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Symmetric Designs without Ovals and Extremal Self-Dual Codes
45 1
Annals of Discrete Mathematics 37 (1988) 451-458 0 Elsevier Science Publishers B.V. (North-Holland)
SYMMETRIC DESIGNS WITHOUT OVALS AND EXTREMAL SELF-DUAL CODES Vladimir D . Tonchev* I n s t i t u t e of Mathematics, S o f i a 1090, P. 0. Box 373, Bulgaria ABSTRACT. Constructions of doubly-even s e l f - d u a l codes from symmetric d e s i g n s a r e discussed. I t i s shown t h a t t h e absence of o v a l s i n a design i s u s u a l l y necessary, and sometimes even s u f f i c i e n t c o n d i t i o n f o r t h e corresponding code t o be extremal.
1. INTRODUCTION The terminology and n o t a t i o n s from design and coding theory used i n t h i s paper a r e i n accordance with those from [ 3 A binary
t o r space V
1,
[ 5 1, and L8] r e s p e c t i v e l y .
(n,k) code C is a k-dimensional subspace of t h e n-dimensional vec-
1
Over GF(2). Given an ( n , k ) code C , t h e (n,n-k) code C
= {xeVn:
yx = 0 f o r each F C ) i s c a l l e d t h e orthogonal, o r dual of C. A matrix with t h e p r o p e r t y t h a t t h e l i n e a r span of i t s rows g e n e r a t e s t h e code C , i s a g e n e r a t o r
1 m a t r i x of C. The g e n e r a t o r m a t r i c e s of t h e d u a l code C a r e c a l l e d p a r i t y check matrices of C. We s h a l l o f t e n r e f f e r t o t h e elements of a code a s codewords, o r words only. The weight of a codeword i s t h e number of i t s nonzero p o s i t i o n s , and t h e minimum weight of a code i s t h e weight of a l i g h t e s t nonzero codeword. An
( n , k , d ) code i s an ( n , k ) code with minimum weight d. A code C i s self-orthogonal
(resp. self-dual)
i f CCC'
(resp. C =
2 ) . The
weights of a l l words i n a s e l f - o r t h o g o n a l code a r e even. I f i n a d d i t i o n a l l weights a r e d i v i s i b l e by 4 , t h e code i s c a l l e d doubly-even. d u a l ( n , n / 2 ) code e x i s t s i f and only if n
=
0 (mod
a),
A doubly-even
self-
and t h e minimum weight
d of such a code i s bounded by d S 4[ 11/24] + 4 , (cf.
[a]).
(1)
A code s a t i s f y i n g t h e e q u a l i t y i n (1) is c a l l e d extremal. The code-
words of minimum weight i n an extremal doubly-even s e l f - d u a l code y i e l d o r 5-design provided t h a t
I-, 3-,
n 5 16, 8 , o r 0 (mod 2 4 ) .
The number of extremal codes i s f i n i t e ( b u t unknown), t h e s m a l l e s t open c a s e being f o r n = 7 2 . The following theorem d e s c r i b e s a c o n s t r u c t i o n of doubly-even s e l f - d u a l
CO-
d e s from symmetric d e s i g n s . THEOREM 1.1. Let A be an incidence matrix of a symmetric 2-(v,k,k)
of an odd o r d e r k-A.
Then
( i )If k E 3 (mod 4 ) , t h e code generated by t h e matrix
*Research p a r t i a l l y supported by t h e
cscMBunder
contract
37/1987-
design
V.D. Tonchev
452
i s a doubly-even s e l f - d u a l
(2v,v) code.
(ii) I f k Z 2 (mod 4 ) , then t h e code generated by t h e matrix
(3)
i s a doubly-even s e l f - d u a l
(2v+2, v + l ) code.
This statement i s a v a r i a n t of a m r e g e n e r a l c o n s t r u c t i o n ( c f . [ 21
,[
4) )
.
I n t h i s paper we a r e i n t e r e s t e d i n necessary and s u f f i c i e n t c o n d i t i o n s f o r a code c o n s t r u c t e d i n t h i s way t o be extremal.
In o t h e r words, what combinato-
r i a l p r o p e r t i e s should a design possess i n o r d e r t o produce an extremal code? A notion p l a y i n g an important r o l e i n t h e study of symmetric d e s i g n s , espe-
c i a l l y p r o j e c t i v e p l a n e s and b i p l a n e s , is t h a t of "arc" o r "oval". An a r c i n a design i s a s u b s e t S of p o i n t s such t h a t each block i n t e r s e c t s S i n a t most 2 p o i n t s . The c a r d i n a l i t y of an a r c S i n a n o n t r i v i a l symmetric 2-(v,k,h)
design
of an odd o r d e r k-h i s bounded by
IS/ S (k+h-l)/h.
(4)
An a r c with maximum number of p o i n t s i n t h e sense of bound
( 4 ) i s c a l l e d an
oval [ 2 ] . I t w i l l be seen by t h e next e x p o s i t i o n t h a t t h e absence of o v a l s i n a sym-
m e t r i c design i s u s u a l l y necessary, and sometimes even s u f f i c i e n t c o n d i t i o n f o r t h e corresponding s e l f - d u a l code t o be extremal. 2 . ARCS AND CODES
Let D be a symmetric 2-(v,k,h) Denoting by ni
design with an a r c S of s i z e s , i . e .
IS/ =s.
( i = 0,1,2) t h e number of blocks having e x a c t l y i common p o i n t s
w i t h S , w e have:
no
+
nl
+
*2 =
vt
Consider t h e code with g e n e r a t o r matrix ( 2 ) defined by D . C l e a r l y , t h e weight of t h e sum (over GF(2)) of s rows of t h e g e n e r a t o r matrix ( 2 ) indexed by t h e p o i n t s of an a r c of s i z e s i s s + n l = s(k+l)-(s-l)A). S i m i l a r l y , t h e sum of s
r o w s of t h e matrix ( 3 ) , indexed by t h e p o i n t s of an a r c of s i z e s has weight
n+ 1 s+nl+(l+(-l) ) /2
=
s ( k + l - ( s - l ) h ) + (1+(-1)'+l) / 2 .
Thus we have t h e following THEOREM
2.1.
Let D be a symmetric 2-(v,k,h)
design s a t i s f y i n g t h e assump-
Symmetric Designs without Ovals
453
t i o n of Theorem 1.1, and admitting an a r c S of s i z e s . Then t h e minimum weight of t h e code d e f i n e d by D a s i n Theorem 1.1 i s bounded a s follows: d S {
s ( k + l - ( s - l ) h i n c a s e (i); s (k+l- (s-1)h ) + ( l + ( - l'+') )
(6)
/2 i n case (ii).
I t t u r n s o u t t h a t i f t h e a r c S i n an o v a l , t h e i n e q u a l i t i e s (6) a r e o f t e n
s h a r p e r than ( 1 ) . Thus a code a r i s i n g from a design with o v a l s i s u s u a l l y n o t extremal. We s h a l l i l l u s t r a t e t h i s by some examples. In Table 1 t h e parameters of symmetric 2-(v,k,A)
d e s i g n s with k - h S 9 ,
y i e l d i n g doubly-even codes by t h e c o n s t r u c t i o n of Theorem 1.1. a r e l i s t e d . TABLE 1 NO.
k-h
1
1
v 3
Codes
h
k 2
1
Extended Hamming ( 8 , 4 , 4 ) code; extremal.
2
3
3
Extended Golay (24,12,8) code; extremal.
3
5
19
10
5
Three extremal (40,20,8) codes.
4
7
31
10
3
Extremal (64,32,12) code from any design without o v a l s (Theorem 2 . 5 ) .
5
7
27
14
7
An extremal
6
9
35
18
9
7
9
36
15
6
8
9
40
27
18
1
1
6
Parameters N o . 2-(4t-l,2t,t). 2t-1.t-1))
(56,28,12) code from t h e q u a d r a t i c r e s i d u e design.
Is t h e r e any extremal (72,36,16) code? ( 8 0 , 4 0 , d > l 2 ) code from any design without ovals.
A
1 , 2 , 3 , 5 , 6 i n Table 1 a r e of Hadamard t y p e , i . e .
The s i z e of an o v a l i n a Hadamard 2 - ( 4 t - l , 2 t , t )
design i s 3 , t h u s t h e bound from Theorem 2.1 f o r t h e minimum weigth
of a code of type (ii) i s 4. However, a Hadamard 2 - ( 4 t - l , 2 t , t ) t
of t h e form
( o r 2-(4t-1, design with odd
> 1 does n o t admit any o v a l s ; t h i s follows from a r e s u l t of Morgan [ 9 1 , and
can be a l s o e a s i l y seen d i r e c t l y . For, i f a 2 - ( 4 t - l , 2 t , t )
design p o s s e s s e s an
a r c of s i z e 3, then t h e sum of t h r e e rows of i t s incidence matrix A indexed by t h e p o i n t s of t h e a r c , as w e l l as t h e sum of a l l remaining rows over GF(2) is t h e z e r o v e c t o r . Consequently, t h e rank of t h e incidence matrix over GF(2) must be l e s s o r equal t o 4t-3. (I,J-A),
where J.denotes
On t h e o t h e r hand, s i n c e t i s odd, t h e matrix
t h e a l l - o n e m a t r i x , g e n e r a t e s a s e l f - d u a l code. Thus
t h e rank of J-A over G F ( 2 ) i s 4t-1, whence t h e rank of A i s 4t-2,
a contradic-
t ion. Estimating t h e weight of a sum of a t most 4 rows of a matrix of t h e form ( 3 ) , where A i s an incidence matrix of a Hadamard 2 - ( 4 t - l , 2 t , t )
odd t , say t = 2 m + l , THEOREM 2 . 2 .
design with
t h e following can be proved:
[13
1.
The minimum weight of a (16m+8,8m+4) code with genera-
454
V.D. Torichav
t o r m a t r i x ( 3 ) d e f i n e d by a Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n i s e q u a l t o 4 if m=O,
and i s a t l e a s t 8 i f m
> 0.
In f a c t , t h e codes d e r i v e d from Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n s f o r
m S 2 a r e a l l e x t r e m a l . I t i s worth n o t i n g t h a t Hadamard 2 - ( 4 t - l , 2 t , t ) d e s i g n s which a r e e x t e n d a b l e i n t o isomorphic Hadamard 3 - ( 4 t , 2 t , t - l )
designs, y i e l d
e q u i v a l e n t codes. For i n s t a n c e , t h e r e e x i s t e x a c t l y 6 nonisomorphic 2 - ( 1 9 , 1 0 , 5 ) d e s i g n s and 3 nonisomorphic 3 - ( 2 0 , 1 0 , 4 ) d e s i g n s , producing 3 i n e q u i v a l e n t extremal (40,20,8) codes [ 1 3 ) . The absence of o v a l s i n a 2-(8m+3,4m+2,2m+l) design with m > 2 i s n o t a s u f f i c i e n t c o n d i t i o n f o r t h e e x t r e m a l i t y of t h e r e l a t e d code.
In t h i s case
some a d d i t i o n a l c o n d i t i o n s are t o be f u l f i l e d , i n c l u d i n g t h e absence of o v a l s i n t h e complementary 2-(8m+3,4m+1,2m) d e s i g n a s w e l l . THEOREM 2.3.
[ 1 3 1 . 3 n e c e s s a r y c o n d i t i o n f o r an (16+8,8m+4) code d e f i n e d
by a Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n D t o have minimum weight d 2 12 i s any t r i p l e of pOints of D t o be c o n t a i n e d i n a t l e a s t 2 b l o c k s , and a t most 2m-1 b l o c k s of D .
-
Formulated f o r t h e complementary 2-(8m+3,4m+2,2m) design D , t h i s c o n d i t i o n s t a t e s t h a t each t r i p l e of p o i n t s must occur i n a t most 2m-2 l e a s t one block of
D.
In particular,
-
b l o c k s , and a t
D cannot have gny o v a l s .
The enumeration of a l l 2-(27,14,7)
ciesigns up t o isomorphism i s n o t y e t
completed. However, t h e only primes which can d i v i d e t h e o r d e r of t h e automorphism group of a Hadamard m a t r i x of o r d e r 28 a r e 13, 7 , 3 and 2 , and t h e m a t r i c e s p o s s e s s i n g automorphisms of o r d e r 13 o r 7 a r e a l r e a d y known [ 1 4 ] , [ 1 5 ] . Among t h e d e s i g n s a r i s i n g from Hadamard m a t r i c e s of o r d e r 28 w i t h automorphisms
of o r d e r 13 o r 7 , only t h o s e r e l a t e d t o t h e Hadamard m a t r i x of q u a d r a t i c r e s i due type y i e l d an extremal (56,28,12) code. Any 2 - ( 3 5 , 1 8 , 9 ) d e s i g n d e f i n e s a doubly-even
(72,36) code with minimum
weight d 2 8. Although t h e e x i s t e n c e of an extremal (72,36,16) code i s s t i l l i n doubt, i t i s known t h a t such a code cannot be o b t a i n e d from a 2 - ( 3 5 , 1 8 , 9 ) des i g n w i t h an automorphism of o r d e r 17 [ 1 6 ] . More g e n e r a l l y , such a code cannot p o s s e s s automorphisms of o r d e r 17 or any l a r g e r prime o r d e r 161, 1111, 1121. Another approach f o r c o n s t r u c t i o n of doubly-even of symmetric 2-(36,15,6)
d e s i g n s ( s e e Theorem 1 . 1 ,
(72,36) codes i s by means
( i ) ) In . t h i s c a s e Theorem
2.1 g i v e s t h e f o l l o w i n g r e s u l t : THEOREM 2.4.
A necessary condition f o r a 2-(36,15,6)
design t o y i e l d a
( 7 2 , 3 6 , d 2 1 2 ) code i s t h e absence of any a r c s of s i z e 3 . The n e x t theorem i s an example of a better-behaved
THEOREM 2.5. A doubly-even s e l f - d u a l
situation.
(64,32) code d e f i n e d by a 2 - ( 3 1 , 1 0 , 3 )
d e s i g n D i s e x t r e m a l ( i . e . has minimum weight 12) i f and only i f D does n o t p o s s e s s any o v a l s . Proof. The f i r s t row of t h e g e n e r a t o r m a t r i x ( 3 1 , where A i s now an i n c i -
S.ririitzetric Dcsigrrs without Ovals
455
d e s i g n D , h a s w e i g h t 32, w h i l e a l l re-
dence m a t r i x of a symmetric 2 - ( 3 1 , 1 0 , 3 )
maining rows a r e of weight. 12. The w e i g h t o f a sum of t w o rows of ( 3 ) i s 24 p r o v i d e d t h a t one of t h e rows i s t h e f i r s t row of
( 3 ) , or 16 i f b o t h rows are
o t h e r t h a n t h e f i r s t row of ( 3 ) . The sum of 3 r o w s of
(3) including t h e f i r s t
row i s 20. C o n s i d e r now t h e s u m of a t r i p l e of rows n o t i n c l u d i n g t h e f i r s t
row. L e t n . d e n o t e t h e number of b l o c k s c o n t a i n i n g e x a c t l y i p o i n t s of t h e t r i p l e of p o i . n t s o f D c o r r e s p o n d i n g t o t h e choosen t r i p l e of rows. W e have:
no + n l +
n2 +
n3
=
31,
n l + 2n2 + 3n3
=
3.10,
n2 + 3n3 = 3.3. T h i s system h a s t h e f o l l o w i n g s o l u t i o n s :
no
nl
10
n2
n3 0
12
9
9 1 5
6
1
8 1 8
3
2
7 2 1
0
3
E v i d e n t l y , t h e w e i g h t o f t h e sum o f t h e c o n s i d e r e d t r i p l e o f r o w s i s 4 + n l + n 3 2 16. F u r t h e r m o r e , t h e w e i g h t of a sum of 4 rows of
(3) including t h e f i r s t r o w
i s 4+n +n +1 2 12. 0 2 Now w e e s t i m a t e t h e w e i g h t of t h e sum of a q u a d r u p l e of rows of ( 3 ) n o t i n c l u d i n g t h e f i r s t row. Denoting by n , ( 0 S i S 4 ) t h e number of b l o c k s of D c o n t a i n i n g e x a c t l y i p o i n t s from t h e q u a d r u p l e of p o i n t s c o r r e s p o n d i n g t o t h e choosen q u a d r u p l e of rows, one h a s : no + n l + n
1
n2 +
n3 +
+ 2n2 + 3n
n4 = 31,
+ 4n
= 4.10, 3 4 n 2 + 3n3 + 6n4 = 6 . 3 ,
= 3n + 8n4 + 4 . 3 The w e i g h t of t h e c o n s i d e r e d sum i s 4+n +n = 8+4n +en4. By o u r a s s u m p t i o n 1 3 3 D d o e s n o t p o s s e s s any o v a l s , hence n +n > 0 , and c o n s e q u e n t l y t h e w e i g h t of
whence n l
t h e sum of f o u r rows i s a t l e a s t 12. Thus t h e w e i g h t of any l i n e a r c o m b i n a t i o n of a t m o s t 4 rows of t h e g e n e r a -
tor m a t r i x ( 3 ) is a t l e a s t 12. A
symmetric d e s i g n of an odd o r d e r k-X c o n t a i n s a n o v a l if a n d o n l y i f t h e
d u a l d e s i g n c o n t a i n s an o v a l ( 2 1 . The f o l l o w i n g m a t r i x
1
At
1 01...1
I
(7)
V.D. Tonchev
456
i s a p a r i t y check matrix of t h e code generated by ( 3 ) , and s i n c e t h e code is s e l f - d u a l , t h e matrix ( 7 ) is a l s o a generator matrix of t h e same code. Since t . is an incidence matrix of t h e d u a l design of D , we can apply t h e same argu-
A
ments f o r t h e weight of t h e sum of a t most 4 rows of ( 7 ) . Since a codeword of weight 8 o r less must be sum of a t most 4 rows of one of t h e matrices ( 3 ) o r
( 7 1 , t h i s completes t h e proof. The 2-(31,10,3)
design l i s t e d i n H a l l book ( 7 1 possesses o v a l s . Since t h e r e
a r e no c y c l i c d i f f e r e n c e - s e t s with parameters ( 3 1 , 1 0 , 3 ) , it can be e a s i l y seen by use a r e s u l t of Aschbacher [ l ] t h a t t h e g r e a t e s t prime which can be an o r d e r of an automorphism of a 2-(31,10,3) design i s 7. We enumerated a l l such designs f i n d i n g e x a c t l y 4 nonisomorphic s o l u t i o n s , one of them being without o v a l s [17]
-
A s s u m e t h a t an automorphism 5 of o r d e r 7 a c t s on t h e p o i n t s and blocks a s follows :
B = (1,2
,...,7 ) ( 8 , 9 ,...,14) (15,16 ,..., 21) (22,23 ,...,28) (29) (30)( 3 1 ) .
Then a 2-(31,10,3)
design without o v a l s i s defined by t h e following base
blocks : B1 = (1,8,13,14,15,18,21,22,25,27),
B22 = (1,4,6,10,14,19,20,22,28,31) I
B8 = (1,6,7,8,11,16,18,24,28,29), B15
= (1,4,7,9,11,15,20,26,27,30),
B29 = (8,9,10,11,12,13,14,29,30,31), B30 = (15,16,17,18,19,20,21,29,30,31), B g l = (22,23,24,25,26,27,28,29,30,31).
The f u l l automorphism group of t h i s design i s of o r d e r 42. We do n o t know whether
t h e code obtained from t h e above design is e q u i v a l e n t t o t h e extremal (64,
32) code c o n s t r u c t e d by Pasquier 1101. F i n a l l y , l e t us consider t h e parameters 2-(40,27,18).
By arguments s i m i l a r
t o those from t h e proof of t h e preceding theorem t h e following p r o p o s i t i o n can be proved. THEOREM 2.6.
The minimum weight d of a doubly-even
(80,401
code obtained
from a 2-(40,27,18) design D i s 8 i f and only i f t h e complementary 2-(40,13,4) design D p o s s e s s e s an oval. Otherwise, d 2 1 2 . L e t us remark t h a t f o r e x t r e m a l i t y one needs d = 16. As a f i r s t c a n d i d a t e ,
w e checked t h e complement of t h e 2-(40,13,4) design formed by t h e hyperplanes i n PG(3,3). However, it t u r n s o u t t h a t t h e r e s u l t i n g code has minimum weight d = 12. REFERENCES [ l ] M. Aschbacher, On c o l l i n e a t i o n groups of symmetric block d e s i g n s , J. Combin. Theory, A 11 (19711, 272-281. [ 2 1 E.F. Assmus, J r . , and J . H . van L i n t , Ovals i n p r o j e c t i v e d e s i g n s , J . Combin. Theory, A 27 (19791, 307-324. 31 Th. Beth, D. Jungnickel, H. Lenz, "Design Theory", B.I. Wissenschaftsverlag,
Symmetric Designs without Ovals
457
Ztirich 1 9 8 5 . Bhargava, J . M . S t e i n , ( v , k , h ) c o n f i g u r a t i o n s and s e l f - d u a l codes, Information and C o n t r o l , 28 ( 1 9 7 5 ) , 352-355. P . J . Cameron and J . H . van L i n t , "Graphs, Codes and Designs", London Math. SOC. L e c t u r e Note Ser. 4 3 , Cambridge U n i v e s s i t y P r e s s , Cambridge 1 9 8 0 . J . H . Conway, V . P l e s s , On primes d i v i d i n g t h e group o r d e r of a doubly-even ( 7 2 , 3 6 , 1 6 ) code and t h e group o r d e r of a q u a t e r n a r y ( 2 4 , 1 2 , 1 0 ) code, D i s c r e t e Math. 38 ( 1 9 8 2 ) , 143-156. M. H a l l , Jr., "Combinatorial Theory", Ginn ( b l a i s d e l l ) , Boston 1967. F.J. MacWilliams and N . J . A . Sloane, "The Theory of E r r o r - C o r r e c t i n g Codes", Noth-Holland, Amsterdam 1 9 7 7 . E . J . Morgan, Arcs i n block d e s i g n s , A r s Combinatoria 4 ( 1 9 7 7 ) , 3-16. G. P a s q u i e r , A b i n a r y extremal doubly even s e l f - d u a l code ( 6 4 , 3 2 , 1 2 ) obt a i n e d from an extended Reed-Solomon code over F 1 6 , IEEE T r a n s , Inform. Theory, 27 ( 1 9 8 1 ) , 807-808. V . P l e s s , 2 3 does n o t d i v i d e t h e o r d e r of t h e group of a ( 7 2 , 3 6 , 1 6 ) doubly even code, IEEE T r a n s . Inform. Theory, 2 8 ( 1 9 8 2 ) , 113-117. V. P l e s s , J . G . Thompson, 17 does n o t d i v i d e t h e o r d e r of t h e group of a ( 7 2 , 3 6 , 1 6 ) doubly even code, IEEE T r a n s , Inform. Theory, 28 ( 1 9 8 2 ) , 537-
[ 41 V.K.
i
51
1
61
1
91 [lo1
[ll] [ 121
541.
V.D. Tonchev, Block d e s i g n s of Hadamard t y p e and s e l f - d u a l codes, Problemi p e r e d a t c h i i n f o r m a t s i i , 19 ( 1 9 8 3 ) , No. 4 , 25-30. V.D. Tonchev, Hadamard m a t r i c e s of o r d e r 2 8 w i t h automorphisms of o r d e r 1 3 , J . Combin. Theory, A 35 ( 1 9 8 3 ) , 4 3 - 5 7 . V.D. Tonchev, Hadamard m a t r i c e s of o r d e r 28 w i t h automorphisms of o r d e r 7 , J. Combin. Theory, A 40 ( 1 9 8 5 ) , 62-81. V . D . Tonchev, R.V. Raev, C y c l i c 2 - ( 1 7 , 8 , 7 ) d e s i g n s and r e l a t e d doubly-even codes, Compt. rend. Acad. bulg. S c i . , 35 ( 1 9 8 2 1 , 1367-1370. V.D. Tonchev, Symmetric 2 - ( 3 1 , 1 0 , 3 ) d e s i g n s with automorphisms of o r d e r 7 , Annals of D i s c r e t e Math. ( t o a p p e a r ) .
Annals of Discrete Mathematics 37 (1988) 451-458 0 Elsevier Science Publishers B.V. (North-Holland)
SYMMETRIC DESIGNS WITHOUT OVALS AND EXTREMAL SELF-DUAL CODES Vladimir D . Tonchev* I n s t i t u t e of Mathematics, S o f i a 1090, P. 0. Box 373, Bulgaria ABSTRACT. Constructions of doubly-even s e l f - d u a l codes from symmetric d e s i g n s a r e discussed. I t i s shown t h a t t h e absence of o v a l s i n a design i s u s u a l l y necessary, and sometimes even s u f f i c i e n t c o n d i t i o n f o r t h e corresponding code t o be extremal.
1. INTRODUCTION The terminology and n o t a t i o n s from design and coding theory used i n t h i s paper a r e i n accordance with those from [ 3 A binary
t o r space V
1,
[ 5 1, and L8] r e s p e c t i v e l y .
(n,k) code C is a k-dimensional subspace of t h e n-dimensional vec-
1
Over GF(2). Given an ( n , k ) code C , t h e (n,n-k) code C
= {xeVn:
yx = 0 f o r each F C ) i s c a l l e d t h e orthogonal, o r dual of C. A matrix with t h e p r o p e r t y t h a t t h e l i n e a r span of i t s rows g e n e r a t e s t h e code C , i s a g e n e r a t o r
1 m a t r i x of C. The g e n e r a t o r m a t r i c e s of t h e d u a l code C a r e c a l l e d p a r i t y check matrices of C. We s h a l l o f t e n r e f f e r t o t h e elements of a code a s codewords, o r words only. The weight of a codeword i s t h e number of i t s nonzero p o s i t i o n s , and t h e minimum weight of a code i s t h e weight of a l i g h t e s t nonzero codeword. An
( n , k , d ) code i s an ( n , k ) code with minimum weight d. A code C i s self-orthogonal
(resp. self-dual)
i f CCC'
(resp. C =
2 ) . The
weights of a l l words i n a s e l f - o r t h o g o n a l code a r e even. I f i n a d d i t i o n a l l weights a r e d i v i s i b l e by 4 , t h e code i s c a l l e d doubly-even. d u a l ( n , n / 2 ) code e x i s t s i f and only if n
=
0 (mod
a),
A doubly-even
self-
and t h e minimum weight
d of such a code i s bounded by d S 4[ 11/24] + 4 , (cf.
[a]).
(1)
A code s a t i s f y i n g t h e e q u a l i t y i n (1) is c a l l e d extremal. The code-
words of minimum weight i n an extremal doubly-even s e l f - d u a l code y i e l d o r 5-design provided t h a t
I-, 3-,
n 5 16, 8 , o r 0 (mod 2 4 ) .
The number of extremal codes i s f i n i t e ( b u t unknown), t h e s m a l l e s t open c a s e being f o r n = 7 2 . The following theorem d e s c r i b e s a c o n s t r u c t i o n of doubly-even s e l f - d u a l
CO-
d e s from symmetric d e s i g n s . THEOREM 1.1. Let A be an incidence matrix of a symmetric 2-(v,k,k)
of an odd o r d e r k-A.
Then
( i )If k E 3 (mod 4 ) , t h e code generated by t h e matrix
*Research p a r t i a l l y supported by t h e
cscMBunder
contract
37/1987-
design
V.D. Tonchev
452
i s a doubly-even s e l f - d u a l
(2v,v) code.
(ii) I f k Z 2 (mod 4 ) , then t h e code generated by t h e matrix
(3)
i s a doubly-even s e l f - d u a l
(2v+2, v + l ) code.
This statement i s a v a r i a n t of a m r e g e n e r a l c o n s t r u c t i o n ( c f . [ 21
,[
4) )
.
I n t h i s paper we a r e i n t e r e s t e d i n necessary and s u f f i c i e n t c o n d i t i o n s f o r a code c o n s t r u c t e d i n t h i s way t o be extremal.
In o t h e r words, what combinato-
r i a l p r o p e r t i e s should a design possess i n o r d e r t o produce an extremal code? A notion p l a y i n g an important r o l e i n t h e study of symmetric d e s i g n s , espe-
c i a l l y p r o j e c t i v e p l a n e s and b i p l a n e s , is t h a t of "arc" o r "oval". An a r c i n a design i s a s u b s e t S of p o i n t s such t h a t each block i n t e r s e c t s S i n a t most 2 p o i n t s . The c a r d i n a l i t y of an a r c S i n a n o n t r i v i a l symmetric 2-(v,k,h)
design
of an odd o r d e r k-h i s bounded by
IS/ S (k+h-l)/h.
(4)
An a r c with maximum number of p o i n t s i n t h e sense of bound
( 4 ) i s c a l l e d an
oval [ 2 ] . I t w i l l be seen by t h e next e x p o s i t i o n t h a t t h e absence of o v a l s i n a sym-
m e t r i c design i s u s u a l l y necessary, and sometimes even s u f f i c i e n t c o n d i t i o n f o r t h e corresponding s e l f - d u a l code t o be extremal. 2 . ARCS AND CODES
Let D be a symmetric 2-(v,k,h) Denoting by ni
design with an a r c S of s i z e s , i . e .
IS/ =s.
( i = 0,1,2) t h e number of blocks having e x a c t l y i common p o i n t s
w i t h S , w e have:
no
+
nl
+
*2 =
vt
Consider t h e code with g e n e r a t o r matrix ( 2 ) defined by D . C l e a r l y , t h e weight of t h e sum (over GF(2)) of s rows of t h e g e n e r a t o r matrix ( 2 ) indexed by t h e p o i n t s of an a r c of s i z e s i s s + n l = s(k+l)-(s-l)A). S i m i l a r l y , t h e sum of s
r o w s of t h e matrix ( 3 ) , indexed by t h e p o i n t s of an a r c of s i z e s has weight
n+ 1 s+nl+(l+(-l) ) /2
=
s ( k + l - ( s - l ) h ) + (1+(-1)'+l) / 2 .
Thus we have t h e following THEOREM
2.1.
Let D be a symmetric 2-(v,k,h)
design s a t i s f y i n g t h e assump-
Symmetric Designs without Ovals
453
t i o n of Theorem 1.1, and admitting an a r c S of s i z e s . Then t h e minimum weight of t h e code d e f i n e d by D a s i n Theorem 1.1 i s bounded a s follows: d S {
s ( k + l - ( s - l ) h i n c a s e (i); s (k+l- (s-1)h ) + ( l + ( - l'+') )
(6)
/2 i n case (ii).
I t t u r n s o u t t h a t i f t h e a r c S i n an o v a l , t h e i n e q u a l i t i e s (6) a r e o f t e n
s h a r p e r than ( 1 ) . Thus a code a r i s i n g from a design with o v a l s i s u s u a l l y n o t extremal. We s h a l l i l l u s t r a t e t h i s by some examples. In Table 1 t h e parameters of symmetric 2-(v,k,A)
d e s i g n s with k - h S 9 ,
y i e l d i n g doubly-even codes by t h e c o n s t r u c t i o n of Theorem 1.1. a r e l i s t e d . TABLE 1 NO.
k-h
1
1
v 3
Codes
h
k 2
1
Extended Hamming ( 8 , 4 , 4 ) code; extremal.
2
3
3
Extended Golay (24,12,8) code; extremal.
3
5
19
10
5
Three extremal (40,20,8) codes.
4
7
31
10
3
Extremal (64,32,12) code from any design without o v a l s (Theorem 2 . 5 ) .
5
7
27
14
7
An extremal
6
9
35
18
9
7
9
36
15
6
8
9
40
27
18
1
1
6
Parameters N o . 2-(4t-l,2t,t). 2t-1.t-1))
(56,28,12) code from t h e q u a d r a t i c r e s i d u e design.
Is t h e r e any extremal (72,36,16) code? ( 8 0 , 4 0 , d > l 2 ) code from any design without ovals.
A
1 , 2 , 3 , 5 , 6 i n Table 1 a r e of Hadamard t y p e , i . e .
The s i z e of an o v a l i n a Hadamard 2 - ( 4 t - l , 2 t , t )
design i s 3 , t h u s t h e bound from Theorem 2.1 f o r t h e minimum weigth
of a code of type (ii) i s 4. However, a Hadamard 2 - ( 4 t - l , 2 t , t ) t
of t h e form
( o r 2-(4t-1, design with odd
> 1 does n o t admit any o v a l s ; t h i s follows from a r e s u l t of Morgan [ 9 1 , and
can be a l s o e a s i l y seen d i r e c t l y . For, i f a 2 - ( 4 t - l , 2 t , t )
design p o s s e s s e s an
a r c of s i z e 3, then t h e sum of t h r e e rows of i t s incidence matrix A indexed by t h e p o i n t s of t h e a r c , as w e l l as t h e sum of a l l remaining rows over GF(2) is t h e z e r o v e c t o r . Consequently, t h e rank of t h e incidence matrix over GF(2) must be l e s s o r equal t o 4t-3. (I,J-A),
where J.denotes
On t h e o t h e r hand, s i n c e t i s odd, t h e matrix
t h e a l l - o n e m a t r i x , g e n e r a t e s a s e l f - d u a l code. Thus
t h e rank of J-A over G F ( 2 ) i s 4t-1, whence t h e rank of A i s 4t-2,
a contradic-
t ion. Estimating t h e weight of a sum of a t most 4 rows of a matrix of t h e form ( 3 ) , where A i s an incidence matrix of a Hadamard 2 - ( 4 t - l , 2 t , t )
odd t , say t = 2 m + l , THEOREM 2 . 2 .
design with
t h e following can be proved:
[13
1.
The minimum weight of a (16m+8,8m+4) code with genera-
454
V.D. Torichav
t o r m a t r i x ( 3 ) d e f i n e d by a Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n i s e q u a l t o 4 if m=O,
and i s a t l e a s t 8 i f m
> 0.
In f a c t , t h e codes d e r i v e d from Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n s f o r
m S 2 a r e a l l e x t r e m a l . I t i s worth n o t i n g t h a t Hadamard 2 - ( 4 t - l , 2 t , t ) d e s i g n s which a r e e x t e n d a b l e i n t o isomorphic Hadamard 3 - ( 4 t , 2 t , t - l )
designs, y i e l d
e q u i v a l e n t codes. For i n s t a n c e , t h e r e e x i s t e x a c t l y 6 nonisomorphic 2 - ( 1 9 , 1 0 , 5 ) d e s i g n s and 3 nonisomorphic 3 - ( 2 0 , 1 0 , 4 ) d e s i g n s , producing 3 i n e q u i v a l e n t extremal (40,20,8) codes [ 1 3 ) . The absence of o v a l s i n a 2-(8m+3,4m+2,2m+l) design with m > 2 i s n o t a s u f f i c i e n t c o n d i t i o n f o r t h e e x t r e m a l i t y of t h e r e l a t e d code.
In t h i s case
some a d d i t i o n a l c o n d i t i o n s are t o be f u l f i l e d , i n c l u d i n g t h e absence of o v a l s i n t h e complementary 2-(8m+3,4m+1,2m) d e s i g n a s w e l l . THEOREM 2.3.
[ 1 3 1 . 3 n e c e s s a r y c o n d i t i o n f o r an (16+8,8m+4) code d e f i n e d
by a Hadamard 2-(8m+3,4m+2,2m+l) d e s i g n D t o have minimum weight d 2 12 i s any t r i p l e of pOints of D t o be c o n t a i n e d i n a t l e a s t 2 b l o c k s , and a t most 2m-1 b l o c k s of D .
-
Formulated f o r t h e complementary 2-(8m+3,4m+2,2m) design D , t h i s c o n d i t i o n s t a t e s t h a t each t r i p l e of p o i n t s must occur i n a t most 2m-2 l e a s t one block of
D.
In particular,
-
b l o c k s , and a t
D cannot have gny o v a l s .
The enumeration of a l l 2-(27,14,7)
ciesigns up t o isomorphism i s n o t y e t
completed. However, t h e only primes which can d i v i d e t h e o r d e r of t h e automorphism group of a Hadamard m a t r i x of o r d e r 28 a r e 13, 7 , 3 and 2 , and t h e m a t r i c e s p o s s e s s i n g automorphisms of o r d e r 13 o r 7 a r e a l r e a d y known [ 1 4 ] , [ 1 5 ] . Among t h e d e s i g n s a r i s i n g from Hadamard m a t r i c e s of o r d e r 28 w i t h automorphisms
of o r d e r 13 o r 7 , only t h o s e r e l a t e d t o t h e Hadamard m a t r i x of q u a d r a t i c r e s i due type y i e l d an extremal (56,28,12) code. Any 2 - ( 3 5 , 1 8 , 9 ) d e s i g n d e f i n e s a doubly-even
(72,36) code with minimum
weight d 2 8. Although t h e e x i s t e n c e of an extremal (72,36,16) code i s s t i l l i n doubt, i t i s known t h a t such a code cannot be o b t a i n e d from a 2 - ( 3 5 , 1 8 , 9 ) des i g n w i t h an automorphism of o r d e r 17 [ 1 6 ] . More g e n e r a l l y , such a code cannot p o s s e s s automorphisms of o r d e r 17 or any l a r g e r prime o r d e r 161, 1111, 1121. Another approach f o r c o n s t r u c t i o n of doubly-even of symmetric 2-(36,15,6)
d e s i g n s ( s e e Theorem 1 . 1 ,
(72,36) codes i s by means
( i ) ) In . t h i s c a s e Theorem
2.1 g i v e s t h e f o l l o w i n g r e s u l t : THEOREM 2.4.
A necessary condition f o r a 2-(36,15,6)
design t o y i e l d a
( 7 2 , 3 6 , d 2 1 2 ) code i s t h e absence of any a r c s of s i z e 3 . The n e x t theorem i s an example of a better-behaved
THEOREM 2.5. A doubly-even s e l f - d u a l
situation.
(64,32) code d e f i n e d by a 2 - ( 3 1 , 1 0 , 3 )
d e s i g n D i s e x t r e m a l ( i . e . has minimum weight 12) i f and only i f D does n o t p o s s e s s any o v a l s . Proof. The f i r s t row of t h e g e n e r a t o r m a t r i x ( 3 1 , where A i s now an i n c i -
S.ririitzetric Dcsigrrs without Ovals
455
d e s i g n D , h a s w e i g h t 32, w h i l e a l l re-
dence m a t r i x of a symmetric 2 - ( 3 1 , 1 0 , 3 )
maining rows a r e of weight. 12. The w e i g h t o f a sum of t w o rows of ( 3 ) i s 24 p r o v i d e d t h a t one of t h e rows i s t h e f i r s t row of
( 3 ) , or 16 i f b o t h rows are
o t h e r t h a n t h e f i r s t row of ( 3 ) . The sum of 3 r o w s of
(3) including t h e f i r s t
row i s 20. C o n s i d e r now t h e s u m of a t r i p l e of rows n o t i n c l u d i n g t h e f i r s t
row. L e t n . d e n o t e t h e number of b l o c k s c o n t a i n i n g e x a c t l y i p o i n t s of t h e t r i p l e of p o i . n t s o f D c o r r e s p o n d i n g t o t h e choosen t r i p l e of rows. W e have:
no + n l +
n2 +
n3
=
31,
n l + 2n2 + 3n3
=
3.10,
n2 + 3n3 = 3.3. T h i s system h a s t h e f o l l o w i n g s o l u t i o n s :
no
nl
10
n2
n3 0
12
9
9 1 5
6
1
8 1 8
3
2
7 2 1
0
3
E v i d e n t l y , t h e w e i g h t o f t h e sum o f t h e c o n s i d e r e d t r i p l e o f r o w s i s 4 + n l + n 3 2 16. F u r t h e r m o r e , t h e w e i g h t of a sum of 4 rows of
(3) including t h e f i r s t r o w
i s 4+n +n +1 2 12. 0 2 Now w e e s t i m a t e t h e w e i g h t of t h e sum of a q u a d r u p l e of rows of ( 3 ) n o t i n c l u d i n g t h e f i r s t row. Denoting by n , ( 0 S i S 4 ) t h e number of b l o c k s of D c o n t a i n i n g e x a c t l y i p o i n t s from t h e q u a d r u p l e of p o i n t s c o r r e s p o n d i n g t o t h e choosen q u a d r u p l e of rows, one h a s : no + n l + n
1
n2 +
n3 +
+ 2n2 + 3n
n4 = 31,
+ 4n
= 4.10, 3 4 n 2 + 3n3 + 6n4 = 6 . 3 ,
= 3n + 8n4 + 4 . 3 The w e i g h t of t h e c o n s i d e r e d sum i s 4+n +n = 8+4n +en4. By o u r a s s u m p t i o n 1 3 3 D d o e s n o t p o s s e s s any o v a l s , hence n +n > 0 , and c o n s e q u e n t l y t h e w e i g h t of
whence n l
t h e sum of f o u r rows i s a t l e a s t 12. Thus t h e w e i g h t of any l i n e a r c o m b i n a t i o n of a t m o s t 4 rows of t h e g e n e r a -
tor m a t r i x ( 3 ) is a t l e a s t 12. A
symmetric d e s i g n of an odd o r d e r k-X c o n t a i n s a n o v a l if a n d o n l y i f t h e
d u a l d e s i g n c o n t a i n s an o v a l ( 2 1 . The f o l l o w i n g m a t r i x
1
At
1 01...1
I
(7)
V.D. Tonchev
456
i s a p a r i t y check matrix of t h e code generated by ( 3 ) , and s i n c e t h e code is s e l f - d u a l , t h e matrix ( 7 ) is a l s o a generator matrix of t h e same code. Since t . is an incidence matrix of t h e d u a l design of D , we can apply t h e same argu-
A
ments f o r t h e weight of t h e sum of a t most 4 rows of ( 7 ) . Since a codeword of weight 8 o r less must be sum of a t most 4 rows of one of t h e matrices ( 3 ) o r
( 7 1 , t h i s completes t h e proof. The 2-(31,10,3)
design l i s t e d i n H a l l book ( 7 1 possesses o v a l s . Since t h e r e
a r e no c y c l i c d i f f e r e n c e - s e t s with parameters ( 3 1 , 1 0 , 3 ) , it can be e a s i l y seen by use a r e s u l t of Aschbacher [ l ] t h a t t h e g r e a t e s t prime which can be an o r d e r of an automorphism of a 2-(31,10,3) design i s 7. We enumerated a l l such designs f i n d i n g e x a c t l y 4 nonisomorphic s o l u t i o n s , one of them being without o v a l s [17]
-
A s s u m e t h a t an automorphism 5 of o r d e r 7 a c t s on t h e p o i n t s and blocks a s follows :
B = (1,2
,...,7 ) ( 8 , 9 ,...,14) (15,16 ,..., 21) (22,23 ,...,28) (29) (30)( 3 1 ) .
Then a 2-(31,10,3)
design without o v a l s i s defined by t h e following base
blocks : B1 = (1,8,13,14,15,18,21,22,25,27),
B22 = (1,4,6,10,14,19,20,22,28,31) I
B8 = (1,6,7,8,11,16,18,24,28,29), B15
= (1,4,7,9,11,15,20,26,27,30),
B29 = (8,9,10,11,12,13,14,29,30,31), B30 = (15,16,17,18,19,20,21,29,30,31), B g l = (22,23,24,25,26,27,28,29,30,31).
The f u l l automorphism group of t h i s design i s of o r d e r 42. We do n o t know whether
t h e code obtained from t h e above design is e q u i v a l e n t t o t h e extremal (64,
32) code c o n s t r u c t e d by Pasquier 1101. F i n a l l y , l e t us consider t h e parameters 2-(40,27,18).
By arguments s i m i l a r
t o those from t h e proof of t h e preceding theorem t h e following p r o p o s i t i o n can be proved. THEOREM 2.6.
The minimum weight d of a doubly-even
(80,401
code obtained
from a 2-(40,27,18) design D i s 8 i f and only i f t h e complementary 2-(40,13,4) design D p o s s e s s e s an oval. Otherwise, d 2 1 2 . L e t us remark t h a t f o r e x t r e m a l i t y one needs d = 16. As a f i r s t c a n d i d a t e ,
w e checked t h e complement of t h e 2-(40,13,4) design formed by t h e hyperplanes i n PG(3,3). However, it t u r n s o u t t h a t t h e r e s u l t i n g code has minimum weight d = 12. REFERENCES [ l ] M. Aschbacher, On c o l l i n e a t i o n groups of symmetric block d e s i g n s , J. Combin. Theory, A 11 (19711, 272-281. [ 2 1 E.F. Assmus, J r . , and J . H . van L i n t , Ovals i n p r o j e c t i v e d e s i g n s , J . Combin. Theory, A 27 (19791, 307-324. 31 Th. Beth, D. Jungnickel, H. Lenz, "Design Theory", B.I. Wissenschaftsverlag,
Symmetric Designs without Ovals
457
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