On the Construction of Some Type II Codes over Z4×Z4

Designs, Codes and Cryptography, 30, 301–323, 2003

# 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.

On the Construction of Some Type II Codes over

Z46Z4

EDERLINA G. NOCON [email protected]

De La Salle University—Manila, 2401 Taft Avenue, Manila, Philippines

Communicated by: E. Bannai

Received May 4, 2001; Revised May 20, 2002; Accepted May 29, 2002

Abstract. We consider here the construction of Type II codes over the abelian group Z46Z4. The

definition of Type II codes here is based on the definitions introduced by Bannai [2]. The emphasis is given

on the construction of these types of codes over the abelian group Z46Z4 and in particular, the methods

applied by Gaborit [7] in the construction of codes over Z4 was extended to four different dualities with

their corresponding weight functions (maps assigning weights to the alphabets of the code). In order to do

this, we use the flattened form of the codes and construct binary codes analogous to the ones applied to Z4codes. Since each duality generates more than one weight function, we focus on those weights satisfying

the squareness property. Here, by the squareness property, we mean that the weight function wt assigns the

weight 0 to the Z46Z4 elements ð0; 0Þ; ð2; 2Þ and the weight 4 to the elements ð0; 2Þ and ð2; 0Þ. The mainresults of this paper are focused on the characterization of these codes and provide a method of

construction that can be applied in the generation of such codes whose weight functions satisfy the

squareness property.

Keywords: duality, Type II codes, self-dual codes, squareness property, flattened and deflattened form of

codes

AMS Classification: 11T71

1. Introduction

Let G be a finite abelian group. By a code over G of length n, we mean an additivesubgroup C of Gn ¼ G6G6 � � �6G (the direct product of n G’s). Each element u [Cis called a codeword of C.To define a self-dual code over a finite abelian group G, we consider a fixed

character table P ¼ ðwxðyÞÞ of G satisfying the condition that tP ¼ P. (The symbol tPhere refers to the transpose of the matrix P. The rows and columns of P are indexedby the elements of G arranged in some fixed order. Hence, the symbol wxðyÞ refers tothe xy-th entry of the matrix P with x; y [G). For a fixed character table P of theabelian group G, we define the inner product, hx; yi, of two elements x ¼ðx1; x2; . . . ; xnÞ; y ¼ ðy1; y2; . . . ; ynÞ of Gn as

hx; yi ¼Yni¼1

wxiðyiÞ; ð1Þ

and the dual, C\, of a subgroup C of Gn as the set

C\ ¼ fy [Gnjhx; yi ¼ 1; Vx [Cg: ð2Þ

We say that a code C is self-dual if C\ ¼ C.For the finite abelian group G of order g with a symmetric character table, P, we

say that a diagonal matrix

T ¼

t0 0 0 � � � 0

0 t1 0 � � � 0

..

.

0 0 0 � � � tg1

0BBBBBB@

1CCCCCCA; ð3Þ

has the modular invariance property if

ðPTÞ3 ¼ lIg; l 6¼ 0; ð4Þ

where Ig is the identity matrix of order g.Here, we remark that for each such G and fixed character table P (with tP ¼ P),

the solutions T of equation (4) are completely determined [3]. Also, if g is even, thenT can be expressed as

T ¼

Za0 0 0 � � � 0

0 Za1 0 � � � 0

..

.

0 0 0 � � � Zag1

0BBBBBB@

1CCCCCCA; ð5Þ

where Z ¼ e2pi ? 1=2l and l, the largest order of the elements of G. If g is odd, we cantake Z ¼ e2pi ? 1=l and get a similar expression for T. Here, we assume a0 ¼ 0.In defining Type II codes over a finite abelian group G of order g, we fix a

symmetric character table P of G and a solution T of equation (4) written in the form(5) with respect to this fixed P. For each a [G, we define the weight wtðaÞ ¼ aa. Also,for each element u ¼ ðuaÞ; a [G of Gn for some positive integer n, we define

wtðuÞ ¼Xa [G

wtðuaÞ:

The above given definition gives us the weight function which assigns a number wtðuÞto every word u of a code C � Gn. Here, we use the same notation ‘‘wt’’ to denote theweight of a component a [G and the weight of a word u [C � Gn. A code C (over afinite abelian group G) is said to be a Type II code if C is self-dual and if 2l jwtðuÞ forevery u [C with l as the largest order of the elements of G.

302 NOCON

It can easily be proved that for a finite abelian group G of order g and self-dualcode C � Gn, the number of codewords in C is given by

jCj ¼ffiffiffiffiffign

p: ð6Þ

Hence, for the case wherein G ¼ Z46Z4, we have jCj ¼ 4n.

2. The Dualities and Weight Functions of Z46Z4

We define here a duality map over any finite abelian group g to be an isomorphismC: x�Cx from g to its character group gg such that CxðyÞ ¼ CyðxÞ for any x; y [g.Hence, it can be said that a duality map is an isomorphism from g onto gg such that,if the elements of g and gg are arranged in rows and columns of the character tableaccording to the correspondence C, then we form a character table which issymmetric.Now, let d be the set of all duality maps over G ¼ Z46Z4 and s be the set of all

nondegenerate symmetric bilinear forms on G over the ring Z4. Define the mapf:s?d by

fðBÞxðyÞ ¼ zBðx;yÞ ¼ hx; yi;

for any B [s, where fðBÞ [d and z is a fixed primitive 4th root of unity. It is easy tosee that the following theorem holds:

THEOREM 2.1. There exists a one-to-one correspondence between the set of alldualities h ; i of G and the set of all nondegenerate symmetric bilinear forms on G overthe ring Z4.

Since for every symmetric bilinear form Bðx; yÞ, we can associate a quadratic formQðxÞ ¼ Bðx; xÞ, the duality maps of Z46Z4 can also be associated with quadraticforms over the ring Z4. In the following discussion, we identify all the distinctdualities of the abelian group Z46Z4 and group them via the equivalence ofquadratic forms associated with them.We now state the following theorem which will enable us to enumerate all the

distinct dualities of the abelian group G ¼ Z46Z4.

THEOREM 2.2. (Equivalence of Dualities over Z46Z4). Let Q1 and Q2 be the twoquadratic forms associated with the dualities h ; iQ1 and h ; iQ2 , respectively.Furthermore, let M be an element of SLð2;Z4Þ such that

Q2 ¼t MQ1M:

CONSTRUCTION OF SOME TYPE II CODES OVER Z46Z4 303

Then

hx; yiQ2 ¼ hpðxÞ; pðyÞiQ1 :

Proof. First we note that every matrixM ¼ ðmijÞ [SLð2;Z4Þ defines a permutationp of the elements of Z46Z4 by ðx1; x2Þ £? ðm11x1 þ m12x2;m21x1 þ m22x2Þ. Now,we consider the nondegenerate symmetric bilinear forms B1;B2 associated with thegiven quadratic forms Q1 and Q2. We then have

B2ðx; yÞ ¼ txB2y

¼ txðtMB1MÞy

¼ ðtxtMÞB1ðMyÞ

¼ tðMxÞB1ðMyÞ

¼ B1ðMx;MyÞ

¼ B1ðpðxÞ; pðyÞÞ:

The above arguments only show that the duality associated with the quadratic formQ1 is just the same as the duality associated with the quadratic form Q2. &

Based on the discussion above, the search for all the dualities of the abelian groupcan be reduced to the case of enumerating all inequivalent quadratic forms (or theirassociated bilinear forms) which can easily be enumerated via the matrices over Z4that represent them.The following result Et. Bannai et al. [5] gives us a way of classifying the dualities

of an abelian group G.

PROPOSITION 2.3. Let C be a duality map of a finite abelian group X and supposethat X is the direct product of X1 and X2. The following conditions are equivalent:

i. Cx1ðx2Þ ¼ 1 for any x1 [X1;x2 [X2.

ii. There exist duality maps Cð1Þ;Cð2Þ of X1 and X2, respectively, satisfying

Cx1þx2ðy1 þ y2Þ ¼ Cð1Þx1ðy1ÞCð2Þ

x2ðy2Þ;

for any x1; y1 [X1 and x2; y2 [X2.

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Definition 2.1. Let C be a duality map of a finite abelian group X, and suppose thatX is the direct product of X1 and X2. If one of the equivalent conditions ofProposition (2.3) is satisfied, we say that C splits over X16X2 and that C is thetensor product of duality maps Cð1Þ and Cð2Þ.

Based on Proposition (2.3), any duality of G can either be a non-tensor product or atensor product of some dualities of the components of G written as a direct productof cyclic groups. We then enumerate the complete listing of the dualities of Z46Z4based on this classification.

2.1. Listing of Nonequivalent Quadratic Forms and their Associated Dualities

We have seen from our discussion above that a given duality map over a finiteabelian group g generates a character table which is symmetric. In the succeedingdiscussion, we shall denote by P the symmetric character table associated with aduality map C [d of G ¼ Z46Z4 so that the ðx; yÞ-th entry of P is given byPðx; yÞ ¼ CxðyÞ. Here, the columns and rows of the matrix P are indexed by theelements of Z46Z4 given in the following order:

ð0; 0Þ; ð0; 1Þ; ð0; 2Þ; ð0; 3Þ; ð1; 0Þ; ð1; 1Þ; ð1; 2Þ; ð1; 3Þ;

ð2; 0Þ; ð2; 1Þ; ð2; 2Þ; ð2; 3Þ; ð3; 0Þ; ð3; 1Þ; ð3; 2Þ; ð3; 3Þ:

If the dualityC is associated with the nondegenerate symmetric bilinear form B, thenwe write the symbol P*B to express the existence of this equivalence. Now, by thedefinition of the ðx; yÞ-th entry of the character table P, we have

Pðx; yÞ ¼ CxðyÞ ¼ffiffiffiffiffiffiffi1

p Bðx;yÞ:

Based on the results obtained on the list of inequivalent nondegenerate symmetricbilinear forms over Z4, the nontensor product dualities are those generated by

b1 ¼1 0

0 1

!

and

b2 ¼1 0

0 3

!:

We shall call their corresponding symmetric character tables as P1 and P2,


respectively. The tensor product dualities are those generated by

b3 ¼0 1

1 0

!

and

b4 ¼2 1

1 2

!:

Their corresponding symmetric character tables will be denoted by P3 and P4,respectively.We then draw the following theorem as an enumeration of all the distinct dualities

of the abelian group Z46Z4. This listing was obtained from the computer algorithmthat the author formulated by using the Mathematica programming language.

THEOREM 2.4. There are exactly four symmetric character tables P1;P2;P3, and P4of Z46Z4 which are distinct up to permutations of columns and rows of the charactertables. For ðx1; x2Þ; ðy1; y2Þ [Z46Z4, we have

P1*Bðx; yÞ ¼ x1y1 þ x2y2; P2*Bðx; yÞ ¼ x1y1 þ 3x2y2;

P3*Bðx; yÞ ¼ x1y2 þ x2y1; P4*Bðx; yÞ ¼ 2x1y1 þ x1y2 þ x2y1 þ 2x2y2:

Note that this theorem also tells us about the four distinct dualities of Z46Z4associated with the four character tables P1;P2;P3, and P4.

3. The T Solutions of the P Matrices

In this section, we enumerate all the T solutions of each of the four duality matricesdescribed in Theorem (2.4). Here, we put emphasis on some of these solutions havingthe ‘‘squareness property’’ which gives a nice result that somehow generalizes thecharacterization of Type II codes over Z46Z4 for any given duality.

THEOREM 3.1. Let P be a symmetric matrix representing a fixed duality of an abeliangroup G of finite order. If T is a solution of the modular invariance equation

ðPTÞ3 ¼ kI ; k=0;

and if T 0 is a diagonal matrix obtained by permuting the rows and columns of Tsimultaneously, then there exists a duality P0 such that P*P0 and T 0 is a solution of theequation ðP0T 0Þ3 ¼ kI .

306 NOCON

Proof. We can write T 0 as

T 0 ¼ Q1TQ;

for some permutation matrix Q.Now, define P0 as

P0 ¼ Q1PQ:

Indeed, P0 and P are equivalent and

ðP0T 0Þ3 ¼ ððQ1PQÞðQ1TQÞÞ3

¼ ½Q1ðPTÞQ�3

¼ Q1ðPTÞ3Q

¼ Q1ðkIÞQ

¼ kI :&

THEOREM 3.2. Let P be a symmetric matrix representing a fixed duality of the groupZ46Z4 and T a solution of the modular invariance equation ðPTÞ3 ¼ kI written in theform

T ¼ DiagðZa0 ; Za1 ; . . . ; Za15Þ;

with a0 ¼ 0 and Z a fixed primitive 8th-root of unity. If T 0 is a diagonal matrix obtainedby changing the primitive 8th-root of unity Z to another primitive 8th-root of unity Z0,then the set of Type II codes that can be obtained with respect to P and T is equal to theset of Type II codes that can be obtained with respect to P and T 0.

Proof. Since thedualitydescribed in the theoremisfixed, then the setof self-dual codeswill be the same in both cases. We then intend to show that if a word w [ ðZ46Z4Þn hasweight divisible by 8 with respect to the weight assignment defined byT, then its weightwith respect to the weight assignment defined by T 0 is also divisible by 8.Note that if Z and Z0 are two distinct primitive 8th-roots of unity, then

Z0 ¼ Za; ð7Þ

with some a [ f3; 5; 7g. Also if equation (7) holds, then

Z ¼ ðZ0Þa: &

3.1. The Weight Assignments

The complete list of weight assignments can be generated by using the formuladerived by Bannai et al. [3]. The solutions T of the four dualities can be reduced by


applying Theorems (3.1) and (3.2). In this paper only one weight function is chosenfor each duality and these weight functions are those satisfying a property which theauthor described as the ‘‘squareness property.’’Here, each of the solutions take the form

DiagðZa0 ; Za1 ; Za2 ; . . . ; Za15Þ ¼ ða0; a1; a2; . . . ; a15Þ:

Thus, with the fixed indexing of the rows and columns of Pi based on the ordering

ð0; 0Þ; ð0; 1Þ; ð0; 2Þ; ð0; 3Þ; ð1; 0Þ; ð1; 1Þ; ð1; 2Þ; ð1; 3Þ;

ð2; 0Þ; ð2; 1Þ; ð2; 2Þ; ð2; 3Þ; ð3; 0Þ; ð3; 1Þ; ð3; 2Þ; ð3; 3Þ

the weight function wt endowed by a solution T is given by

wtð0; 0Þ ¼ a0 ¼ 0; wtð0; 1Þ ¼ a1; wtð0; 2Þ ¼ a2; . . . wtð3; 3Þ ¼ a15:

The chosen weight functions are listed below:

Tensor product dualities

Duality Weight Function

P1 0 1 4 1 1 2 5 2 4 5 0 5 1 2 5 2

P2 0 3 4 3 1 4 5 4 4 7 0 7 1 4 5 4

Non-tensor product dualities

Duality Weight Function

P3 0 2 4 6 2 6 2 6 4 2 0 6 6 6 6 6

P4 0 4 4 0 4 2 4 2 4 4 0 0 0 2 0 2

We note here that the evaluation of the weight of each of the elements of Z46Z4 iscomputed modulo 8. This will be considered all throughout the paper.

4. The Construction Theorems

We shall consider any of the dualities P1;P2;P3;P4 as described in the precedingdiscussion. We use the symbol h ; i to describe any of these four dualities.Every self-dual code c over Z46Z4 of length n may be classified as a code of type

4k12k2 for some integers k1; k2 with 2k1 þ k2 ¼ 2n. If c is generated by some set Mcontaining noneven words (words containing at least one odd component) and evenwords (words whose components are all even), then k1 and k2 may be viewed as thenumber of noneven and even words in the generator set M, respectively.Since a code c over Z46Z4 is an abelian group itself, then c can be written as a

direct product of the form c¼ ho1i6ho2i6 � � �6hok1i6he1i6he2i6 � � �6hek2i.

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Hence, c here is a code of type 4k12k2 and the transpose of the matrix

tM ¼ o1 o2 . . . ok1 e1 e2 . . . ek2ð Þ;

will be called a generator matrix of code c. Indeed, for any k1 þ k2-tuplec ¼ ðc1; c2; . . . ; ck1þk2Þ [Zk1þk2

4 , the word W descibed by

W ¼ cM ¼ ðc1; c2; . . . ; ck1þk2Þ

o11 o12 o13 . . . o1n

o21 o22 o23 . . . o2n

..

. ... ..

. ... ..

.

ok11 ok12 ok13 . . . ok1n

e11 e12 e13 . . . e1n

e21 e22 e23 . . . e2n

..

. ... ..

. ... ..

.

ek21 ek22 ek23 . . . ek2n

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA

;

is a codeword in c.Note that in describing a code c over Z46Z4, we can write its generator matrixM

in such a way that the first k1 rows are the noneven generator words and the last k2rows are the even generator words of c.If x ¼ ðx1; x2Þ; ðx3; x4Þ; . . . ; ðxn1; xnÞð Þ is a codeword of length n=2, we define the

flattened form of x by

flat½x� ¼ ðx1; x2; x3; x4; . . . ; xn1; xnÞ:

Hence, a code over Z46Z4 of length n=2 (n is even) generates a code over Z4 oflength n. Note that the action of the map flat can also be reversed so that if c0 is acode over Z4 and x0 ¼ ðx1; x2; x3; x4; . . . ; xn1; xnÞ [ c

0, then the function

deflat½x0� ¼ ðx1; x2Þ; ðx3; x4Þ; . . . ; ðxn1; xnÞð Þ

produces a code over Z46Z4 of length n=2.By way of introducing the flat function, we can then standardize the generator

matrix of a code c over Z46Z4, that is, each code c over Z46Z4 has a ‘‘flattened’’generator matrix of the form

Ik1 A B

0 2Ik2 2C

!;

where A;B are matrices over Z4 and C is f0; 1g-matrix. The code c defined by thismatrix is then of type 4k12k2 .


4.1. A Characterization of Self-dual Codes over Z46Z4

Any code c over Z46Z4 can be associated with two codes C1 and C2 over Z26Z2.In fact, the flattened form c

0 of c can be associated with two binary codes C01 and C0

2

which are the flattened forms of C1 and C2. To show this, we consider the ‘‘modulo2’’ map c1 : Z4?Z2 which assigns 1 and 3 to 1 and 0 and 2 to 0. Note that the kernelof c1 is isomorphic to Z2 via the isomorphism c2 which maps 0 to 0 and 2 to 1. Thetwo maps c1 and c2 are then the maps that we can use in generating the binary codesC01 and C0

2 associated with c.

THEOREM 4.1. Let c be a code over Z46Z4 with flattened form c0. Then there are

two binary codes C01 and C0

2 associated with c given by

C01 ¼ fc1ðuÞ j u [c0g; ð8Þ

C02 ¼ fc2ðuÞ j u [c;c1ðuÞ ¼ 0g: ð9Þ

If c is self-orthogonal with respect to any fixed duality h ; i, then C01 is a self-

orthogonal doubly-even binary code and C01(C0

2(C01\. Furthermore, if c is self-dual,

then C02 ¼ C0

1\.

In the following discussions, we use the symbol ? to denote the ‘‘inner product’’of two words a ¼ ðða1; a2Þ; ða3; a4Þ; . . . ; ða2n1; a2nÞÞ; b ¼ ððb1; b2Þ; ðb3; b4Þ; . . . ;ðb2n1; b2nÞÞ in ðZ46Z4Þn associated with the nondegenerate symmetric bilinearform B (and hence, with the character table P linked with some duality h ; i. We usethis symbol to refer to any of the inner products corresponding to P1, P2, P3, and P4and the context of its use will then depend on the specified duality.Thus, the product a ? b is computedfor P1 as

a ? b ¼ ða1b1 þ a2b2Þ þ ða3b3 þ a4b4Þ þ � � � þ ða2n1b2n1 þ a2nb2nÞ;

for P2 as

a ? b ¼ ða1b1 þ 3a2b2Þ þ ða3b3 þ 3a4b4Þ þ � � � þ ða2n1b2n1 þ 3a2nb2nÞ;

for P3 as

a ? b ¼ ða1b2 þ a2b1Þ þ ða3b4 þ a4b3Þ þ � � � þ ða2n1b2n þ a2nb2n1Þ;

and for P4 as

a ? b ¼ð2a1b1 þ a1b2 þ a2b1 þ 2a2b2Þ þ ð2a3b3 þ a3b4 þ a4b3 þ 2a4b4Þ

þ � � � þ ð2a2n1b2n1 þ a2n1b2n þ a2nb2n1 þ 2a2nb2nÞ:

Note that the effect of c1 and c2 is similar to the construction of residue andtorsion codes in the case of codes over the ring Z4. Here, the doubly-evenness of the

310 NOCON

corresponding binary codes depend on the specified duality. But in general, abinary code c associated with a code c is doubly-even if for every word y [c;y ? y:0ðmod 4Þ where the operation ? is taken based on the context of the specifiedduality of the code c. For example, if the duality concerned is the one associatedwith P2, then we evaluate for each y ¼ y1y2; . . . ; y2n1y2n in c the value of y ? y by

y ? y ¼ ðy21 þ 3y22Þ þ � � � þ ðy22n1 þ 3y22nÞ:

If all such words y [c satisfy the condition y ? y ¼ 0 ðmod4Þ, then it can be said thatc is doubly-even with respect to the duality associated with P2.The next theorem enables us to construct codes over Z46Z4 by starting with two

binary codes C01 and C0

2 that are related such that C01(C0

2.

THEOREM 4.2. If C01 and C0

2 are binary codes of length 2n with C01(C0

2, then there isa code c over Z46Z4 associated with these two binary codes. Moreover, if C0

1 is self-orthogonal and doubly even with respect to a duality h ; i and C0

2(C01\, then there is a

self-orthogonal code c over Z46Z4. Also, if C02 ¼ C0

1\, then c is self-dual.

Theorems (4.1) and (4.2) suggest a way of identifying the self-dual codes of Z46Z4by way of working on their flattened forms. In order to enumerate all the Type IIcodes over Z46Z4 with duality h ; i, all the possible binary codes C0

1 must beidentified. Then, the binary code C0

2 is computed based on the property thatC02 ¼ C0

1\. It should be noted that each code C0

1 generates one or more self-dualcodes over Z46Z4 and hence, to require these codes to be Type II, the correspondingweight assignments must be considered. The proofs of Theorems (4.1) and (4.2) arediscussed in a detailed manner in the paper Nocon [8].Note that if the weight function chosen for any given duality has the property that

wtð0; 0Þ ¼ 0; wtð0; 2Þ ¼ 4; wtð2; 0Þ ¼ 4; wtð2; 2Þ ¼ 0; ð�Þ

then the following hold:

THEOREM 4.3. If the weight assignment of duality h ; i has the property ð�Þ, then anyType II code c over Z46Z4 with respect to this duality and weight function contains aword whose components are from the set fð1; 1Þ; ð1; 3Þ; ð3; 1Þ; ð3; 3Þg.

Proof. It is sufficient to show that the binary code C01 that can be derived from c

contains the all 1’s word 1 ¼ f1; 1; 1; 1; . . . ; 1; 1g. If the weight function has propertyð�Þ, then every word x [C0

2 must contain an even number of 1’s. Note that the word 1must be in C0

2? since for any word x [C0

2,

1 ? x:0 ðmod2Þ:

But by Theorem (4.1), C01 ¼ C0

2? and hence, 1 [C0

1. &

Note. From hereon, we shall name property ð�Þ as the ‘‘squareness property’’.


Since Type II codes over Z46Z4 contain words whose components are among ð1; 1Þ,ð1; 3Þ, ð3; 1Þ and ð3; 3Þ, it will be important for us to consider the weights of thesecomponents in the choice of weight assignments. In this paper, we limit ourselves tothe weight functions corresponding to the four dualities P1; P2; P3 and P4 as listed in3.1.

4.2. Constructing Type II Codes over Z46Z4

The following theorems give a method of identifying among the self-dual codes (withrespect to the duality h ; i) those that are Type II. The weight functions chosen foreach duality here are those that satisfy the squareness property.For a code c over Z46Z4 of length n=2, let C0

1 and C02 be the associated binary

codes. Note that if x [C01, then there exists at least one element x þ 2y in the code c0

which is the flattened form of c. Also, if x [C02, then 2x must be in c

0.Now, we consider all such y associated with an element x [C0

1 and define the mapr:C0

1? Fn2jC0

2. Here,

x £? fy [ Fn2jx þ 2y [c0g;

and we set

c0 ¼ fx þ 2yjx [C0

1; y [ rðxÞg:

Then if c0 has generator matrix

Ik1 A B1 þ 2B20 2Ik2 2C

!;

where Ik1 and Ik2 are the identity matrices of order k1 and k2, respectively, A;B1;B2,and C are all matrices over f0; 1g, then C0

1 has generator matrix

Ik1 A B1ð Þ;

and C02 has generator matrix

Ik1 A B1

0 Ik2 C

!:

Indeed, the function r characterizes the f0; 1g-matrix B2. Thus, every code c can beassociated with the triple ðC0

1;C02; rÞ via its flattened form c

0 where C01 and C0

2 are theassociated binary codes and r is the map described above. Also, every such tripleðC01;C

02; rÞ can define a code c over Z46Z4. We then draw the following theorem.

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THEOREM 4.4. Let C01 and C0

2 be two binary codes such that C01(C0

2 and r be thefunction r : C0

1? Fn2jC0

2 defined by

x £?rðxÞ ¼ fy [ Fn2 j x þ 2y [c0g;

such that c0 ¼ fx þ 2y j x [C01; y [ rðxÞg. If the code c over Z46Z4 is the deflattened

form of c0, then the correspondence f between c and the triple ðC0

1;C02; rÞ is a

bijection.

It can be said that this theorem is analogous to the theorem drawn by Gaborit [7] inhis paper wherein every self-dual code over Z4 can be associated with two binarycodes with similar properties. However, in this case, we need to consider first theflattened form of the code over Z46Z4 and construct the corresponding binarycodes. Based on this theorem, we can construct self-dual codes over Z46Z4 fromtwo binary codes C0

1 and C02 satisfying the condition that C0

1 is self-orthogonal andC01 ¼ C0

2?.

LEMMA 4.5. Let B1 ¼ fe1; e2; . . . ; ek1g be a basis of C01. Then, the set fe�1 þ C0

2; e�2 þ

C02; . . . ; e

�k1þ C0

2g with the condition ei ? e�j ¼ dij forms a basis for Fn

2jC02.

Proof. The basis B1 of C01 can be extended to form a basis of the vector space F

n2.

Now, since C01 ¼ C0

2\, the set fe�k1þ1; e

�k2þ2; . . . ; e

�ng forms a basis for C0

2.Consequently, the set Fn

2jC02 is generated by fe�1 þ C0

2; e�2 þ C0

2; . . . ; e�k1þ C0

2g. &

In the following theorem we use the one-to-one correspondence between the tripleðC01;C

02; rÞ and the code c over Z46Z4.

THEOREM 4.6. Let the triple c ¼ ðC01;C

02; rÞ be a self-dual code over Z46Z4 and let

M ¼ ðmijÞ, with 1 � i; j � k, be the matrix representation of the function r withrespect to the basis fe1; e2; . . . ; ek1g and fe�1; e�2; . . . ; e�k1g such that ei ? e

�j ¼ dij, then the

code c is self-dual if and only if the following conditions are satisfied

i. C01 is doubly even (with respect to any of the four dualities of Z46Z4);

ii. C02 ¼ C0

1?;

iii. ei ? ej:2ðmij þ mjiÞ ðmod 4Þ; 1 � i; j � k1:

Proof. Conditions (i) and (ii) were shown to be true in Theorem (4.1). Note that forany yi ¼ f ðeiÞ [ rðeiÞ and yj ¼ f ðejÞ [ rðejÞ, the words

ei þ 2f ðeiÞ ¼ ei þ 2Xk1p¼1

mipe�p


and

ej þ 2f ðejÞ ¼ ej þ 2Xk1s¼5

mjse�s

are in the code c0. Thus, c is self-dual if and only if for any 1 � i; j � k1

ðei þ 2f ðeiÞÞ ? ðej þ 2f ðejÞÞ ¼ ei ? ej þ ei ? 2Xk3s¼1

mjse�s þ ej ? 2

Xk1p¼1

mipe�p

: 0 ðmod 4Þ:

Hence,

ei ? ej : 2 ei ?X

mjse�s þ ej ?

Xmipe

�p

� ðmod 4Þ

: 2ðmji þ mijÞ ðmod 4Þ: &

In the following discussions, we use the symbols c1 and c2 to denote the two binarycodes associated with the given code c over Z46Z4. Moreover, for the discussion ofTheorems (4.7) through (4.8) we let fe1; e2; . . . ; ek1g be a basis of c1 and M ¼ ðmijÞ,with 1 � i; j � k, be the matrix representation of the function r with respect to thebasis fe1; e2; . . . ; ek1g and fe�1; e�2; . . . ; e�k1g such that ei ? e

�j ¼ dij.

We include here more detailed discussions on choosing the suitable M matricesthat can be attached to a self-orthogonal binary code c1 with respect to a chosenduality h ; i.

4.2.1. The Tensor Product Cases

THEOREM 4.7. Let c be a self-dual code over Z46Z4. Then c is a Type II code withrespect to duality h ; i (P1 or P2) and weight function wt with the squareness property ifand only if the following conditions hold:

i. c2 ¼ c?1

ii. ei ? ej:2ðmij þ mjiÞ ðmod 4Þ V1 � i; j � k,

iii. c1 is a doubly even binary code (that is, x ? x:0 ðmod 4Þ; Vx [c1 with respect tothe specified duality) containing 1.

iv. ei ? ei þ 4ðmii þ mi1Þ:0 ðmod 8Þ; V2 � i � k1.

314 NOCON

Proof. We have seen from the previous theorems the equivalence between the self-duality of a code c over Z46Z4 (whose flattened form is associated with the binarycodes c1 and c2) and the three conditions (i), (ii), (iii). Hence, we will just considerthe weight conditions for each duality h ; ii in order to prove the theorem. Also,since the word 1 ¼ ð1; 1; 1; 1; . . . ; 1; 1Þ must be in c1, we may set e1 ¼ 1.If e�1 ¼ ða�1; a�2; a�3; a�4; . . . ; a�n1; a�nÞ, then it can easily be seen that for any of the

tensor product cases, e�1 has odd weight and e�j has even weight for all j=1. Now, weconsider each of these two cases:

Case 1. h ; i1

Here, hðx1; x2Þ; ðy1; y2Þi ¼ffiffiffiffiffiffiffi1

p x1y1þx2y2. Note that the weight function

corresponding to h ; i1 assigns to each component ðx1; x2Þ [Z46Z4, the weightwtðx1; x2Þ ¼ x21 þ x22 ðmod 8Þ.Now, for i=1, the word xi ¼ deflat½wi� [c with wi ¼ ei þ 2f ðeiÞ and

ei ¼ ða1; a2; a3; a4; . . . ; an1; anÞ and f ðeiÞ ¼ ðb1; b2; b3; b4; . . . ; bn1; bnÞ;

must have weight divisible by 8.Thus,

wtðxiÞ ¼ wi ?wi ¼ ½ei þ 2f ðeiÞ� ? ½ei þ 2f ðeiÞ�

¼ ei ? ei þ 4ei ? f ðeiÞ þ 4f ðeiÞ ? f ðeiÞ

¼ ei ? ei þ 4mii þ 4Xk1j¼2

mije�j ? e

�j þ 2

X1�s<t�k1

mismite�s ? e

�t

!

:ei ? ei þ 4mii þ 4Xk1j¼1

mije�j ? e

�j ðmod 8Þ:

Hence,

0:ei ? ei þ 4mii þ 4mij ðmod 8Þ: ð10Þ

Now, for any word X ¼Pk1

i¼1 yixi [c, where yi [Z4, we show that wtðXÞ is divisibleby 8 if and only if (10) holds, i.e., c is Type II if and only if the conditions mentionedin the theorem are all satisfied. Here, we use the following notations (which we will


also use in the succeeding discussions).

e1 ¼ ða11; a12; a13; a14; . . . ; a1n1; a1nÞ

¼ ð1; 1; 1; 1; . . . ; 1; 1Þ

e2 ¼ ða21; a22; a23; a24; . . . ; a2n1; a2nÞ

. ..

ei ¼ ðai1; ai2; ai3; ai4; . . . ; ain1; ainÞ

..

.

ek1 ¼ ðak11; ak12; ak13; ak14; . . . ; ak1n1; ak1nÞ;

f ðe1Þ ¼ ðb11; b12; b13; b14; . . . ; b1n1; b1nÞ

f ðe2Þ ¼ ðb21; b22; b23; b24; . . . ; b2n1; b2nÞ

..

.

f ðeiÞ ¼ ðbi1; bi2; bi3; bi4; . . . ; bin1; binÞ

f ðek1Þ ¼ ðbk11; bk12; bk13; bk14; . . . ; bk1n1; bk1nÞ;

and for each ei; 1 � i � k1, we take e�i ¼ ðci1; ci2; ci3; ci4; . . . ; cin1; cinÞ so that ei ? e�j ¼

dij for all 1 � i; j � k1.

wtðXÞ ¼Xk1i¼1

yiwi ?Xk1j¼1

yjwj

¼Xk1i¼1

y2i ðwi ?wiÞ þ 2X

1�i<j�k1

yiyjðwi ?wjÞ

¼Xk1i¼1

y2i ðei ? ei þ 4ei ? f ðeiÞ þ 4f ðeiÞ ? f ðeiÞÞ

þ 2X

1�i<j�k1

yiyjðei ? ej þ 2ei ? f ðejÞ þ 2ej ? f ðeiÞ þ 4f ðeiÞ ? f ðejÞÞ

316 NOCON

:Xk1i¼1

y2i ðei ? ei þ 4mii þ 4mi1Þ

þ 2X

1�i<j�k1

yiyjðei ? ej þ 2ðmij þ mjiÞÞ ðmod 8Þ:

We see from the last expression that the weight of any codeword X in c (generatedby the xi’s), is divisible by 8 if and only if conditions (ii) and (iv) of Theorem (4.7) aresatisfied.

Case 2. h ; i2

For this duality,

hðx1; x2Þ; ðy1; y2Þi ¼ffiffiffiffiffiffiffi1

p x1y1þ3x2y2:

Also, wtðx1; x2Þ ¼ x21 þ 3x22 ðmod 8Þ Vx ¼ ðx1; x2Þ [Z46Z4.The word xi ¼ deflat½ei þ 2f ðeiÞ� [c has weight divisible by 8 and thus,

wtðxiÞ ¼Xn=2r¼1

ðai2r1 þ 2bi2r1Þ2 þ 3ðai2r þ 2bi2rÞ2h i

¼Xn=2r¼1

ða2i2r1 þ 3a2i2rÞ þ 4Xn=2r¼1

ðai2r1bi2r1 þ 3ai2rbi2rÞ

þ 4Xn=2r¼1

ðb2i2r1 þ 3b2i2rÞ

¼ ei ? ei þ 4ei ? f ðeiÞ þ 4f ðeiÞ ? f ðeiÞ

¼ ei ? ei þ 4mii þ 4Xk1j¼1

mije�j ? e

�j þ 2

X1�s<t�k1

mismite�s ? e

�t

!

¼ ei ? ei þ 4mii þ 4mi1 ðmod 8Þ

:0 ðmod 8Þ:

Again, for any word X ¼Pk1

i¼1 yixi [c, we compute the value of wtðXÞ and requirethis to be divisible by 8.


Now,

wtðXÞ ¼Xn=2j¼1

Xk1i¼1

yi ai2j1 þ 2bi2j1� �" #2

þ3Xk1i¼1

yi ai2j þ 2bi2j

� �" #28<:

9=;

:Xk1i¼1

y2i

Xn=2j¼1

a2i2j1 þ 3a2i2j�

þ 4 ai2j1bi2j1 þ 3ai2jbi2j

� �h(

þ 4 b2i2j1 þ 3b2i2j� io

þ 2X

1�i<l�k1

yiyl

Xn=2j¼1

ai2j1al2j1 þ 3ai2jal2j

� ��(

þ 2 ai2j1bl2j1 þ 3ai2jbl2j

� �þ 2 al2j1bi2j1 þ 3al2jbi2j

� ��)ðmod 8Þ

:Xk1i¼1

y2i ei ? ei þ 4ei ? f ðeiÞ þ 4f ðeiÞ ? f ðeiÞ½ �

þ 2X

1�i<l�k1

yiyl½ei ? el þ 2mil þ 2mli� ðmod 8Þ

:Xk1i¼1

y2i ½ei ? ei þ 4mii þ 4mi1�

þ 2X

1�i<l�k1

yiyl½ei ? el þ 2mil þ 2mli� ðmod 8Þ:

The last expression tells us that the weight of the codeword X is divisible by 8 if andonly if conditions (ii) and (iv) of this theorem are satisfied. &

4.2.2. The Non-Tensor Product Cases

In the following discussion, we make use of the symbol oðaÞ to denote the number of1’s in the binary word a ¼ ða1; a2; . . . ; anÞ. Thus, oðaÞ ¼

Pni¼1 ai.

THEOREM 4.8. Let c be a self-dual code over Z46Z4. Then c is a Type II code withrespect to duality h ; i (P3 or P4) and weight function wt with the squareness property ifand only if the following conditions hold:

i. c2 ¼ c?1

ii. ei ? ej:2ðmij þ mjiÞ ðmod 4Þ V1 � i; j � k,

iii. c1 is a doubly even binary code (that is, x ? x:0 ðmod 4Þ; Vx [c1 with respect tothe specified duality) containing 1.

318 NOCON

iv. ei ? ei þ 4mii þ 4mi1 þ 2oðeiÞ:0 ðmod 8Þ; 2 � i � k1.

Proof. With respect to the notations used in the previous theorem, it is clear that e�1has odd weight and e�j has even weight for all j=1 for any of the two non-tensorproduct cases.

Case 1. h ; i3

For this duality, for any two components x ¼ ðx1; x2Þ; y ¼ ðy1; y2Þ [Z46Z4


p x1y2þx2y1;

and the weight of x is given by

wtðxÞ ¼ 2ðx1 þ x2 þ x1x2Þ ðmod 8Þ:

Here, we also use the same notations in the proof of Theorem (4.7).

wtðxiÞ ¼Xn=2r¼12 ðai2r1 þ 2bi2r1Þ þ ðai2r þ 2bi2rÞ þ ðai2r1 þ 2bi2r1Þðai2r þ 2bi2rÞ½ �

:2Xn=2r¼1

ðai2r1 þ ai2rÞ þ 4Xn=2r¼1

ðbi2r1 þ bi2rÞ þ 2Xn=2r¼1

ðai2r1ai2rÞ

þ 4Xn=2r¼1

ðai2r1bi2r þ ai2rbi2r1Þ ðmod 8Þ

:2oðeiÞ þ 4o f ðeiÞ½ � þ ei ? ei þ 4ei ? f ðeiÞ ðmod 8Þ

:ei ? ei þ 4mii þ 4mi1 þ 2oðeiÞ ðmod 8Þ:

Here, we consider the fact that o f ðeiÞ½ �:mi1 ðmod 2Þ.Thus, if each xi ¼ deflat½ei þ 2f ðeiÞ� has weight divisible by 8, we have

ei ? ei þ 4mii þ 4mi1 þ 2oðeiÞ:0 ðmod 8Þ:

We also note that for any xi ¼ deflat½ei þ 2f ðeiÞ�; 1 � i � k1 and any y [Z4, wehave

wtðyxiÞ ¼ wt y ðeiÞ þ 2f ðeiÞð Þð Þ

:y 2oðeiÞ þ 4mi1ð Þ þ y2ðei ? ei þ 4miiÞ ðmod 8Þ:


Thus,

wtðyxiÞ:

yð2oðeiÞ þ ei ? ei þ 4mi1 þ 4miiÞ ðmod 8Þ if y ¼ 0; 1;

6oðeiÞ þ ei ? ei þ 4mi1 þ 4mii ðmod 8Þ if y ¼ 3;

4oðeiÞ ðmod 8Þ if y ¼ 2:

8>><>>:

We see that if ei ? ei þ 2oðeiÞ þ 4mi1 þ 4mii:0 ðmod 8Þ, then for any y [Z4 and1 � i � k1,

wtðyxiÞ:y 2oðeiÞ þ 4mi1ð Þ þ y2ðei ? ei þ 4miiÞ ðmod 8Þ

:0 ðmod 8Þ:

Now, for any word X ¼Pk1

i¼1 yixi [c, where xi ¼ deflat½wi� [c with wi ¼ ei þ 2f ðeiÞand yi [Z4, we have

wtðXÞ ¼ 2Xn=2j¼1

Xk1i¼1

yiðai2j1 þ 2bi2j1Þ þXk1i¼1

yiðai2j þ 2bi2jÞ"

þXk1i¼1

yiðai2j1 þ 2bi2j1Þ ! Xk1

p¼1ypðap2j þ 2bp2jÞ

!#

:Xk1i¼1

yi 2Xn

j¼1aij

" #þ 4

Xn

j¼1bij

" # !þ 2

Xk1i¼1

y2i

Xn=2j¼1

ðai2j1ai2j þ 2ai2j1bi2j

þ 2ai2jbi2j1Þ þ 2X

1�i<l�k1

yiyl

Xn=2j¼1

ðai2j1al2j þ ai2jal2j1Þ�

þ 2ðai2j1bl2j þ ai2jbl2j1Þ þ 2ðal2jbi2j1 þ al2j1bi2jÞ�ðmod 8Þ

:Xk1i¼1

yi 2oðeiÞ þ 4o½f ðeiÞ�ð Þ þXk1i¼1

y2i ei ? ei þ 4ei ? f ðeiÞð Þ

þ 2X

1�i<l�k1

yiyl ei ? el þ 2ei ? f ðelÞ þ 2el ? f ðeiÞð Þ ðmod 8Þ:

320 NOCON

The last expression tells us that wtðXÞ:0 ðmod 8Þ for any codeword X [ \cal c if andonly if conditions (ii) and (iv) of this theorem are satisfied.

Case 2. h ; i4

Here, for x ¼ ðx1; x2Þ; y ¼ ðy1; y2Þ [Z46Z4


p 2x1y1þx1y2þx2y1þ2x2y2

and

wtðx1; x2Þ ¼ 2ðx21 þ x22 þ x1 þ x2 þ x1x2Þ ðmod 8Þ:

For xi ¼ deflat½ei þ 2f ðeiÞ�,

wtðxiÞ ¼Xn=2r¼12 ðai2r1 þ 2bi2r1Þ2 þ ðai2r þ 2bi2rÞ2 þ ðai2r1 þ 2bi2r1Þh

þðai2r þ 2bi2rÞ þ ðai2r1 þ 2bi2r1Þðai2r þ 2bi2rÞ�

:2Xn

j¼1aij þ 4

Xn

j¼1bij þ

Xn=2r¼12ða2i2r1 þ ai2r1ai2r þ a2i2rÞ

þ 4Xn=2r¼1

ð2ai2r1bi2r1 þ ai2r1bi2r þ ai2rbi2r1 þ 2ai2rbi2rÞ ðmod 8Þ

:2oðeiÞ þ 4oð f ðeiÞÞ þ ei ? ei þ 4ei ? f ðeiÞ ðmod 8Þ

:2oðeiÞ þ 4mi1 þ ei ? ei þ 4mii ðmod 8Þ:

Thus, wtðxÞ is divisible by 8 if and only if ei ? ei þ 4mii þ 4mi1 þ 2oðeiÞ:0 ðmod 8Þ.It can be shown that for any xi ¼ deflat½ei þ 2f ðeiÞ�; 1 � i � k1 and y [Z4,

wtðyxiÞ:y 2oðeiÞ þ 4mi1ð Þ þ y2ðei ? ei þ 4miiÞ ðmod 8Þ

:0 ðmod 8Þ:


Again, for any word X ¼Pk1

i¼1 yixi [c, where 1 � i � k1; yi [Z4,

wtðXÞ ¼ 2Xn=2j¼1

Xk1i¼1

yiðai2j1 þ 2bi2j1Þ !2

þXk1i¼1

yiðai2j þ 2bi2jÞ !22

4

þXk1i¼1

yiðai2j1 þ 2bi2j1Þ þXk1i¼1

yiðai2j þ 2bi2jÞ

þXk1i¼1

yiðai2j1 þ 2bi2j1Þ ! Xk1

p¼1ypðap2j þ 2bp2jÞ

!#

:Xk1i¼1

yi 2Xn

j¼1aij þ 4

Xn

j¼1bij

!þXk1i¼1

y2i

Xn=2j¼12ða2i2j1 þ ai2j1ai2j þ a2i2jÞ

"

þ 4Xn=2j¼1

ð2ai2j1bi2j1 þ ai2j1bi2j þ ai2jbi2j1 þ 2ai2jbi2jÞ#

þ 2X

1�i<l�k1

yiy2Xn=2j¼1

ð2ai2j1al2j1 þ ai2j1al2j þ ai2jal2j1 þ 2ai2jal2jÞ"

þ 2Xn=2j¼1

ð2ai2j1bl2j1 þ ai2j1bl2j þ ai2jbl2j1 þ 2ai2jbl2jÞ

þXn=2j¼1

ð2al2j1bi2j1 þ al2j1bi2j þ al2jbi2j1 þ 2al2jbi2jÞ!#

ðmod 8Þ

:Xk1i¼1

yi 2oðeiÞ þ 4mi1ð Þ þXk1i¼1

y2i ðei ? ei þ 4miiÞ

þ 2X

1�i<l�k1

ei ? el þ 2ðmil þ mliÞð Þ ðmod 8Þ:

Thus, we see that X has weight divisible by 8 if and only if conditions (ii) and (iv) aresatisfied. &

Remark 4.1. We see from the last two theorems that for a given binary code c1 ofdimension k1 satisfying all the required conditions for the self-duality and Type IIproperty of the associated code c ¼ ðc1;c2; rÞ over Z46Z4, the number of

322 NOCON

functions r with matrix representation M is given by

2ðk21þk1Þ=2; if c is self-dual and

2ðk21k1Þ=2þ1; if c is Type II:

Acknowledgments

The author is greatly indebted to Prof. Eiichi Bannai, Prof. Etsuko Bannai, Dr.Philippe Gaborit, and Dr. Patrick Sole for their guidance and encouragement. Theirwritten papers as experts in the field of coding theory indeed helped the author in herresearch endeavor.

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models, J. of Combinatorics, Vol. 8 (1998) pp. 223–233.

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On the Construction of Some Type II Codes over Z4×Z4

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