k-1 + 3 1 0 1 1

2k-1 + 4 1 1 0 0

2k-1 + 5 1 1 0 1

• . • •

• . • •

• • • •

• • . •

k 2 - 1 1 1 1 1

0 0 0 0 0

1 0 0 0 1

2 0 0 1 0

3 0 0 1 1

4 0 1 0 0

5 0 1 0 1

1 1

0 0

1 1

1 1

0 0

1 1

0 0

2 k 1

• • • • 4

• • • • •

• • • • •

• • • • •

0 1 1 1 0

>4

n

166.

O ave.

functions Wal(1,9), Wal(3,9), Wal(7,9), It should be

noted that the result of multiplying together two Walsh functions,

when transformed by replacing all the positive elements by 0's

and the negative elements by l's, is identical to that produced

by similarly transforming the original functions and performing

a simple modulo-2 addition(97). Since all the redundant digits

of any (n,k) linear code are a consequence of the modulo-2

addition of some of the k information digits, it follows that

all the n columns of the above array are Walsh functions.

Every Walsh function has equal numbers of l's and 0's and hence

the columns of any code array will have equal l's and 0's;

therefore the total number of l's in the array is equal to

(n.2k-1). Since there are 2

k - 1 nonzero codewords in V, the

average weight of codewords in V is given by:*

n.2k-1

Wave.

2-1 (6.9)

Now consider an (n,k) linear code (V) of minimum

Hamming distance (d) equal to average weight (Wave.). Since the

minimum Hamming distance a of any (n,k) linear code is equal to

the minimum weight(7), (i.e. in the case of the code under

considerationMd=Wrian.= W ), it follows that as d tends

to Wave.

the maximum weight of M' codewords of V will be equal

to the minimum weight. Therefore the code V is an equidistance

code with all its 2k 1 nonzero codewords of weight

Wave., where

Wave. =n.2k-1

/(2k - 1). For large values of k, Wave.

tends to

n/2. The weight distribution of V is shown diagrammatically below. k 2 -1

fw

The same results may be obtsline identities, see also Peterson(?

167.

by using Pless(100)

page 70.

2ve

1

On the other hand, consider an (n,k) linear block

code (V) of minimum Hamming distance equal to unity. Such a

code has no redundant digits and therefore its weight distrib-

ution is binomial, as shown below:

0 W. VV

Wave,

For other linear block codes of minimum Hamming

distance (d), where 1<d<-1:1- 2' the weight distribution would lie

between the distributions of equidistance and unit distance

codes. However, we could assume that the distribution of an

(n,k) non-repetitive linear block code could have a maximum

value at a point corresponding to the average weight of the

code. Intuitively, this assumption seems reasonable; in fact

the weight distribution of the known binary linear block codes

meets the above assumption(3,7,94,98-100) and furthermore,

Peterson(101) established that the class of BCH codes also

fulfils the above assumption. The correctness of the above

statement, for certain values of k and n, however, can be seen

from the following argument. Consider the sequency* of the 2k

Walsh functions; Wal(1,9) up to Wal(2,9) have a sequency equal

to 2k-1, whilst Wal(3,9) up to Wal(6,0) have a sequency equal

to in in general Wal(2j - 1,9) up to Wal(234-1-2,9) have a

sequency equal to 2k 3. This implies that of the group of the

2k Walsh functiong having a sequency2, one quarter of the

The sequency of Wal(i,9) is defined as the largest number

of adjacent l's or 0's occuring in the given Wal(i,9);

whilst the integer i is called the normalized sequency of

the function Wal(i,9).

168.

functions of the group have a sequency<4, and in general,

the (20th part of the Walsh functions of the group have a

sequency <C23, such that only one Walsh function has a sequency

equal to 2k. Since the code array of an (n,k) code may be

arranged in such a way that each column of the array represents

a Walsh function and if the (n,k) code is non-repetitive, then

each row of the array forms a part of a distinct Walsh function.

This suggests that if n is not negligible compared with 2k, then

half of the rows of the (n,k) code array have a sequency <2,

one quarter of the rows have a sequency <4, ki rows have a

sequency of 2j and two rows may have a sequency of n. Then

since the largest number of codewords has a weight equal to the

average value, it follows that the weight distribution of an

(n,k) non-repetitive linear block code may have a maximum value

at a point corresponding to the average weight of the code.

The proposed improvement on the Varshamov-Gilbert

lower bound on d for linear block codes may be evaluated for

codes, fulfilling the above assumptions, as follows:-

(i) Assume that the minimum number of codewords of

weight (w) where d <w< wave in an (n,k,d) linear code,

is as low as zero; and

(ii) the minimum number of codewords of average weight

(fwave) tends to the average value of fw. The correct-

ness of this statement can be seen from the fact that

since fwave

is the maximum value for fw, (where w = 1, 2,

n), then the lowest value of fwave cannot be less

than the average value of fw. That is:

2k

1 fwave.

(6.10)

where Z is the total number of the non-zero fw and in

general Z‹.;:n-d. However, for an (n,k) code with even

169.

Hamming distance d, derived from an (n-1,k) code with

odd Hamming distance (d-1) by annexing an overall

parity check digit, fw, equal to zero for w odd and

therefore:

Z <

(6.11)

Using equations (6.1), (6.8) and (6.10), the Varshamov-Gilbert

bound and its improvement may be evaluated. The improvement

is found to be significant for n<10. However, as the code

length increases, the numerical evaluation of the proposed:

improvement using the above suggested bound on the lowest

value of fwave is found to be negligible. Nevertheless, the

proposed improvement can be of great importance if a tighter

bound on the lowest values of fw (where d<w<d-2 + 2 ) is

known.

In spite of the fact that a considerable amount of

research has been devoted to the study of weight distribution

of linear block codes, no work has been done to study the

bounds on the weight distribution (fw).

170.

6.4 MAXIMUM HAMMING DISTANCE BOUNDS FOR LINEAR ANTICODES

In Section 3.4.2 a systematic procedure for the

construction of an (m,k,6) anticode over GF(q) for given values

of 6 and (m-k) has been proposed. Thr procedure is based on

the characteristics of the parity-check matrix EL: of the anti-code. It has been established that the maximum Hamming distance

of an (m,k) anticode is 6, if and only if all combinations of

6+i, i = 1, 2, ..., m-6 columns of EL: are quasi-linearly independent. From this the following procedure for constructing

the m columns of the ELImatrix is formulated as follows:-

(1) The first (m-k) columns of EL:1 matrix form an identity matrix of order (m-k).

(2) The (m-k+1)th column is chosen arbitrarily from the

vector space (Vm_k) over GF(q), subject only to the condition

that the chosen column must not be the inverse of any of the

vectors obtained from the quasi-linear combinations of every

6+1, i = 0, 1, (m-k) of the previously chosen columns.

(3) The remaining columns are chosen so that they are

different from the inverse of any of the vectors obtained from

the quasi-linear combinations of every 6+1, i = 0, 1 ... of

the previously chosen columns.

This method of generation guarantees the required

independence of the columns. The method cannot be completed

successfully unless (m-k) is sufficiently large. To find the

minimum value of (m-k), consider the worst possible case for

an arbitrary choice of columns of the ELD matrix, viz., the inverses of all the vectors of the quasi-linear combinations

might be distinct. Since the inverse vectors of the quasi-

linear combinations of the mth stage include all these vectors

in the previous stages, then to ensure success in the face of

171.

•

•

this worst case, the number of qm-k vectors of the vector space

Vi-k need only be greater than or equal to the total number of

inverses of all vectors of the quasi-linear combinations of the

last stage plus a vector for the last chosen column. That is,

if:

qm-k!l+ )(c1-65 +(rac-1)(q-1)

6+1 + • •• (m-1).c1-1) 0+1

m m-1 , .111-1

(6.12)

there exists an anticode with m digits and minimum-maximum

Hamming distance (5). Now let m be the largest value of m for

which inequality (6.12) holds. Then an (m,k) anticode with

minimum-maximum distance (5) exists which satisfies the

inequality:

(7.1)(q-1)i (6.13)

This provides a lower bound on the minimum-maximum

Hamming distance obtainable with an (m,k) anticode.

For large m, asymptotic results for binary anticodes

may be obtained using the following inequalities:

n

. ‹; (X)-Xn (0-1111 (6.14)

i= n

providing X >k, where 1.1= 1 -X. (See Peterson(7), page 468.)

(6.15)

Taking the logarithm of both sides, assuming 2m-k%>1,

and dividing by m, we obtain:

6 1 - k6 log2 m dl) - (1 -m) —) log2 (1 -b-) (6.16)

m m

This lower bound is plotted below. Note that the

shape of this bound is the mirror image of Varshamov and

Gilbert's(60,27) lower bound for linear block codes of minimum

distance (d), where d is replaced by 6 and m by n.

qm-k 1 +

Inequality (6.14) becomes:

2m-k - 1 ‹_ (5/m)-6 (114)-(m-6)

172.

An upper bound on the minimum-maximum distance of

anticodes may be established by using the Plotkin(63) principle

of "average distance". That is, the minimum weight of a code-

word in an (n,k) linear code is at most as large as the average

weight of the code. Due to the similarities between linear

codes and linear anticodes, it follows that the Plotkin(63)

average distance bound is also true for linear anticodes, and

that it is true to say that the maximum weight of an anticode-

word in an (m,k) linear anticode is never smaller than the

average weight of the anticode.

Consider the array of an (m,k) linear anticode over

GF(q). Since each field element appears qk-1 times in each

column(14)(86), the number of non-zero elements in each column

is (q-1)qk-1

, and since there are m columns, the sum of the

weights of all anticodewords in the anticode is m(q-1)qk-1

Also, since there are qk-1 non-zero anticode words, it follows

that the average weight of an (m,k) anticode over GF(q) is

mqk-1(q-1)/(qk-1). This is also found by Farrell(102) in

unpublished work. The asymptotic form of the Plotkin bound

on 6 for linear anticodes is given by:

6 < m/q (6.17)

For the binary case, this bound is plotted below.

173.

0.9

0-8—

0.7

0.6

"f • 0-5

I 0.4

0.3

0.2

0.1

0

0 • 0

Ac c

o •

istan

ce u

p per

bou

nd

•■•••••••1

.1011.1101.1•M

•■•••••■1

Bounds on minimum maximum-distance for Linear Anticodes

1 1 1 1 1 1 al 0.2 0.3 0.4 0.5 0.6 0.7 0-8 09 1.0

6 /rn

174.

7. SUMMARY OF CONTRIBUTIONS AND SUGGESTIONS FOR FURTHER RESEARCH

7.1 Summary of Contributions.

7.2 Suggestions for Further Research.

175.

7.1 SUMMARY OF CONTRIBUTIONS

In this section the original aspects of the work

contained in the thesis are summarized. These may be listed

as follows:-

(1) Asomputerised search for linear codes: Certain

properties of the parity-check matrix Da of (n,k) linear codes have been used to establish a computerised search procedure for

new binary linear codes. Of the new error-correcting codes

found by this procedure, two codes are capable of correcting up

to two errors, three codes up to three errors, four codes up to

four errors, and one code up to five errors. Two meet the lower

bound given by Helgert and Stinaff(68)

, and seven codes in fact

exceed it. In addition, one meets the upper bound. Of the even

Hamming distance versions of these codes, eight meet the upper

bound, and the remaining two exceed the lower bound. Moreover,

the following codes have been found whose parameters (n,k,d)

are as good as those of best previously known codes of identical

Haunting distances and the same number of parity-check digits.

These are three codes capable of correcting two random errors,

three codes up to three errors, four codes up to four errors

and three codes up to five errors. Corresponding to these

there are thirteen more codes having even Haunting distance.

(2) Applications of Walsh functions in constructin linear

block codes: A class of algebraic linear codes has been intro-

duced in which the parity-check matrix of the code is

constructed by using a sub-set of the Abelian group of Walsh

functions. These codes meet the Helgert and Stinaff upper

bounds on minimum Hamming distance and all the codes of this

class are easily decodable by a one step majority-logic algorithm.

176.

(3) Aulication of the concept of anticodes: A math-

ematical analysis of linear anticodes has been established

from which a systematic procedure for generation of such anti-

codes is proposed.

(4) Two procedures of linear block code modification:

(a) Code puncturing.

(b) Code lengthening.

Two procedures for modifying linear block error-

correcting codes have been suggested. The first is based on

the deletion of x rows and y columns from the parity-check

matrix of a given (n,k) code in such a way that the minimum

Hamming distance of the resulting (n-y, k+x-y) code remains

equal to that of the original. It has been shown that if x is

unity, then y may be as low as the minimum Hamming distance of

the (n, n-k) dual of the original (n,k) code. This procedure

of deleting rows and columns when applied to the known linear

binary block code yields a family of codes, some of which have

better rates than those of the best previously known codes of

identical Hamming distance and the same number of parity-check

digits. Seventeen examples of such new codes have been given.

It has also been shown that, apart from appropriate slight

modification, the coding and decoding algorithm for this family

of modified codes are similar to those of the original code.

The second proposed procedure of code modification

entails lengthening the original (n,k) linear block code by

annexing k' message digits. If the original code is capable

of correcting t random errors or less, then the resulting

modified (n+k', k+k') code has a rate higher than that of the

original (n,k) code. Moreover, its error-correcting capability

177.

is such that it will correct t random errors or less if at

least one of these occurs in the block of k message digits

and s random errors or less, where 14;s<t, if none of the

errors occur in any of the k message digits. Five examples

of such codes have been given.

(5) A new class of nested codes and their decoding

algorithm: A new class of multiple error-correcting linear

block codes has been introduced. The codes of this class,

referred to as nested codes, cover a wide range of code length

and rates. The parity-check matrix of such an (n,k) nested

code is derived by the construction of a nest of parity-check

matrices of previously known linear block codes. Although

the rates of the resulting nested codes are not optimum, these

codes exhibit, nevertheless, some useful properties and a well

defined mathematical structure which leads to a simple decoding

algorithm. This algorithm, referred to as the nested decoding

algorithm, makes use of the fact that the syndromes of the

nested codes, corresponding to patterns of errors in the

information digits, are codewords of another nested code. Thus,

the calculated syndrome of the corrupted received codewords

gives rise to a nest of error-correcting codes. This unique

characteristic results in the complexity of the decoder

increasing only linearly with the code length and error-

correcting capability. Moreover, the nested decoder has been

shown to be capable of correcting some of the errors of weight

greater than the number of errors, t, to be corrected.

(6) Block codes for non-independent errors: A computerised

search procedure based on certain properties of the parity-check

matrix of those block codes capable of correcting non-independent

errors has been described which yields a considerable number of

178.

codes that are capable of correcting burst, burst-and-random,

and multiple-burst-errors. These new codes were used,

together with other known block codes for compound channels,

to construct a new class of burst-and-random-error correcting

codes. It was shown that this class of code exhibits the

properties of the nested codes, i.e. the syndromes corresponding

to errors in the information digits are codewords of other

nested codes. Moreover, it was also established that codes

of this class are decodable by a simply implementable decoding

algorithm, the decoder for which exhibits the same features

and merits as the random-error nested decoder.

Finally, in Section 6 a lower bound on the minimum

Hamming distance (d) for linear block codes and a lower bound

on the maximum Haulaing distance (6) for linear anticodes has

been introduced. Moreover, the Plotkin upper bound on d for

linear block codes has been modified to establish an upper

bound on 6 for linear anticodes. These new bounds are:

(7) A possible improvement on the Varshamov-Gilbert

lower bound on the minimum Hamming distance of linear block

codes:

The improved bound is based on the assumption that,

for an (n,k) group code, the number of distinct vectors

resulting from the linear combination of every (d-2) columns

of the parity check matrix is much less than the total number

of vectors generated from such linear combinations. An

expression for the largest possible number of distinct vectors

obtainable from any (n,k) group code can therefore be

introduced and shown to be a function of the weight distribution

of the code. It has been demonstrated that this functions could

lead to a considerable improvement on the Varshamov-Gilbert

bound for certain ranges of n and k.

179.

(8)

Maximum Hammin distance bounds for linear anticodes:

The mathematical analysis of linear anticodes given

in Section 3.4.1 and the systematic procedure of generating

linear anticodes proposed in Section 3.4.2 have both been used

to obtain a lower bound on the maximum Hauaiiing distance , 6,

of linear anticodes. For the case of binary anticodes, the

asymptotic form of this bound has been established. In

addition, the Plotkin principle of "average distance" yields

an upper bound on 6 for linear anticodes.

180.

e

7.2 SUGGESTIONS FOR FURTHER RESEARCH

The various possibilities which appear to deserve

consideration are discussed below:

(1) In Section 3.4.2, a systematic procedure has been

introduced for generating linear anticodes. This procedure

may be used as a basis for a computerized search for "good"

anticodes and these anticodes in turn may lead to other new

families of linear block codes.

(2) The measure of complexity for a given decoder is

still not a rigorously defined concept. Research is therefore

needed to evaluate numerically the complexity of the Berlekamp

decoding algorithm for BCH codes, the majority logic decoding

algorithm for cyclic codes, and the nested decoding algorithm

for nested codes, in terms of:

0 The number of operations needed for decoding a single

information digit.

ii) The total time required for these operations.

iii) The complexity of the hardware circuitry; that is,

the number of components needed and the resulting wiring

complexity.

iv) The overall cost evaluation.

The outcome of such research of course would be very

beneficial from an engineering point of view.

(3) In Section 4 the nested decoding algorithm has been

introduced. This is capable of correcting some errors of

weight greater than the error-correcting capability of the

nested code. In consequence, the following questions arise:-

(a) What are the effects of this property of the nested

decoder on the average probability of error?

181.

(b) What characteristics of a nested code give the lowest

probability of error?

(c) What are the relationships between the probability of

error of a given nested code and that of the codes of the

construction?

(4) In Section 5 two computerised search procedures have

been introduced to search for burst-and-random and multiple-

burst-error-correcting codes, the results of these being given

in Tables 5.4.1 and Table 5.5.1 respectively. However, these

searches are not exhaustive and the computer limits have not

yet been reached. More codes therefore may be found by

extending the proposed computer searches.

Enlarging Table 5.4.1 is of particular importance due

to the fact that the extension of the nested burst-and-random-error-

correcting codes (of Section 5) depends on the existence of

such a table of codes.

(5) In Section 6 an improvement on the Varsh2mov-Gilbert

lower bound on the minimum Hauaing distance of linear codes has

been introduced. This improvement was found to be a function

of the weight enumerator (fw) of the codes. In spite of the

fact that a considerable amount of research has been devoted

to the study of weight enumerators, no work has, however, been

published to study the "lower bound" on these enumerators;

"lower bound" theory being used here with reference to the -

lowest possible value of fw for the (n,k) linear code. Research

is therefore needed to examine and suggest such a bound. A

lower bound on fw

may lead to a considerable improvement on the

Varshamov-Gilbert bound on d.

(6) In the last few years, due to the increasing applic-

ations for non-binary codes, a great deal of attention has been

devoted to this subject (see references 141 and 142).

182.

Since the problems introduced in Sections 3, 4 and 5

of this thesis have been formulated in a generalized manner,

that is, the codes are considered over a field of q elements,

the techniques and methods given in these sections may

consequently be used to construct and decode non-binary codes.

The immediate and reasonable suggestions for further research

on this subject apppear to be as follows:

(a) A computer programme may be developed along the lines

of the computerised search procedure described in Sections 3

and 5. The resulting computer search will then lead to new

families of non-binary, random, burst, burst-and-random, and

multiple-burst-error-correcting codes.

(b) The codes found by the technique described above may

also be used together with other known non-binary codes to

construct new classes of non-binary simply decodable nested

codes.

(7) Disjoint Codes:

Consider an (n,k1,t) linear block code V over a field

of q elements. Assume that there exists an (n,k2) linear block

code U over the same field, such that it is disjoint with V;

that is, no non-zerp vector in V is in U and therefore the

intersection between V and U is equal to zero vector. If the

minimum Hamming distance of U is equal to or greater than unity,

the error-correcting capability (t) of V may be increased to

any value by annexing more parity-check digits to V as follows:

(1) Add an overall parity check digit (in the usual

manner) to V to increase its Hamming distance by one. The

resulting annexed code will therefore have an even Hamming

distance.

(2) Add (n-k2) redundant digits to each code vector

of V such that the annexed (n-k2) digits of any code

183.

vector of V (say v) are those digits of the syndrome

vector of U corresponding to the code vector v. Since

none of the code vectors of V are in U, the syndromes of

U corresponding to the n-tuple vectors of V must there-

fore be non-zero vectors. Thus the weight of each

annexed code vector must have been increased by at least

one.

The newly annexed (2n- k2+ 1, kl) code is therefore

capable of correcting (t+ 1) random errors or less. This error-

correcting capability can be increased by another one by again

adding an overall parity check digit, this time to the newly

formed code and the same set of (n-k2) redundant digits as was

added in step (2). This procedure can be repeated indefinitely

to achieve any arbitrary error-correcting capability.

If (n-k2) is much less than k1, the above method

should yield families of asymptotically or nearly asymptotically

good codes. Research devoted to finding codes such as U, which

are disjoint with single-error-correcting Hamming codes, would

therefore be very fruitful.

Consider now the case where the codes V and U are not

disjoint and let the intersection between them be a subspace Z.

If Z is capable of Correcting (t+j) random errors or less, the

procedure described above may be repeated j times to generate

((j +1)n- jk2+j, kl) block code cap able of correcting (t+j)

random errors or less. Consequently the search for subspaces

which satisfy the properties of Z and which could be used to

generate new annexed codes should prove to be a profitable line

of research.

It is appropriate to mention that Andryanov and

Saskovets's(87) procedure for annexing BCH codes can be

considered as a special case of the latter described idea.

184.

Finally, the classical problems of coding theory as

outlined in Section 1 still remain as the main source of

stimulation for research work in this field. Although these

problems have been recognised now for many years and much

classical work has been produced in an attempt to find solutions,

the main questions still remain unanswered. Coding theory has

reached maturity but this maturity does not preclude future

research in this field from being any less scientifically

rewarding than it has been in the past.

185.

APPENDIX 1 -

Mathematical Background

The purpose of this appendix is .to list some mathem-

atical definitions of the terms used in this thesis. All

definitions are taken from standard text-books, such as

references (3), (5) and (7) and others dealing witn algebra

(e.g. references (16), (17) and (18)).

Set:

A set S is a well-defined collection of entities.

Null Set:

A null (empty) set is that set with no elements.

Binary Operation:

A binary operation represented by ,*, on a set is a

rule which assigns to each ordered pair of elements of the set

some element of the set.

Commutative and Associative Binary Operation:

A binary operation * on a set is cotiauutative if and

only if a*b=b*a for all a,b € S. The operation * is associative

if and only if (a*b)*c= a*(b*c) for all a,b,c S.

Group:

A group G is a set G, together with a binary operation

* on G, such that the following axioms are satisfied:

i) The binary operation * is associative.

ii) There is an element e in G such that e*x=x*e = x

for all xeG. This element e is the identity element for

* on G.

186.

iii) For each a in G, there is an element a' in G with the property that a`lra=a*a t = e. The element a' is the inverse of a with respect to *.

Note: The group G is closed under the operation *, since by the definition of a binary operation on G, (a*b)GG.

* Abelian Group:

A group is Abelian if its binary operation * is colianutative.

Subset:

A set B is a subset of set A, if every element of

B is in A.

Improper Subset:

If A is any set, then 0 and A are improper subsets of

A. Any other subset of A is a proper subset of A.

Subgroul:

A group H is a subgroup of G, if H is a subset of

group G such that the binary operation of G is closed with

respect to H, and if H is itself a group under this binary

operation.

Improper Subgroup:

If G is a group, then G and "e" are improper subgroups

of G. All other subgroups are proper subgroups.

Cyclic Subgroup:

If H is a subgroup of G and is the smallest subgroup

of G which contains the element a, aeG such that,

H = {- n/x = an, n(3A1

H is the cyclic subgroup of G generated by a.

187.

Cyclic Grou :

An element a of a group G generates G and is a

generator for G if<a>= G. A group G is cyclic if there is some

element a in G which generates G.

Ring:

A ring R is a set R together with two binary operations

to be known as addition and multiplication defined on R such that

the following axioms are satisfied:

i) <R, +:> is an Abelian group.

ii) Multiplication is associative.

iii) For all a,b,c(;11, the left distributive law,

a(b+c) = (ab) + (ac), and the right distributive law:

(a+b)c = (ac) + (bc), hold.

Field:

A field is an Abelian group under two binary operations,

multiplication as well as•addition.

Vector Space:

A vector space consists of an Abelian group V under

addition of elements called vectors, and a field F of scalar

elements satisfying the following axioms:

(A) To every pair, a and x, where cc. is a scalar in F and

x is a vector in V, there corresponds a vector ax in V,

called the product of 0. and x, in such a way that:

i) multiplication by scalars is associative,

a(Px) = (a0)x, and,

ii) lx = x for every vector x.

(B) i) Multiplication by scalars is distributive with

respect to vector addition, OC(x+y) = Ctn + C(y, and

188.

ii) multiplication by vectors is distributive

with respect to scalar addition, (a+ 13)x= ax+ Px.

The relation between a vector space v and the under-

lying field F is usually described by saying that V is a vector

space over field F.

Subspace:

Let V be a vector space over F. The vectors in a

subset S = fa. I ie 1i of V is called a subspace if, (i) the all-zero vector in S and, (ii) the sum of any two vectors in S is

also in S, i.e., the set of all linear combinations of subset S

of V is a subspace of V.

Definition (17) Linear independence and linear

dependence: the vectors in a subset S = fail iGij of a vector space V over a field F are linearly independent over F if n

T1ij a.ai0forscalarsa.j= 1, 2, ..., n, not all zero. I. J

j,

j=1 If the vectors are not linearly independent over F, they are

linearly dependent over F.

Span a Vector Space:

Let V be a vector space over F. The vectors in a

subsets=fa.lif3I1 of V span (or generate) v if for every REV,

we have:

13= alail + ct 2 ai2 + + an ain

for some a. e F and a. 3 . G S, j = 1, , n. A vector 1 is a linear combination of the a... 13

Dimension:

a. a,13..

=1

In any space, the number of linearly independent

vectors that span the space is called the dimension of the

space. A vector space V over a field F is finite dimensional

189.

I

if there is a finite subset of V whose vectors span v.

Basis:

If V is a vector space over a field F, the vectors

in a subset f3= {: . 1.613 of V form a basis for V over F if

they span V and are linearly independent. It follows that if V

is a k-dimensional vector space, any set of k linearly independent

vectors in V is a basis for V.

Inner Product and Ortho:onal Vectors:

If a= E a1, a2' .. an] and p. £ I1, p2... on] are two vectors of vector space V over field F, , V, and their

inner product defined to be the scalar:

a43 = al P l a2 P2 + a Pri; the inner product is commutative and associative:

a. p 13. a and that )5.( + p). )s.c1L + ?5.p

a and 13E v. The two vectors a and p , are said to be ortho-

gonal if their inner product is zero.

Transpose Matrix:

The transpose of a'matrix is that matrix of order

n x m obtained by interchanging the rows and columns of an m x n

matrix A is called the "TRANSPOSE" of A and is denoted by AT.

Symmetric Matrix:

Symmetric matrix is a square matrix A such that AT=A.

Matrix Overfield:

When all of the elements of a matrix A are in a field

F, it is called matrix A over field F.

190.

Row Space:

Let tGi an "k x n" matrix over field "F" if the rows k

are less than the columns n, and are linearly independent, then

all linear combinations of rows of G form a k-dimensional subspace

of the vector space Vn over field F, which is called the row space

of G.

Row Rank Column Rank and Matrix Rank:

The dimension of the row space is called the row rank.

Similarly, the set of all linear combinations of column vectors

of the matrix forms the column space, whose dimension is called

the column ranks, it can be shown that row rank equals column

rank; this value is referred to as the rank of the matrix.

Echelon Canonical form of a Matrix:

The Echelon Canonical form of a matrix is a standard

simplified form of a matrix which has the following properties:

i) Every leading term of a non zero row is 1.

ii) Every column containing such a leading term has all

its other entries zero.

iii) The leading term of any row is to the right of the

leading term in every preceding row. All zero rows are

below all nonzero rows.

The Dimension of Echelon Canonical Form Matrix:

If A is a matrix in echelon canonical form, then the

nonzero rows of A are linearly independent, and thus the number

of nonzero rows is the dimension of the row space. There is

only one matrix in echelon canonical form for any given row

space.

191.

•

Nonsingular

A nonsingular matrix is a square matrix with all rows

linearly independent. The echelon canonical form of a non-

singular matrix is an identity matrix I.

Null Space:

Let V be a vector space over F, of n-tuples. The

vectors in a subset S =fa. iGI1 of V is called the null space of a subset u =. ell of V, if all vectors of S are 011 orthogonal to all vectors of u.

A Null Space of a Matrix:

For any k x n matrix G with k linearly independent

rows, there exists an (n-k) x n matrix H with (n-k) linearly

dependent rows and any vector v in the row space of G is

orthogonal to all the rows of H. The row space of G is the

null space of H or vice versa.

192.

APPENDIX 2

Computer Programmes:

Programme 1 is a computer search for a new binary

linear block code according to the procedure proposed in

Section 3.2, where t and r are given to be 3 and 15 respectively.

The starting parity-check matrix is. given by Karlin(47)

:

(30,16,3) code EH] matrix with an extra dummy parity-check digit.

Programme 2 is written to generate the parity-check

matrix [H] of (255,187,19) BCH code and to determine the lowest

weight of this [H] matrix.

Programme 3 is written to determine all possible

vectors of the vector space Vn-k which satisfy the condition

set by the lengthening procedure proposed in Section 3.5.4.

The code before lengthening in this programme has the parameters

(7,1,2).

All programmes are written in FORTRAN IV and executed

on London University computer CDC 7600.

193.

PROGRAMME

CODE

DIMENSION IC(40),MT(780),IP(22),II(15),IS(60),NC(547),ME(9880) COMMON IC,MT,IP,II,IS,LC I IIND,NO,LR,L1,L2,L3,L4,L5,J,K,NOP,L11, 1L22,LIS,LE,NC,ME

INTEGER SHIFT LC = 31 LIS = 32767 NP = 15 DO 100 I = 1,60 ND = I - 1 IS(I) = SHIFT(1,ND)

100 CONTINUE DO 200 I = 1,547 NC(I) = 0

200 CONTINUE DO 196 I=1,NP IC(I) = IS(I)

196 CONTINUE READ (5,1000) (IC(I),I=16,LC)

1000 FORMAT (1605) WRITE (6,4000) LC

4000 FORMAT (26H1 (THE INPUT DATA FOR LC=,13) DO 156 I=1,LC DO 951 J=1,NP IP(J) = IC(I).AND.IS(J) IF(IP(J).NE.0) IP(J) = 1

951 CONTINUE WRITE (6,5000) II IC(I),IC(I),(IP(J),J=1,NP)

5000 FORMAT (5X,13,18,2X,08,2X,15I4) 156 CONTINUE

LR = 0 LE = 0 • DO 300 I=1,LC Ll = (IC(I)-1)/60 L2 = IC(I) L1*60 LI = L1 + 1 NX = NC(L1).AND.IS(L2) IF (NX.NE.0) WRITE (6,2233) IC(I)

2233 FORMAT (31H0 THERE IS AN ERROR AT IC(I),I8) NC(L1) = NC(L1).O.IS(L2) L3 = I + 1 IF (L3.GT.LC) GO TO 300 DO 310 J=L3,LC LR = LR + I MT(LR) = L1 = (MT(LR)-1)/60 L2 = MT(LR) = L1*60 LI = LI + 1

194.

NX = NC(L1).AND.IS(L2) IF(NX.NE.0) WRITE (6,2233) IC(I) NC(L1) = NC(L1).0.IS(L2) L4 = J + 1 IF (L4.GT.LC) GO TO 310 DO 320 K=L4,LC LE = LE + 1 ME(LE) = .N.MT(LR).A.IC(K).0.MT(LR).A..N.IC(K) Ll = (ME(LE)-1)/60 L2 = ME(LE) = L1*60 Ll = Ll + 1 NX = NC(L1).AND.IS(L2) IF (NX.NE.0) WRITE (6,2233) IC(1) NC(L1) = NC(L1).0.IS(L2)

320 CONTINUE 310 CONTINUE 300 CONTINUE

NOP = 33 L2 = LC - 24 J = L2 - 1 II(L2) = 1 DO 357 I=1,J II(I) = IC(25+1)

357 CONTINUE 40 DO 400 L1=L2,15

L3 = 25 + Ll 30 L = II(L1)

CALL TEST IF (LC.LT.L3) GO TO 10 IF (LC.EQ.NOP) CALL OUTPUT II (L1+1) = II(L1) + 1 IF (II(L1+1).GT.LIS) GO TO 20 GO TO 400

10 LL(L1) = II(L1) + 1 IF (II(L1).GT.LIS) GO TO 20 GO TO 30

400 CONTINUE GO TO 60

20 L4 = L3 - 1 IF (L4.EQ.25) GO TO 60 DO 500 J=L4,LC L11=IC(J)-1)/60 L22=IC(J) - L11*60 L11=L11 + 1 NC(L11) = NC(L11).A..N.IS(L22)

500 CONTINUE LC = L4 - 1 L5 = LC*(LC-1)/2 + 1 DO 600 J=L5,LR

195.

L11 = MT(J)-1)/60 L22 = MT(J) - L11*60 L11 = L11 + 1 NC(L11) = NC(L11).A..N.IS(L22)

600 CONTINUE LR = L5 - 1 L5 = LR*(LC-2)/3 + 1 DO 700 J=L5,LE L11 = (ME(J)-1)/60 L22 = ME(J) L11*60 L11 = L11 + 1 NC(L11) = NC(L11).A..N.IS(L22)

700 CONTINUE LE = L5 - 1 L2 = L3 - 26 II(L2) = II(L2) + 1 GO TO 40

60 CALL OUTPUT STOP END

196.

SUBROUTINE TEST DIMENSION IC(40),MT(780),IP(22),I1(14),IS(60),NC(547),ME(9880) COMMON IC,MI,IP,II,IS,LC,I,ND,NO,LR,L1,L2,L3,L4,L5,J,K,NOP,

1L11,L22,LIS,LE,NC,ME

L11 = (1-1)/60 L22 = I = L11*60 L11 = L11 + 1 NO = NC(L11).A.IS(L22) IF (NO.NE.0) RETURN DO 100 J=1,LC • ND = L11 = (ND-1)/60 L22 = ND - L11*60 L11 = L11 + 1 NO = NC(L11).A.IS(L22) IF (NO.NE.0) RETURN

100 CONTINUE DO 200 J=1,LR ND = L11 = (ND-1)/60 L22 = ND = L11*60 L11 = L11 + 1 NO = NC(L11).A.IS(L22) IF (NO.NE.0) RETURN

200 CONTINUE DO 300 J=1,LR LE = LE + 1 ME(LE) = L11 = (ME(LE)-1)/60 L22 = ME(LE) L11*60 L11 = L11 + 1 NC(L11) = NC(L11).O.IS(L22)

300 CONTINUE DO 400 J=1,LC LR = LR + 1 MT(LR) = .N.IC(J).A.I.O.IC(J).A..N.I L11 = (MT(LR)-1)/60 L22 = MT(LR) - L11*60 LII = L11 + 1 NC(L11) = NC(L11).0.IS(L22)

400 CONTINUE LC = LC + I IC(LC) = L11 = (1-1)/60 L22 = I - L11*60 L11 = L11 + 1 NC(L11) = NC(L11).0.IS(L22) CALL SECOND (T)

197.

IF (T.GT.1180.0) GO TO 17 RETURN

17 WRITE (6,3300) 3300 FORMAT (1H0 / 27H0 LC II(1 )

WRITE (6,1020) LC,(II(JI),JI=1,L1) 1020 FORMAT (5H0 , 1116)

STOP END

SUBROUTINE OUTPUT

SUBROUTINE OUTPUT DIMENSION IC(40),MT(780),IP(22),11(15),IS(60),NC(547)-IME(9880) COMMON IC,MT,IP,II,IS,LC,I,ND,NOILR,L1,L2,L3,L4,L5,J,KI NOP,L11,

1 L22,LIS,LE,NC,ME NOP = NOP + 1 WRITE (6,1000) LC,LR,LE

1000 FORMAT (38H0 THE TOTAL NUMBER OF CODEWORDS IS ,17 / 48H0 THE 1 TOTAL NUMBER OF COMBINATION IN TWOS IS ,18 / 52H0 THE TOTAL 2 NUMBER OF COMBINATIONS IN THREES IS ,18) DO 100 J=1,LC DO 200 N0=1,22 IP(NO) = IC(J).A.IS(NO) IF (IP(NO).GT.0) IP(N)=1

200 CONTINUE WRITE (6,2000) J,IC(J),IC(J),(IP(NO),N0=1,22)

2000 FORMAT (5H0 ,IB„18,2X,08,2X,2214) 100 CONTINUE

RETURN END

198.

PROGRAMME 2

PROGRAM CODE (INPUT,OUTPUT,TAPE5=INPUT,TAPE6=OUTPUT) DIMENSION IP(100),NU(100),IM(100),IS(100) INTEGER SHIFT N = 255 K = 187 NT = 9 M = N K Ll = M - 1 DO 200 1=1,100 IP(I) = 0 NU(1) = 0 J = L 1 IS(I) = SHIFT(1,J)

200 CONTINUE I = 57 WRITE (6,5000) I

5000 FORMAT (31HO THE POSITIONS OF ZEROS ARE / 10X,I5) DO 310 J=11M NU(J) = NU(J) + 1

310 CONTINUE DO 400 I-1,K DO 410 J=1,M IP(J) = 0

410 CONTINUE DO 420 J=I1K IF (J.EQ.1) GO TO 10 KON = IP(M)

20 IF (KON.EQ.I) GO TO 30 DO 430 L=1,L1 IP(M+1-0 = IP(M-L)

430 CONTINUE IP(1) = 0 GO TO 420

30 DO 440 L=2,M IM(L) = .NOT.IP(L-1) IM(L) = IM(L).AND.1

440 CONTINUE IM(1) = 1 DO 450 L=1,L1 IP(M+1-L) = IP(M-L)

450 CONTINUE IP(1) = IM(1) IP(4) = IM(4) IP(6) = IM(6) IP(7) = IM(7) IP(8) = IM(8) IP(11) = IM(11) IP(12) = IM(12) IP(13) = IM(13)

199.

IP(14) = IM(14) IP(17) = IM(17) IP(20) = IM(20) IP(23) = IM(23) IP(25) = IM(25) IP(26) = IM(26) IP(28) = IM(28) IP(42) = IM(42) IP(43) = IM(43) IP(45) = IM(45) IP(46) = IM(46) IP(47) = IM(47) IP(49) = IM(49) IP(52) = IM(52) IP(53) = IP(53) IP(55) = 114(55) IP(57) = IM(57) IP(58) = IM(58) IP(60) = IM(60) IP(61) = IM(61) IP(62) = IM(62) IP(63) = IM(63) IP(65) = IM(65) IP(67) = IM(67) GO TO 420

10 KON = .NOT.IP(M) KON = KON.AND.1 GO TO 20

420 CONTINUE DO 460 J=1,M IF (IP(J).EQ.0) GO TO 460 NU(J) = NU(J) + 1

460 CONTINUE IF (IP(57).EQ.1) WRITE (6,6000) I

6000 FORMAT (10X,I5) 400 CONTINUE

WRITE (6,1000) N,K,M,NT 1000 FORMAT (42H1 THE PARITY CHECK MATRIX H OF BCH CODE / 18H0

1CODE LENGTH IS,I6 / 29H0 NUMBER OF MESSAGE BITS IS,I6 / 31H0 2NUMBER OF PARITY CHECKS IS ,I6 / 34H0 ERROR CORRECTING 3CAPABILITY T=3,I2 / 5H0 ) WRITE (6,3000) (NU(J),J=1,M)

3000 FORMAT (4(5H0 ,30I4 / )) STOP END

200.

PROGRAMME 3

PROGRAM OCDEXPN(INPUT,OUTPUT,TAPE5=INPUT,TAPE6=OUTPUT) DIMENSION IS(60),IC(520),NC(1093),NR(1093),IP(18),IM(18) INTEGER SHIFT DO 114 L=1,60 J = I - 1 IS(I) = SHIFT(1,J)

114 CONTINUE N = 6 LC = 7 DO 109 I=1,6 IC(I) = IS(I)

109 CONTINUE IC(7) = 15 NT = 2

70 NN = LC JI = NN + 1 KK = NN - N MLC = N + 1 LIS = 2**N - 1 WRITE (6,2000) NN,KK,NT

2000 FORMAT (33H1 THE CODE EXPANSION TECHNIQUES / 30 HO THE lORIGINAL CODE LENGTH IS,I3 / 36H0 NUMBER OF INFORMATION 2DIGITS IS,I3 / 34H0 ERROR CORRECTING CAPABILITY 18,13 #54H0 3THE COLUMNS OFTHE PARITY CHECK MATRIX IS GIVEN BY) DO 110 L=1,LC DO 120 J=1,N IP(J) = IC(I).AND.IS(J) IF(IP(J).GT.0) IP(J)=1

120 CONTINUE-: WRITE (6,3000) I,IC(I),IC(I),(IP(J),J=1,N)

110 CONTINUE DO 200 1=1,1093 NR(I) = 0 NC(I) = 0

200 CONTINUE = 0

J = 0 DO 300 I=1,LC LF = (IC(I)-1)/60 LS = IC(I) - LF*60 LF = LF + 1 NE = NC(LF).AND.IS(LS) IF(NE.NE.0) WRITE (6,1234) I,J NC(LF) = NC(LF).0R.IS(LS) NR(LF) = NR(LF).OR.IS(LS) JJ = I+ 1 IF (JJ.GT.LC) GO TO 300 DO 310 J=JJ,LC

201.

ND = .NOT.IC(I).AND.IC(J).0R.IC(I).AND..NOT.IC(J) LF = (ND-1)/60 LS = ND - LF*60 LF = LF + 1 NE = NC(LF).AND.IS(LS) IF(NE.NE.0) WRITE (6,1234) I,J NC(LF) = NC(LF).OR.IS(LS) IF (I.GT.N.OR.J.GT.N) NR(LF) = NR(LF).OR.IS(LS)

1234 FORMAT (36H THERE IS AN ERROR IN THE DATE AT,2I4) 310 CONTINUE 300 CONTINUE

WRITE (6,1111) 1111 FORMAT (105H0 THE EXPANDED COLUMNS TO CORRECT ONE ERROR AND

1SOME OF THE TWO ERRORS AND DETECTING OTHER ERRORS / 4H0 ) DO 400 I=1,LIS LF = (I-1)/60 LS = I - LF*60 LF = LF + 1 NE = NC(LF).AND.IS(LS) IF (NE.NE.0) GO TO 400 KL = 0 DO 410 J=1,N ND = I.AND.IS(J) IF(ND.NE.0) KL = KL + 1 IF (KL.GT.3) GO TO 400

410 CONTINUE IF (KL.LT.3) GO TO 400 DO 415 J=1,N ND = .NOT.IC(J).AND.I.OR.IC(J).AND..NOT.I LF = (ND - 1) / 60 LS = ND - LF*60 LF = LF + 1 NE = NR(LF).AND.IS(LS) IF (NE.NE.0) GO TO 400

415 CONTINUE DO 420 J=MLC,NN ND = .NOT.IC(J).AND.I.OR.IC(J).AND..NOT.I LF = (ND-1)/60 LS = ND - LF*60 LF = LF + 1 NE = NC(LF).AND.IS(LS) IF (NE.NE.0) GO TO 400

420 CONTINUE LF (LC.LT.JI) GO TO 15 DO 425 J=JI,LC ND=.NOT.IC(J).AND.I.OR.IC(J).AND..NOT.I LF = (ND - 1) / 60 LS = ND - LF*60 LF = LF + 1

• 202.

NE = NR(LF).AND.IS(LS) IF (NE.NE.0) GO TO 400

425 CONTINUE 15 DO 510 J=1,N

ND = .NOT.IC(J).AND.I.OR.IC(J).AND..NOT.I LF = (ND - 1) / 60 LS = ND - LF*60 LF = LF + 1 NC(LF) = NC(LF).0R.IS(LS)

510 CONTINUE DO 520 J=MLC,NN ND = .NOT.IC(J).AND.I.OR.IC(J).AND..NOT.I LF = (ND - 1) / 60 LS = ND - LF*60 LF = LF + 1 NC(LF) = NC(LF).OROIS(LS) NR(LF) = NR(LF).OR.IS(LS)

520 CONTINUE IF (LC.LT.JI) GO TO 16 DO 530 J=JI,LC ND = .NOT.IC(J).AND.I.OR.IC(J).AND..NOT.I LF = (ND - 1) / 60 LS = ND - LF*60 LF = LF + 1 NC(LF) = NC(LF).OR.IS(LS)

530 CONTINUE 16 LC = LC + 1

IC(LC) = I LF = (I-1)/60 LS = I - LF*60 LF = LF + 1 NC(LF) = NC(LF).0R.IS(LS) NR(LF) = NR(LF).OROIS(LS) DO 610 J=1,N IP(J) = I.ANDOIS(J) IF (IP(J).GT.0) IP(J) = 1

610 CONTINUE WRITE (6,3000)LC9J9I,(IP(J),J=1,N)

3000 FORMAT (5X,I3,I9,4X,07,2014) 400 CONTINUE

STOP END

203.

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DIGITAL CODE DIVISION MULTIPLEXING Mr. A.A. Hashim, B.Sc. (Eng.), M.Sc.

Dr. A.G. Constantinides, B.Sc. (Eng.), Ph.D. Imperial College of Science and Technology

London, England.

Summary

The theory of Code-Division-Multiplexing is developed in this paper. The development of this theory is effected on the inter-pretation of the system in terms of the concept of code-division tree based on the amplitude distribution of code-words at the output of the multiplexer. It is shown that in the Digital-Majority-Logic-Multiplexing system the length L of the codeword carriers is equal to the number of channels n for n=3 and 7 only. For all other values of n, the length L is considerably greater than n.

In addition to the above, a method of multiplexing is proposed, designated as Time-Code-Division-Multiplexing (TCDM) which is based on the combination of Time-Division-Multiplexing (TDM) and Code-Div-ision-Multiplexing (CDM). The binary input data in TCDM is modulated onto codewords of three levels, i.e. of levels +1, 0, and -1. A majority-logic gate forms the transmitted binary signal from the modulated codewords. Detection is achieved by correlating the received binary signal with the appropriate locally generated three-levelled codewords. Such a system has the advantages of both TDM and CDM systems in the sense that the length of the codewords is equal to the number of channels. It is necessary, how-ever, to have the total number of channels in the system a multiple of 3 or 7. Auto-matic random and effective burst error correcting capability at the multiplexer is one of the desirable features of TCDM. Moreover, there exists a tradeoff between the number of channels in use at any one time and the error correcting capabilities of the system

1. Introduction

The basic n-channel CDM system is founded on the orthogonality of the re-channel carriers (1) and (2). The input binary data di=1 or -1 is modulated onto

periodic binary orthogonal codewords W.= w. 1 , w.2' —.. Iw.L' wherewii =1 or -1 11 is the j-th element of the i-th codeword carrier. The modulated binary codeword is summed algebraically to form a multi-level transmitter signal S = s , s2,

2' ''''' sL' the elements of which are given by n

s. i = E1 j d.w. (1) = 1 i

At the receiver, the incoming signal is correlated with the orthogonal codeword carriers. Assuming a noise free channel the correlator output for the k-th channel Rk is given by

R = E sw (2) j k j=1 kj On substituting equation (1) into equation (2) and no ing the orthogonality of the carriers ( wk.wi.=L for k=i otherwise the sum is zero), equation (2) may be rewritten in the form

Rk = L dk (3)

Thus Rk has an amplitude which is L times greater than that of the k-th channel

data dk.

A modification of the basic CDM system has been proposed in (3), which is referred to as Boolean-Function Multiplexing, where-by the transmitter signal S of the CDM system before transmission is passed through a majority-logic function generator the output of which St

= stl, st2,...pstL is binary. The elements of St are given by

St. = Sgn1=1

d.w..) (4) ij At the receiver, the correlator output

for the k-th channel is given by:

Rtk

= sgn ( E S .) (5) j=1 tJ kJ

Moreover, it has been shown that the crosstalk between channels is zero when(3):

i) The codewords form an algebraic group and L = 2n, or

ii) The codewords are the first n time-shifts of a pseudorandom binary

1974 ZURICH SEMINAR F5 (1)

sequence of length L = 2n - 1.

If the system parameters are chosen corr-ectly and the errors in the received signal do not exceed the permitted level, then Sgn(Rt0=dk and the channels are separated correctly.

The error-correcting capability of the system is characterised by the minimum "Reassurance" defined as

L r =d ( 2: s oa ) =d R (6) min k j=1 tj kj min k tk(min) The reassurance of a system must remain

positive for correct separation of all channels. The minimum reassurance is equivalent to the minimum Hamming distance of an error correcting code. The number of random errors t that can be corrected by the system is given by

t = (rmin

- 1)/2 (7)

In (4), (5) and (6) a Digital Majority-logic Multiplexing system is developed which is based on the Boolean-function multiplexing system of (3). By suitable computer search the authors of the above papers propose several sets of suitable codeword carriers that give minimum reass-,'rance of unity for the case when the number of channels is equal to the length of the codewords. These are,

i) All shifts of length n=L=3 and n=L=7 of maximal length sequences, and

ii) the three truncated Walsh functions Wal(1,8), Wal(2,9) and Wal(309) with the first character omitted in each case, and the seven truncated Walsh functions Wal(1,9), Wal(2,9), ....,Wal(7,9).

It appears that the Digital-Majority-logic multiplexing system has inherent limitations. These limitations relate to the type of sets of codeword carriers and to the maximum number of channels which can be multiplexed without errors occurring. We offer in this paper a theoretical basis for the understanding of these limitations. Furthermore, we propose a new system, developed from these theoretical consider-ations, which overcomes the limitations and increases the advantages of digital multiplexing.

2. CDM System Analysis

Consider a basic n-channel CDM system of n=23-1, j is a positive integer. Let the n rows of the orthogonal Hadamard matrix of length L=23 (the first row being omitted) be used as the n codewords carrier. The Hadamard matrix of seven rows and eight columns is given below:

If all the n input binary data are +1, i.e. di=+1, i=1,2,....,n, then the L digit multi-level transmitter signal S, which is formed by the algebraic sum of the n modulated carrier codewords, contains one digit of amplitude level n, and n digits of amplitude level -1. Now, if the j-th row is replaced by its binary complement, i.e. di -=-1, then S contains one digit of level (n-2), (L/2)-1 digits of level -3 and L/2 digit of level 1. This follows from the fact that all two rows of Hadamard matrix have equal numbers of +1's and -1's. As the Hadamard matrix is orthogonal, then, if the j-th and k-th rows are replaced by their complements, i.e. d7•=dk=-1, S contains one digit of amplitude level (n-4), (L/4)-1 digits of amplitude level 3, and (2(L/4)) digits of amplitude level -1, provided that the number of channels with -1 information is less than L/4.

On this basis a code division tree can be easily constructed as can be seen in Fig. 1. Taking any node in the tree at a given amplitude level a, two branches (left and right) lead to two further nodes whose amplitude levels are respectively (a-2) and (a+2). Each amplitude level occurs ((n+1)/21) times if it lies to the right of the dotted line in Fig. 1 and ((n+1)/22,1.1 times if it lies to the left, i being the number of levels down the tree from the initial level of node -1. To complete the representation of S, an amplitude level of (n-2i) is added to the i-th level of the tree. Obviously the code division tree is applicable to any set of binary orthogonal

F5 (2)

NUMBER OF EVENTS (i) Nodes No.

0

1

2

3

-3

NUMBER OF EVENTS

n+ 1 - 1 I

n+1

2i 2i

n-8 A A A A 10, -\\ - - - - - - - - 3 - 3 4

n- 3 A \ 3 \AAA/V\ AA 5 -7 - -7 -3 -3 -7 - -3 -3 1 1 5-7 -3 -3 -3 1 5-3 1 1 1 5 5 9

Fig. 1 Code-Division Tree

functions which form a matrix of columns having equal numbers of +1's and -1's (excluding the first column) as for example, the set of codewords formed by the n phase shift of maximal pseudo-random binary sequence of length n, with +1 added at the beginning of each row.The transmitter signal S has 2n unique codewords. The frequency of occurrence of the modules of amplitude levels in each codeword is identical to the frequency of one of the first L/4 code division tree levels, if L/4 c log2L.

However, if log2L > L/4, the first (l+log2L) levels represent the frequency distribution of E 2Cin code,,,ove6.

i=log2L

The frequency distribution of the mod-ulus of the amplitude level of codewords of the correlator output at the receiver is identical to S. However, the amplitude level signs may take any pattern so that the algebraic sum of all (L) digits of the codeword yield a reassurance of value +L. The sign patterns of the digits which have maximum number of amplitude level 1 with opposite sign to the channel information and maximum number of high amplitude level with identical sign to channel information give minimum reassurance in the CDM system using majority techniques. The minimum reassurance was calculated for all the levels of the tree when L/4 4C- log2L. For

L/4 > log2L, the minimum reassurance assumed to occur at least once in the distribution

n n i=lo

Eg2L C. /2n-1 of the total 2n unique

codewords of the correlator output. The value of reassurance for different values of n were found to be:

L 4 8 16 32 64 128 256 512 1024 2048

n 3 7 15 31 63 127 255 511 1023 2047

R 2 2 -2 -6 -18 -38 -82 -166 -388 -678

From the above results, it can be seen that the CDM system, using majority-logic techniques, works only when L=4 and 8, n=3 and 7. The rows can be truncated by 1, so that at the worst possible case the minimum reassurance reduces to 1. It is also clear that the CDM system using majority techniques breaks down for L=n or n+1 at n>7. In the following section a system which makes use of the codewords L=n=3 and L=n=7 was designed to achieve the value of L=n for large number of channels.

3. TCDM System

Let H21 denote the second order TCDM matrix of first degree, given by:


Li 1

• H21 0

The second order TCDM matrix of second degree is defined by the Kronecker products of H21 with itself, which yields:

H22

= —H21 H21..]

H21 -H21 = 0 [4.1

+1 0

0 +1 0 +1

+1 0 -1 0

+1 0 -1

The i-th order TCDM matrix of the first degree, Hil, is a square matrix of i rows and columns. The i-th order TCDM matrix of the k-th degree, Hik, is a square matrix of (ix2k-1) rows and columns, which is formed by the Kronecker products of Hi(k-1) with itself.

Consider a CDM system using a set of codewords given by higher degree TCDM matrix Hik with the first i's rows omitted; the transmitter signal S has a frequency distr-ibution of the modulud of the amplitude level given by the first L/4k code division tree levels providing the degree of the matrix is less than 6. Tice reason for this is that higher degree i-th order TCDM matrices are orthogonal matrices with columns of equal +1's and -1's excluding the first i columns.

Now consider a COM system which uses a set of codewords given by the third degree, i-th order TCDM matrix with the first i's rows omitted. The signal S has frequency distribution of the amplitude levels identical to the CDM system using a Hadamard matrix of three rows and four columns. Therefore a CDM system using majority techniques can employ a set of codewords given by the third degree i-th order TCDM matrix with first and last i rows and columns omitted respectively so as to achieve a minimum reassurance of +1 when L=n=3i where i=2,3

Similarly a CDM system using majority techniques employing a set of codewords given by fourth degree i-th order TCDM matrix with first and last i rows and columns omitted (called TCDM system) will achieve minimum reassurance of +1, when L=n=7i, where i=2,3,....

Computer programmes were written to simulate a TCDM system for L=n=6,9,12,14 and 15 bits. The results were found to

agree with the above analysis, which gave a minimum reassurance of +1.

Error correcting capabilities

Consider a TCDM system using a set of codewords given by the fourth order TCDM matrix of the third degree, with the first four rows omitted (matrix dimension (12x16), L=16, n=12). When the system is operating to maximum capacity, the 12 rows of the matrix provide carrier codewords for 12 channels. Such a system will have a reassurance of 2. When fewer channels are being used, the required number of codewords is generated by a specific linear addition of some of the rows in the matrix to form new codewords. The number of new codewords generated is equal to the number of channels in use. The rows to be added are chosen such that the resultant codewords are still in the tertiary form, i.e. contain only +1, 0, or -1 levels. The reassurance for channels having carrier codewords formed from linear summation of m rows obviously has the magnitude of 2m. The reassurance distribution in the channels was tabulated in Table (1).

A binary error in the received signal in the worst case will cause the reassurance to be reduced by 2. The number of random binary errors t which can be corrected and the number t of errors that can be detected for a given reassurance r, is given by:

t + t = r - 1

Since the carrier in each channel is made up of information digits interlaced with additional zeros in such a way that between each information digit and the next (i) zeros are added, where i is the order of the TCDM matrix, the TCDM system is capable of correcting ((r-1)/2) bursts of length i or less, where r is the reassurance of the system.

F5 (4)

Number of channels in use

n = 12

Carrier codewords used for each channel, number indicates the number of the rows

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

R= 2

12

R = 4 R = 6 R = 8

11 1+2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 10 1

10 1+2, 3+4, 5, 6, 7, 8, 9, 10, 11, 12 8 2

9 1+2, 3+4, 5+6, 7, 8, 9, 10, 11, 12 6 3

8 1+2. 3+4, 5+6, 7+8, 9, 10, 11, 12 4 4

7 1+2, 3+4, 5+6, 7+8, 9+10, 11, 12 2 5

6 1+2, 3+4, 5+6, 7+8, 9+10, 11+12 6

5 1+2+3+4, 5+6, 7+8, 9+10, 11+12 4 1 1+2+3, 4+5+6, 7+8, 9+10, 11+12 or 3 2

4 1+2+3+4, 5+6+7+8, 9+10, 11, 12 2 2 1+2+3, 4+5+6, 7+8+9, 10+11+12 or 4

3 1+2+3+4, 5+6+7+8, 9+10+11+12 3

Table 1 Number of channels with reassurance

References

1. Harmuth, H.F.: "Transmission of inform-ation by orthogonal functions". (Springer, 1970, 2nd. edition).

2. Judge, W.J.: "Multiplexing using quasi-orthogonal binary functions", Trans. Amer. Inst. Elect. Engineers, 1962, 81, part 1, pp. 81-83.

3. Titsworth, R.C.: "A Boolean-function-multiplexed-telemetry data system", IEEE Trans., 1963, SET-9, pp. 42-45.

4. Gordon, J.A. and Barrett, R.: "Correlation-recovered adaptive majority multiplexing", Proc. IEE, Vol. 118, No. 3/4, March/April 1971, pp. 417-422.

5. Gordon, J.A. and Barrett, R.: "On Coordinate Transformations and Digital Majority Multiplexing", Symp. Theory and Applications of Walsh Functions, Hatfield, June 1971.

6. Gordon, J.A. and Barrett, R.: "Digital Majority Logic-Multiplexer using Walsh Functions", IEEE Trans.,.1971, EMC-13, Special Edition on Walsh Function, Symp., Washington 1971.


Paper Included HASHIM, A. A. and A.G. CONSTANTINIDES "Digital Code Division Multiplexing" The proceedings of the International Zurich Seminar on Digital Communications, Zurich, March 1974.

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