Orthogonal geometries over finite fields with characteristic ≠2 and block designs

Vol.9 No.3 ACTA M A T H E M A T I C A E A P P L I C A T A E SINICA July, 1993

ORTHOGONAL GEOMETRIES

OVER FINITE FIELDS WITH

CHARACTERISTIC =/=2 AND BLOCK DESIGNS*

YANC BENFU (~ ~,-~-)

(Teacher-Training College of Chengdu, Pengxian County 611930, China)

wv.i WAND~ ( ~ :~ ~_)

(Sichuan University, Chengdu 610064, China)

A b s t r a c t

By taking as blocks certain subspace-pairs of an orthogonal geometry over a finite field with characteristic ~ 2 we construct some new types of BIB designs and PBIB designs whose parameters are also given.

1. Introduction

Let q be an odd prime power, and F, t the finite field with q elements. Denote by OV,~(Fq) the n-dimensional or thogonal geometry over F,l, and by O,,(F,I) the o r thogona l group of degree n over F,t.

Dal and Feng [1] give 'a formula for the number (denoted by N ( m + r, 2s + r, s, F; n)) of (m + r, 2s + r, s, F)- type subspaces in OV,,(F,~). Wan{ 7'°] gives a formula for the number (denoted by N ( r n l + h , 2s~ +r~,s~,rl ; m + r 2 s + r , s, r; ~))of (ml +r l , 2s+ rl, s~,r~)- type subspace included in a given (rn + r, 2s + r, s, F)- type subspace in OV,,(F,j). Wan et al. [t-13] also s tudy the transit ivity of O,,(F,~) on some sets of subspaces in OV,,(F,~) as well as on some sets of subspace-pairs. On the basis of these results, they cons t ruc t a number of BIB designs and PBIB designs by suitably taking some subspaces as t rea tments and taking some subspaces as blocks, and give the parameters of these designs. In the present paper , based on the conjugation relationship between the subspaces in OVr,(F~), which has been investigated by the anthors [11], we shall take as blocks some subspace pairs and then cons t ruc t new types of BIB designs and PBIB designs, and also give their parameters .

The concepts and notat ions used but not defined are adopted from [11].

2. BIB Designs

T h r o u g h o u t this section, let n = 2v q-8, v = 1, and 8 = 2. Tha t is, we will conduct our

*Received June 12, 1990.

270 ACTA MATHEMATICAE APPLICATAE SINICA Vol.9

discussion in OV4(Fq) defined by

$ 4 = 0 1

Z

We first prove Theorem 1. Taking as treatments the (I, 0, 0)-type subspaces in OV4(Fg), taking as

blocks the subspace pairs (wl, w2) where w~ and w2 denote two (I, 1, 0, 1)-type subspaces orthogonal to each other, and defining a treatment V to be arranged in a block (wl, w2) if V is orthogonal to wl but not to w2, we obtain a BIB design with parameters either

U = q2 _}. 1,

- 1 ) q : / 4 ,

b---- (q2+ 1 ) ( q - 1)g2/4,

~ = q + 1, A = (q2 _ 1)q/4

or

u = q 2 + l , , = ( q 2 - I)q2/4,

b= (q2+ 1)(q+1)q2/4, k = q - 1, A = (q2 _ 1)(q -2)/4

according to whether or not - 1 is a square element of Fa. Proo/. By Theorem 4 in Chapter 4 of [9], the orthogonal group 04 (Fq) acts transitively

on the set of (1,'0, 0)-type subspaces. And by Theorem 5 at the same place, noting that two (1,0,0)-types subspace are not orthogonal we easily know that 04(Fq) acts transitively on the set of (l,0,0)-type subspace pairs as well as on the set of (l,l,0,1)-type subspace pairs (wl, w2) with the property wl .L w2. Therefore, we certainly obtain a BIB design. We now calculate its parameters.

Clearly, the number of treatments is

v = N(I,O,O;4) = q2 + 1.

In order that a subspace pair (w~, w2) is a block, wl has exactly N(1, 1, 0, 1; 4) choices, and for awl properly chosen, w2 has as many choices as a (1,1,0,1)-type subspace which is orthogonal to wl. Noting that the conjugation of wl is a (3, 3, 1, r)-type subspace where

r= ~ z, if - l is a square element of F,I,

( 1, otherwise,

we conclude that w2 has

( (q- 1)q/2, N(1, 1,0, 1; 3 ,3 ,1 , r ; 4) =

(q + 1)q/2, t otherwise

choices. Hence the number of blocks is

b = N(1, 1,0, 1; 4) N(1, 1,0, 1; 3,3,1, I"; 4)

S (q2 + 1)(q - 1)q2/4, i/ - 1 is a square element of/~],, / (q2 + 1)(q + 1)q2/4, otherwise.

No.3 ORTHOGONAL GEOMETRIES AND DESIGNS 271

Let V be a given treatment, in order that V is arranged in a block (wl, w2), then w~ must be a (1,1,0,1)-type subspace not orthogonal to the (1,0,0)-type subspace V. Noting that the conjugation of a (1, 0, 0)-type subspace is a (3,2,0)-type subspace, we conclude that w2 has

N(1, 1, 0, I; 4} - N( I , 1, 0, 1; 3, 2, 0; 4} = (q - 1}q2/2

choices. For a tu2 properly chosen, Wl must be a (i, 1,0, 1)-type subspace orthogonal to both V and w2. Since V u w2 is a (2,2, 1)-type subspace and its conjugation is a (2, 2, 0)-type subspace, it follows that wl has

N(1, 1,0, 1; 2,2,0; 4) = ( q + 1)/2

choices. Therefore, each treatment appears exactly in

• = I ( q - I)q2/2] " (q-F I ) /2 ---- (q2 _ 1)q2/4

blocks. In virtue of the relations bit = ru and ~(v - 1) = r(]c - 1), we immediately have the

values of parameters k and ~:

k - - { q-t- l ,

q - 1, otherwise.

= ~ (q2_ 1)q/4, ( (q2 1 ) ( q - 2)/4,

This completes the proof. T h e o r e m 2. Take

if - 1 is a square element of Fq,

if - 1 is a square element of Fq,

otherwise.

(1, 0, 0)-type subspaces as treatments, take the set

'={'wi w2'l°l 'lIOl'tYPesubs c° I w 2 is a (1,0,0)-type subspace with the property wl _L w 2

as block set, and define a treatment V to be arranged in a block {wl, w2) if V is orthogonal to wi but not orthogonal to w2. Then we obtain a BIB design with parameters;

v = q 2 + l ,

• = (q + 1 )q~ /2 ,

b = (q~ + 1)(q+1)q/2, k--q, A--(q 2- I)/2.

Proo[. It is explained in the proof of Theorem 1 that the orthogonal group O4(Fa) acts transitively on the set of treatments as well as on the set of t reatment pairs. And by Theorem 5 in Chapter 4 of [9], we easily see that O4(F,~) acts transitively on the block set B. Hence we certainly obtain a BIB design. Its parameters can be calculated as in the proof of Theorem 1:

u = N(1,0,0; 4) = q2+ 1,

b-- N(1, 0,0; 4)- N(1, 1,0, 1; 3,2,0; 4)

= (q~ + 1 ) ( q + 1)q/2 = (q~ + 1)(q + 1)d2, r = IN(I ,0 ,0; 4 ) - N(1,0,0; 3,2,0; 4)]. N(1, 1,0, 1; 2,2,0; 4)

_-- [q2 + 1 - 1]. (q + 1)/2 = (q + 1)q2/2,

k = rv /b = q,

), = r ( k - l) l(v- I ) = (q2 _ i)12.


This completes the prooL

3. P B I B Des igns w i t h T w o A s s o c i a t e C lasses

Throughout this section, let n = 2v + 5, v _> 2, and 6 = 0, 1 or 2. A known association scheme {sl is needed for our purpose, which will be called Association Scheme A and stated as follows.

Association Scheme A : Taking as treatments the (t, 0,O)-type subspace in OV,~(Fq) and defining two treatments to be the first (resp. second) associates of each other if they are orthogonal (resp. nonorthogonal) to each other, one obtains an association scheme with two associate classes and with parameters:

v = {qV _ 1)(qv+,-t + 1)/(q- 1),

nl = (q~-t _ 1){qv+,-2 + 1)q/(q - 1),

Pit = ( q - 1)(q v-2 - 1)(q v+6-3 + 1)q21(q- 1), p21 = (qV-I _ 1) + (qV+,-2 + 1)/(q - 1).

(i)

Using this association scheme and suitably taking blocks, we can construct PBIB designs.

Theorem 3. Adopt Association Scheme A. Take as blocks the subspace pairs (wt, w2) where wt denotes an (m + r, 2s + r, s, I') -type subspace and in2 denotes a (1,0,0,)-type subspace with the property w2 2. tot and w2 n wl = [0 I. And defined a t reatment g to be arranged in a block (wt, to2) if V is orthogonal to Wl but not to to2. Then we obtain a PBIB design with two associate classes. Its basic parameters are those in (1) and the following

b =N(rn + ~, 28 + r, s, r; 2v + 5, A)

• [(#(1,0,0; rnl + ~1,2st + r1,,1,1'1;2v + 6, A) - N(1, rn - 2s)],

r =n2" N ( r n + , , 2s + , , s , r ; (2w- 2) + 6 , 2 ( v - 1) + 6 , v - 1, a ; 2 v + 6, A),

where ml, rl, st, rl are such that the conjugation of a (m + r, 2s + r, 8, [')-type subspace is an (rnt + *'t, 2st + rt, st,rt)-type subspace.

Proof. Let {tat, w2) be a block. Then (~) is an (m + • + 1) x n matrix of rank rn + r + 1. Without loss of generality, we may assume

0 1{') ) wtS~w~ = l(S) 0

r

Thus

w2 \ w2 1 r " o

From this and Theorem 5 in Chapter 4 of [9J, we know that the orthogonal group O,,(F#) acts transitively on the set of blocks. Hence we certainly obtain a PBIB design, its parameters can be calculated as follows.

For (wl,t#2) to be a block, wt has N ( m + r, 2s + r,s,r;2v + 6,A) choices, and for a properly chosen wl, w2 must be a (1, 0, 0)-type subpace which is included in the conjugation


w~ of wl but not included in w~ . Let w~ be of {ml + rl,2s + rx,sl,rx)-type which is determined by the type of wl (see Theorem 1 in [11]}. And it is known that [111 w~ f3 wl is an (m- 2s, 0, 0)-type subspace. Therefore, w2 has

N{1, 0, 0; rnl -I- r~, 2s~ + r~, sl, I"i; 2 v+ 6, A} - N(1, m - 2s)

choices, and then the number of blocks is

b =N(rn + r, 2s + r, s, r ; 2 u + 6 , A)

0, 0;. i + 28, + ,,, rl; + s, a) - .. - 28)I.

Let V be a given treatment. In order that V is arranged in a block (wl, wz), then w2 must be a {1, 0,0)-type subspace not orthogonal to V {thus, w2 has n2 choices). For each w2 so chosen, wl must be an (rn + r, 28 + r, s, r)-type subspace orthogonal to both V and w2. Note that such (m + r, 2s + r, s, r}-type subspaces must not include w z because w 2 is not orthogonal to V. Since V u w 2 is a (2,2,1)-type subspace and its conjugation is a ((2v - 2} + 6, 2(v - 1) + 6, u - 1, A; 2u + 6, A}-type subspace, it follows that wt has

N ( m + r , 2 s + r , s , r ; ( 2 v - 2 ) + g , 2 ( v - 1 ) + 6 , u - l , A ; 2 u + 6 , A)

choices. Hence each treatment appears exactly in

r = n 2 . N ( m + r , 2 s + r , s , r ; ( 2 u - 2 ) + 6 , 2 { u - 1 ) + 6 , v - 1, A; 2u+g, A}

blocks. Let VI and V2 be two (1, 0,0}-type subspaces orthogonal to each other. Now we calculate

the number of blocks (w~,w2) containing both treatments V, and V2. Clearly, w~ has P~2 choices. And for each w~ properly chosen, wt has

• N ( m + f , 2 s + f , s , F; ( 2 u - 3 ) + 6 , 2 ( u - 2 ) + 6 , v - 2 , A; 2v+6, A)

choices, because 111 U V~ U w2 is a (3,2,1)-type subspace and its conjugation is a ((2u - 3) + 6, 2(v - 2) + 6, u - 2, A; 2v + 6, A}-type subspace. Therefore, there are exact

A ~ = p ~ 2 . N ( m + r , 2 8 + r , s , r ; ( 2 v - 3 ) + S , 2 ( v - 2 } + S , v - 2 , ~X;2v+~, A}

blocks containing any two given treatments which are subspaces orthogonal to each other. Hence the theorem.

T h e o r e m ,1. Adopt Association Scheme A. Take as blocks the subspace pairs (wl, w2} where w~ denotes an (m + r, 2s + r, s, r}-type subspace and w2 denotes a (1, 1,0, r '}-type subspace with the property w2.1_ wl. And define a treatment V to be arranged in a block (wt, w2) if V is orthogonal to wl but not to w2. Then we obtain a PBIB design with two associate classes, and its basic parameters besides those in {1) are

b = N ( m + r, 2s + r, 8, r; 2r + g, A)

-N(1 ,1 ,0 , F ' ; m l + r x , 2 s t +rx , sl , F 1 ; 2 v + 6 , A},

r = IN( l , 1, 0, r ' ; 2v + 6, A)

- [N(1, 1, 0, r '; (2v - I) + 6, 2(u - I) + 6, v - 1, A; 2u ,-i- 6, A)]

• N ( m + r , 2 s+r , s ,F ; ( 2 v - 2 ) + 6 , 2 ( v - 1 ) + 6 , v - l , A ; 2 v + 6 , A), A1 =[N(1, 1, 0, r ' ; 2v + 6, A}

- 2 N ( 1 , 1,0, F ' ; ( 2 v - 1 ) + 6 , 2 ( v - 1 ) + g , v - 1 , A;2v+6 , A)

+ N ( 1 , 1,0, r ' ; (2v-- Z) + 6, 2(v -- 2) + 6, v - -2 , A ; 2 v + 6 , A}]

• N ( m + r , 2 s + r , s , F ; ( 2 v - 3 ) + g , 2 ( v - 2 ) + 6 , u - 2 , A;2v+6 , A).


where rnl, rl , s l , r l are such that the conjugation of an ( rn+ r, 2s + r, s, F) - type subspace is an (rnl + rl , 2sl + rl, s l , r l}- type subspace.

Proo/'. The proof is similar to that of Theorem 3, hence omitted.

4. M o r e P B I B De s i gns

In Theorem 11 of Chapter 9 of [9] Wan et al. give the.following association scheme which will be called Association Scheme B.

Association Scheme B: Let n = 2v + 6, and 6 = 0, 1 or 2. Take as t rea tments the (u, 0, O)-type subspaces in OV,,(Fq), and define two treatments to be the i th associates of each other if they span a (u +/)-dimensional subspace (i = 1, 2 , . - . , u). Then we obtain an association scheme'with v associate classes and with parameters given in formula (9) of Chapter '9 of [9].

Using this association scheme and suitably taking subspace pairs as blocks, we can obtain some PBIB designs.

We start with the case 6 = 0.

T h e o r e m 5. Adopt association Scheme B. Taking as blocks subspace pairs (wl, w2) where wl denotes an (m, 0, 0)-type subspace (1 _< m < w) and w2 denotes a (1 ,0 ,0) - type subspace with the property wz .l. w I and wl A w2 = I 0 [, and defining a t rea tment V to be arranged in a block (wl,wz) if V is orthogonal to w 1 but not to w2, we obtain a PBIB design with v associate classes. Besides the parameters of Association Scheme B with 6 = 0, its basic parameters are

v v

FI (¢ - i)(q'-' + i) FI (q' - i) "= - i = w - m q U - I b = ' " "" q " , F =

{q-- l} f i (qi_ I) (q-- I) fi {q'-- i) i = l i=l

v - i q ~ , - , . { q ~ + , - , n - I - q , - q , - I + 1) ]'[ (q' - 1)

t . . ~ m - - m - - i - - I , i < _ v - m - 1 ,

(q- i) (q,- i) t----.l

q ,n {qU-m - 1}(q~,-,~-I - 1)/(q- I), i = w - m,

O, i > w - - m .

Proo/. From'the proof of Theorem 3 we know that the orthogonal group 02v(Fq) acts transitively on the set of blocks. By Lemma 4 in Chapter 9 of [9], the orthogonal group O2v(Fq) acts transitively on the set of (v, 0, 0}-type subspace pairs the two subspaces of each of which span a (u +/)-dimensional subspace (i -- 1, 2, . . . , v). Hence we certainly obtain a PBIB design. We now calculate its parameters.

It is knownllZl that in OY2,,(Fq) the conjugation of an (m, 0,0)-type subspace is a ( 2 u - m , 2 ( v - r n), v - m J - t y p e subspace, and the conjugation of a ( v + 1, 2, 1)-type subspace


is a (v - i, 0, 0)-type subspace. Similar to the proof of Theorem 3, we have

b = N(m, 0,0;2~). IN( l , 0, 0; 2 u - m, 2 ( v - m), v - m; 2v) - N(1, m)] M

= ' ~,II + l (q ' - 1)(q i -1 + 1) [q,, _ 1 (q . . . . 1)(q v-m-1 + 11 qm

l-I(q i 1) [ q - 1 + m _ q - 1

i = 1 i l

l'l ( q i _ 1)(q,-1 + 1)

(q- 11 ~ ( ¢ - 11 i = 1

r = [N(1,0,0;2u) - N(1, v) ] . N(m, 0 , 0 ; u - 1,0,0;2v) v - 1

1"I (q'- 1) = [ i=V-- ~rL ( ¢ ' - 1)(¢ ' -1 + 1) _ ~ - 1 ] .

q-1 q- l J l] (q'- l) i=1

M

( q - "1) I-[ (qi _ 1) i---1

q V - - 1 •

q~ t _ 1

~ z T ]

Let VI and I/2 be two given treatments that are the ith associates of each other. Let

T = {(wi, w2) ] V1 and V2 are arranged in .(wl, w2)},

rl = {(w~, w~) ~ r I ~2 c v~ u v~}, T~ = S\ r l .

Suppose (w~,wz) ~ T1, it is easily seen that V1 u V z is a ( v+ 1, 2i , / ) - type subspaces, and among the N(1 ,0 ,0 ;v + i ,2i, i;2u) (1,0,0)-type subspaces included in V~ u Vz, the H(1, u) (1, 0, 0)-type subspaces included in v, are orthogonal to V~ (i = 1, 2). These can not be taken as w2. On the other hand, among the N(1, 0, 0; v + 1, 2i, i; 2v) (1, 0, 0)- type subspaces included in gl u V2, those (1, 0, 0)-type subspaces a in included in neither V~(i = 1, 2) must be orthogonal to neither Vj.(] = 1, 2), because VI and V2 are both maximal totally isotropic subspaces. These and only these can be taken as wz and their number is

N(1, 0, 0; v Jr i, 2 / , / ; 2~) - 2N(I, v) Jr N(I , v - / } .

For such a w2 chosen, wl has as many choices as the (m, 0, 0)-type subspaces orthogonal to t h e (~, + i , 2i, i)-type subspace Vt u V2. Since the conjugation of a (v + i, 2i , / )- type subspace is a (v - i, 0, 0)-type subspace, it follows that for each w2 properly chosen, wl has

N(m, 0, 0; ~ - / , 0, 0; 2~) -- N(m, ~ - / )


choices. Therefore

ITll = [N(I, 0, 0; ~+~, 24 ~; 2~) - 2~(1, ~)+ ~(1,~ - 0 ] ~(m,~ - 0

1-[ (q' - 1) _ [ q V - L - - 1 (q i -1 ) (q ' -1+1) L.-, 2 q v - 1 q'-'-i I , = . - . . - , + i

[ q _ l + q - + j " q - 1 q - 1 q - - ~ I I (¢ - I)

/----I I u--i J H (q' - i)(¢ ~ - i (q'- I)

=v- - i+I qU-i, 1 <_ v - rn,

= (q - I) II (¢ - I) t----I

O, 1 > v - r e .

Suppose {wl, w~) E T2. As a|] the (1, 0, 0)-type subspaces not included in VI u V~ can be taken as w2, then w2 has

N(I,O,O; 2 ~ ) - N(I,O,O; ~+~,2~,~; 2~)

choices. For each w2 properly chosen, wl has as many choices as the (m, 0, 0)-type subspaces orthogonal to Vl u V2 u w2. Because of Vl u V2 = Vi* u V2* = (Vt n V2)', w2 is not orthogonal to V INV2. Then it is easy tosee [12] that V i u V 2 u w 2 is a ( w + i + 1, 2 ( i + l ) , i + 1)-type subspace with a (v - i - 1, 0, 0)-type subspace as its conjugation, Therefore, w I has

N(m, O, O; v - i - l ,O,O;2v)=N(m, v - i - l )

choices. Hence

IT2I = N(1, o, 0; 2v) - N(1, 0, 0; w + i , 2i, i, 2w)] . N ( r n , u - i - I)

,r(q ~ _ 1)(q~-,~:~ + i) (q~-, _q _ 1 I) (¢_ i)(¢-,q= i ÷ I) q~_,~] /* s I

= N ( ¢ - I) t = u - , , , - ~ , i _ < u - m - I ,

~(¢-I) O, i >_ v - m ,

[~ (qt - 1)

t=~,-,,,-i qU+i-l, i <_ w - m - 1

= ( q - l ) ( q - l ) t = l

O, i > v - m .

Combining the above results, we have the value of the parameter ki:

~ =IT~I + I~I I v_, qV-i(qV+, ..... I _ q, _ q,-I + 1) ~ {qt _ 1)

t = u - , n - i - i , % __~ b' - - r r t - - 1,

= (q-l) (¢-I) t----I

q,,,{qV .... _ 1)(qV ..... t _ l)/(q - 1), i = u -- rn,

O, i > b ' - -m.

No.3 ORTHOGONAL GEOMETRIES AND DESIGNS 2,77

The proof is completed. T h e o r e m 6. Adopt Association Scheme B. Take as blocks the sul)space pairs (wl, w2)

where wl denotes an (m, 0,0)-type subspace and w2 denotes a (1, ~,0, / )- type subspace orthogonal to wl, and define a treatment V to be az'ranged in a block (wl, w2) if V is orthogonal to wl but not to w2. Then we obtain a PBIB design with z)' associate classes, its parameters besides those of Association Scheme B with 6 = 0 are

q~-~(q~-~- I) ~ (q'- 1)(q'-~ + 1) q~-i ~ (q'- 1) • v rt~ b -~- i = u - n ~ + l , r ---~ I

2 f i ( q ' - 1) 2 f i ( q i - 1)" i = 1 i = I

v - - i

q ~ - ~ ( q ~ - ~ - 1) I ] (q~ - 1) t = u - - i - - m + l J ~ - - < 1* - - D~ - - l ,

~, = 2 ~ ( q ' - 1) t = l

q ~ - l ( q ~ - ~ _ 1) /2 , ~ = ~ - m,

O, i > v - r n .

Proof. Similarly to the proof of Theorem 5, we know that this is certainly a PBIB design, and that the parameters b and r are

b = N(m, 0, 0; 2v) N(1, 1, 0, 1; 2 v - m, 2 ( u - m), v - ~ ; 2v)

f l ( q ' - 1)(q'-~ + 1) = ~ = , , , - , ~ + I (qV-,-,~ _ i)q~-i

2 ~I ( q ' - I) i = l

q ~ - ~ ( q . - " ~ - I) ~ (q' - 1)(q'-~ + I) i = v - - m + l

2 f i ( @ - l ) i = l

r = N(I , i, 0, 1; 2v) N(rn, 0, 0; v - 1, 0, 0; 2v) L,--J.

l] (q' - i) = (q~ - 1 ) q ~ - 1 , . . . . . .

2 f i (q' - 1) i : l

qL,-i fl (ql 1) i ----- $ , - - r r ~

2 fi(q'- 1) i=1

Let V1 and V2 be two given treatments that are ith associates of each other, and T, T1, "{2 be defined as in the proof of Theorem 5.

Let (wt, w2) E Tt, we are going to prove that the (1, 1, 0, 1)-type subspaces ~ included in Vt uV 2 are orthogona] to neither V/(i = 1, 2), and then can be taken as w 2. Let ~ = fll +/~2, where ~1 c V1, ]~2 c V2. If ~ _L V1, then ~2 .L V1. But ]~2 ~ V1, so (p~)v, is a (v + 1, 0, 0)-type subspace. This is impossible. Then ~, is not orthogonal to VI. Simila?ly, ~ is not orthogonal to V2 either. Therefore, w2 has

N(1, 1, o, 1; u + i, 2i, i; 2u)

27S ACTA MATHEMATICAE APPLICATAE SINICA Vol.9

choices. Similarly to the proof of Theorem 5, we know that for each w2 properly chosen, wr has N(n, v - i) choices. Thus

I r , l = N ( 1 , l, o, 1; u + i , 2i, i; 2u) . N ( m , u - i )

q"-'(q' - l lq i - I I-I (q' -- 11 t = v - - i - - m - P l

O,

m

2 I-I (q' - 1) t = 1

v - - i

qW-l(qi _ 1) ~I __ t : u - - i - - m + l

= 2 ~ ( q ' - 1) i = 1

0,

(qt _ 1)

i < _ v - m ,

i > u - m ,

, i < _ u - m ,

'/ > t , - - m.

Let (wl, w2) E T2. Similarly to the proof of Theorem 5, we have

IT=I = I N ( l , 1, o, 1; 2v) - N(1, 1, 0, 1; v + £, 2£, i; 2~) ] . N (m , u - ~ - i)

f / [(qV _ 1)q~.-1/2_ ( q i 1 )q , - t / 2 ]

{ O,

qV+i-1 - 1) _ _ - - t = u - i - - . , ~

= 2 U1(qt -- I)

O,

u - i - I

I-[ (q' - 1)

t=1

i<v-rn-1,

i>_v-m.

i < _ v - m - 1 ,

i > _ v - r n ,

Therefore, the parameter Ai is

_ I) 11 t = v - - i - - m + l

A, = 2 f i ( q ' - 1) t = l

qV-l(qV-.~ _ !)/2 '

0,

(qt _ 1)

, i<_u-rn-l,

i > ~ - m .

This completes the proof.

In the sequel we let 6 = 2 and we proceed in OV2,.+2(Fq). T h e o r e m 7. Adopt Association Scheme B with 6 = 2. Take as blocks the subspace

pairs (w~,w2) where wl denotes an (m + r, 2s + r, s, F) - type subspace (r = 0 or 2, and s = 0 or 1) and to2 denotes a (1,0,0)-type subspace with the p r o p e r t y ' w 2 _C w~\w~, and define a t r e a t m e n t V to be arranged in a block (wl, w2) if V is ortkogon.41 to wl but not to w2. Then we obta in a PBIB design with v associate classes, its pa r ame te r s besides those m

No.3 O R T H O G O N A L G E O M E T R I E S AND DESIGNS 279

~i.ssociation Scheme B with 6 = 2 are

b = N ( m + r, 2s + r, s; 2v + 2)

• [ N ( I , 0, 0; rnl + rl , 2sl + r l , s l , r l ; 2v + 2) - N ( 1 , r n - 2s)] ,

r = [N(1 , 0, 0; 2v + 2) - Y(1, 0, 0; v + 2, 2, 0; v + 2)]

• N ( r n + r , 2 s + r , s; ( v - 1 ) + 2 , 2, 0; 2 v + 2 ) ,

A, = [ N ( I , 0, 0; v + i , 2i, i; 2v + 2) - 2 N ( I , v) + N ( I , v - i)]

• N ( m + r , 2 s + r , s ; (u-i)+2, 2 , 0 ; 2 v + 2 )

+ [ N ( 1 , 0 , 0 ; 2 v + 2 ) - N ( 1 , 0 , 0 ; v + i + 2 , 2 i + 2 , i ; 2 v + 2 ) ]

• N ( m + r , 2 s + r , s ; ( v - i - 1 ) + 2 , 2 , 0 ; 2 v + 2 )

+ [ N ( 1 , 0 , 0 ; (v+i)+2, 2 i + 2 , i ; 2 v + 2 )

- N ( 1 , 0, 0; v + i , 2i, i; ,2v + 2)] • N(m. + r, 2 s + , , s, r ; ( v - i ) + 1, 1, o, r ; 2v + 2),

where rex, r i , ' 1 , F1 are such tha t the conjugat ion of an (rn + r, 2s + r, s, 2v + 2) - type subspace, is an (rn 1 + rl , 2s 1 + rl , Sl, F1; 2v + 2)- type subspace.

Proof. The proof is similar to tha t of Theorem 5 above and to t ha t of T h e o r e m 3 of [12], hence omit ted•

T h e o r e m 8. Adopt Association Scheme B with 3 = 2. Take as blocks the subpace pairs (wl, w2) where wi denotes an ( re+r , 2 s + r , s, r ) - t y p e subpace (r = 0 or 2, and s = 0 or 1) and w 2 denotes a (1, 1, 0, 1)-type subspace or thogonal to wl, and define a t r e a t m e n t V to be a r ranged in a block (wl, w2} if V is or thogonal to wl but not to w2. Then we obta in a P B I B design with v associate classes. Its pa ramete r s besides those in Associat ion Scheme B wi th 6 = 2 are

b = N ( m + r, 2s + r, s F;

• N ( 1 , 10 , 1 ; m l + r l

r = [N(1 , I, 0, i; 2v + 2)

2v + 2)

, 2 s i + r l , s l , FI; 2v + 2),

- N ( I , 1 ,0 , 1; v + 2 , 2 , 0 ; 2 v + 2 ) ]

• N ( m + ~, 2s + ~, s, r; (~ - 1) + 2, 2, o; 2~ + 2),

A~ --[N(1, 1, 0, 1; v + i, 2i, i; 2v + 2)

• N ( m + r , 2 s + r , s , r ; ( v - i ) + 2 , 2 , 0 ; 2 v + 2 )

+ I N ( l , 1 , 0 , 1 ; 2 v + 2 ) - N ( 1 , 1 ,0 , 1 ; v + i + 2 , 2 i + 2 , i ; 2 v + 2 ) ]

• N ( r n . + , , 2 s + r , s , F ; ( v - i - l ) + 2 , 2, 0 ; 2 v + 2 )

+ [ N ( I , I , 0 , 1 ; v + i + 2 , 2 i + 2 , i ; 2 v + 2 ) ]

- N ( I , I, 0, 1; v + i , 2i , i; 2v + 2)]

• N ( r n + r , 2s + r, s, F; (v - i) + l, 1 ,0 , r ; 2 v + 2 ) ,

where rnl , ~'i, s l , FI are such tha t the conjugat ion of an (rn + r, 2s + r, s, r , 2v + 2 ) - type subspace, is an (rn I + r l , 2~i + r l , s l , r l ; 2u + 2)- type subspace.

Proof. The proof is similar to tha t of Theorem 6 above and to t ha t of T h e o r e m 4 in [12], hence omi t ted .

R e f e r e n c e s

[1] Dai Zongduo, Feng Xuning, Studies in Finite Geometries and the Construction of incomplete Block Designs IV. Some "Anzahl" Theorems in Orthogonal Geometry Over Finite Fields of Characteristic

2, Acta. Math. Sinica, 15 (1965), 545-558 (in Chinese).


[2] Y. Huo and G. Xie, PBIB Designs with Several Association Classes Constructed from Orthogonal

Geometry Over Finite Fields (II), Advances in Mathematics, 15 (1986), 195-199 (in Chinese).

[3] H. Shen, An Enumeration Theorem in the Orthogonal Geometry Over a Galois Field with Charac-

teristic p-~ 2 and the Construction of PBIB designs, Jourual of Shanghai Jiaotong University, 2 1983,

121-130 (in Chinese, with English Abstract).

[4] H. Shen The Construction of Association Schemes with 1-dimensional Non-isotropic Subspaces in the

Orthogonal Geometry aJ Elements, Journal of Shanghai Jiaotong University, 18 (1984), 1-12 (in

Chinese, with English abstract).

[5] H. Shen, Construction of Block Designs by Using Finite Geometries, Journal of $hanghai Ji~otong

University, 31 (1987), 29-37 (in Chinese).

[6] P. Sun and X. Zhu, PBIB Designs Constructed with a Class of Hyperbolic, Planes in Finite Orthogonal

Geometries, Journal of Northeast Normal University, 2 (1984), 11-15 (in Chinese).

[7] Wan Zhexian, Finite Geometries and Block Designs, Presented On " Chinese-Germmn Conference on

Combinatorics", Shijia~huang, China, 1989.

[8] Z. Wan, Notes on finite Geometries and the Construction of PBIB Designed, VI Some Association

Schemes and PBIB Designs Based on Finite Geometries, Scientist Sinica, 14 (1965), 1872-1875.

[9] Z. Wan, Z. Dai, X Fang and B. Yang, Studies on Finite Geometries and the Construction of Incomplete

Block Designs, Science Press, Beijing , 1966 (in Chinese).

[10] Wei Wandi, Yang Benfu, Orthogonal Geometries Over Finite Fields with Characteristic ~ 2 and Blocks

Designs (II), Linear Algebra App]., 166 (1992), 1-19.

[11] Yang Benfu, Wei Wandl, Orthogonal Geometries Over Finite Fields with Characteristic ~ 2 and PBIB

Designs (1), Linear Algebra App]., 134 (1990), 1-23.

[12] Yang Benfu, Wei Wandi, Finite Unitary Geometries and PBIB Designs (III) (submitted).

[13] X. Zhu and P. Sun, Association Schemes Constructed with a Class of Isotropic Planes in Finite Or-

thogonal Geometries, Suniao Journal, 2 (1985), 9-12 (in Chinese).

Orthogonal geometries over finite fields with characteristic ≠2 and block designs

Documents

Transcript of Orthogonal geometries over finite fields with characteristic ≠2 and block designs