5-Designs with Three Intersection Numbers - Core

JOURNALOF COMBINATORIALTHEORY,Series A 69, 36-50 (1995)

5-Designs with Three Intersection Numbers YURY J. IONIN AND MOHAN S. SHRIKHANDE*' ~ Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859 Communicated by V. Pless Received April 21, 1992

It is well known that the Witt 5-(24, 8, 1) design and its complement have exactly three block intersection numbers. This paper is concerned with the converse question: Is the Witt 5-(24,8, 1) design the unique 5-@,k,h) design (up to complements) with exactly three intersection numbers? Using the annihilator polynomial of such a design, we prove that the answer is affirmative if k - 1 is a prime or h _< 4 or at least two intersection numbers are less than 5. Numerical evidence using MAPLE seems to support that the answer is always yes. © 1995 Academic Press, Inc.

INTRODUCTION W e follow Beth et al. [2] for basic design theory. Let D be a t-(v, k, A) design. For distinct blocks B i and B i the n u m b e r s [Bi A B/[ are known as the (block) intersection numbers. Intersection numbers, as is well known, provide a valuable tool in design theory. For instance, t-(u, k, A) designs with exactly one intersection n u m b e r are precisely the symmetric 2-(v, k, A) designs. Designs with exactly two intersection n u m b e r s are the so called quasi-symmetric designs and are of m u c h current interest [4], [12], [16]. T h e Witt designs with p a r a m e t e r s 5-(24, 8, 1) and 4-(23, 7, 1) are classical objects in design and coding theory. Their intersection n u m b e r s are respectively 0, 2, 4 and 1, 3. T h e c o m b i n e d efforts of Ito and co-workers [6],[10] and B r e m n e r [3] p r o v e d that the Witt 4-(23, 7, 1) design is the unique 4-design (up to c o m p l e m e n t s ) with exactly two intersection numbers. In personal c o m m u n i c a t i o n s in 1985 and 1992 with the second author, Bannai [1] m e n t i o n e d the p r o b l e m of classification of 5-designs with three intersection numbers. A reference to this p r o b l e m can be also f o u n d in H o b a r t ' s thesis [8]. T h e complete solution of this p r o b l e m appears to be very difficult. T h e present p a p e r contains some results *Supported by Central Michigan University Summer Fellowship Award 42137. *E-mail: 3IAMXMZ @ CMUVM.CSV.CMICH.EDU. 36 0097-3165/95 $6.00 Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

5-DESIGNS WITH 3 INTERSECTION SIZES

37

towards our assertion that the Witt 5-(24, 8, 1) design is the only 5-design (up to complements) with exactly three intersection numbers. Let D be a 5-(v, k, A) design having exactly three intersection numbers x i (i = 1,2,3), where x~ < x 2 < x3 < k. Two special cases are worth noting. If x~ = 0, then by the results of Ito et al. and Bremner, it follows that D or its complement is the Witt 5-(24, 8, 1) design. Also, in case D is a Steiner system S(5, k, u) with exactly three intersection numbers D, must be the Witt 5-(24, 8, 1) design. This follows from the results of Gross [7] and Noda [11]. The basic tool used in the present paper is the Delsarte annihilator polynomial P ( z ) = 1-[~=l(z - y i ) . These types of polynomials occur in the work of Delsarte [5] in the context of association schemes of coding theory. Section 1 is devoted to preliminary results. In Lemma 1.1, we use Ionin and Shrikhande [9] to obtain some relations involving v, k, A, P ( k ) and certain linear combinations of the coefficients of the Delsarte polynomial. Section 2 derives some necessary conditions which must be satisfied by the design D. For instance, Theorem 2.7 gives the inequality 2A(~,- 3) (~, - 4) > (k - 1)(k - 2)(k - 3)(k - 4)with equality if and only if x 1 = 0. This improves the results of Shrikhande [15]. In Section 4, we obtain (Theorem 4.1) the following characterization of the Witt 5-(24, 8, 1) design: Let D be a 5-(v, k, A) design with exactly three intersection numbers. If k - 1 is a prime, then D is the Witt 5-(24, 8, 1) design. An obvious equivalent restatement of the results of Gross and Noda is the following: Let D be a 5-(v, k, A) design with intersection numbers xi (i = 1, 2, 3), where x I < x 2 < x 3. If D is not the Witt 5-(24, 8, 1) design, then x 3 >__5. In Section 3, we use this restatement to give another proof of the above-mentioned results of Gross and Noda. In Section 5, Theorem 5.1 obtains the improvement to x 2 > 5. Our proof of Theorem 5.1 is somewhat technical and lengthy and is therefore given in the last section. We give some lower bounds implied by Theorem 5.1: Corollary 5.2 asserts that if D is not the Witt 5-(24,8, 1) design, then x 1 + x 2 + x 3 > 12; Corollary 5.3 states that if D or its complement is not the Witt 5-(24, 8, 1) design, then k > 11 and u > k + 11. Section 6 contains some finiteness results. Theorem 6.1 shows that there is no 5-(u, k, 2) design with exactly three intersection numbers. By a slight modification, we prove (Theorem 6.2) the same result for A = 3 and 4. Theorem 6.3 asserts that for a fixed A, there exist at most finitely many 5-(u, k, A) designs with three intersection numbers. Theorem 6.4 gives the same conclusion for a fixed k. All our results give some evidence that the only 5-design (up to complements) with exactly three intersection numbers might be the Witt 5-(24, 8, 1) design. In addition, we have the following numerical evidence obtained

38

IONIN AND SHRIKHANDE

using M A P L E : t h e r e is no 5-(~, k , A ) d e s i g n with t h r e e i n t e r s e c t i o n n u m b e r s for A _< 50 o r k _< 500 ( a n d A 4= 1).

1. PRELIMINARIES T h r o u g h o u t this p a p e r , D d e n o t e s a 5-(u, k, A) d e s i g n with exactly t h r e e i n t e r s e c t i o n n u m b e r s x i (i = 1, 2, 3). W e a s s u m e t h a t x 1 < x 2 < x 3 < k a n d k > 6. N o t e t h a t such a d e s i g n m u s t satisfy t h e i n e q u a l i t y v>_ k + 3. I n d e e d , if B 1 a n d B z a r e two blocks o f D such t h a t IB 1 c3 Bzl = Xl, t h e n IB~ U B21 = 2 k - x 1. Since x I < k - 3, this implies u > k + 3. W e assume, as is s t a n d a r d , t h a t t h e d e s i g n D is n o t c o m p l e t e , i.e., n o t every k - s e t o f p o i n t s is block. W e d e n o t e by Ai t h e n u m b e r o f blocks c o n t a i n i n g any fixed i - t u p l e o f points, 1 < i < 5. W e d e n o t e t h e ( D e l s a r t e ) a n n i h i l a t o r p o l y n o m i a l P ( z ) of t h e d e s i g n D by P ( z ) = F I 3 = l ( Z - X i ) . Then P(z)=z(z1)(z-2)-FlZ(Z1)+ F z z - F3, w h e r e F 1 = x 1 + x 2 + x 3 - 3,

F2

= X l X 2 q'- X l X 3 q- X 2 X 3 - - X 1 - - X 2 - - X 3 q-

1,

F3

= XlX2X

3.

(1)

T h e next r e s u l t collects s o m e tools, w h i c h will b e o f t e n u s e d in t h e sequel. LEMMA 1.1. L e t D be a 5-(u, k, A) design with exactly three intersection numbers x i (i = 1, 2, 3). Then (i) T h e following f o u r e q u a t i o n s hold: 2 A ( v - k ) ( v - k - 1 ) [ 3 ( k - 2) 2 - ( u - 4 ) F 1 ] = (k - 2)(k - 3)(k - 4)P(k). (v

-

(2)

3)(~ - 4)F 2 - 2(k - 1)(k - 2)(v - 4)F 1 + 3 ( k - 1 ) ( k - 2 ) 2 ( k - 3) = 0.

3(v - 2)(v - 3)F 3 - 2k(k

(3)

- 1 ) ( v - 3 ) F 2 + k ( k - 1 ) 2 ( k - 2 ) F 1 = 0. (4)

2(v - 3)[(3F1F 3 - F ~ ) k 2 + ( F 2 - 9FIFB)k + 3F3(2F 1 + F2) ] = (k - 1)(k - 2)[(9F 3 - FIF2)k 2 +(FIF 2 - 45F3)k + 6F3(F 1 + 9)].

(5)


39

(ii) If the smallest intersection n u m b e r is equal to l, then 2 ( v - 3 ) [ ( F 2 - 3 F 1 ) k 2 + ( 9 F 1 - F 2 ) k - 3 ( 2 F l + F2) ]

(6)

= (k - 1)(k - 2)(k - 3)[(F I - 9)k + 2(F I + 9)]. (iii) If 3 F 1 F 3 > F22, then k < 6 F I F 3 / ( 3 F I F 3 - F 2 ) .

Proof. (i) Specializing equations derived in Ionin and Shrikhande [9, T h e o r e m 3.6], we obtain Eqs. (2)-(4). T o obtain (5), multiply (3) by 3 F 3 and then subtract (4) multiplied b y / v 2.

(ii) If the smallest intersection n u m b e r is equal to 1, then F 2 = F 3 ~ 0 in (5). This implies (6). (iii) Consider (4) as a quadratic equation with respect to u - 3. Since the discriminant of this equation must be non-negative, this yields the inequality 4F2k2(k

-

1) 2 + 9 F 2 > 1 2 F ~ F 3 k ( k - 1)2(k - 2) + 1 2 F 2 F 3 k ( k

-

1).

If 9 F 2 < 1 2 F 2 F 3 k ( k - 1), then 4 F 2 k 2 ( k - 1) 2 > 1 2 F 1 F 3 k ( k - 1) 2 (k - 2), so F 2 k > 3 F I F 3 ( k - 2) and k < 6 F I F 3 / ( 3 F I F 3 - F 2 ) . Thus, it is sufficient to prove that 9 F 2 < 1 2 F 2 F 3 k ( k - 1), i.e., 3 F 3 < 4 F 2 k ( k - 1). Obviously, 3 F 3 = 3 X l X 2 X s < ( x I + x 2 + x 3 ) ( X l X 2 + X l X 3 + x2x3). Since x 1 + x 2 + x 3 < 3(k - 1), we shall prove that x ~ x 2 + x l x 3 + x 2 x 3
F 2, F 3 =g 0. W e can assume x I > 1, x 2 _> 2, and x 3 > 3. T h e n (x 1 - 1)(x 2 - 1) + (x l - 1)(x 3 - 1) + (x 2 - 1)(x 3 - 1)_> 2, so 4 F 2 >__ F 1 + 3. S i n c e k >_ 6, gFzk >_ 8 F 2 > 2 F 2 > F 1 + F 2 + 2 = XlX 2 + XaX 3 + X z X s. T h e p r o o f is now complete. Equations (2)-(4) imply several useful necessary conditions on the p a r a m e t e r s o f a 5-(u, k, A) design with intersection n u m b e r s xl, x2, and x 3. These conditions are given in the next section.

2. SOME NECESSARY CONDITIONS E q u a t i o n (3) implies immediately LEMMA 2.1. I f D is a 5-(v, k , A) design with exactly three intersection n u m b e r s , then ~ - 4 divides 3(k - 1)(k - 2)2(k - 3). COROLLARY 2.2. I f D is a 5-(v, k , ~) design with exactly three intersection n u m b e r s , then k - 4 divides 36A.

40

IONIN

AND

SHRIKHANDE

P r o @ Since /~4 = /~(/2 - - 4 ) / ( k - 4), t h e r e f o r e k - 4 divides A(u - 4). By L e m m a 2.1, k - 4 m u s t divide 3A(k - 1)(k - 2)2(k - 3). Since k 1-3mod(k-4),k-2-2mod(k-4),andk-3-= lmod(k-4),this implies k - 4 divides 36A. LEMMA 2.3. I f D is a 5-(v, k, A) design with exactly three intersection numbers, then v - 3 divides 3 k ( k - 1)2(k - 2)2(k - 3 ) / 2 .

Proof. E q u a t i o n (4) implies that v - 3 divides k ( k - 1)2(k - 2 ) F 1. Since v - 4 - - - l m o d ( u 3), Eq. (3) implies that v - 3 divides ( k 1)(k - 2)[2F 1 + 3(k - 2)(k - 3)]. T h e r e f o r e , u - 3 divides ( k - 1)(k 2)g, w h e r e g is the greatest c o m m o n divisor of k ( k - 1)F 1 a n d 2F~ + 3(k - 2)(k - 3). Since g divides 2F~ + 3(k - 2)(k - 3), it divides k ( k - 1 ) F 1 + 3 k ( k - 1)(k - 2)(k - 3 ) / 2 . Since g divides k ( k - 1)F~, it m u s t divide 3 k ( k - 1)(k - 2)(k - 3 ) / 2 . T h e r e f o r e , ~ - 3 divides 3 k ( k 1)2(k - 2)2(k - 3 ) / 2 . E q u a t i o n (2) implies LEMMA 2.4. I f D is a 5-(u, k, A) design with exactly three intersection numbers, then (u - 4 ) F 1 < 3(k - 2) 2. LEMMA 2.5. I f D is a 5-(v, k, A) design with exactly three intersection numbers, then (u - 4 ) F 1 >_ 2(k - 2)(k - 3), and the equality holds if and only if one o f the intersection numbers is zero.

Proof.

U s i n g (3) a n d (4), we o b t a i n

(u - 2)(u - 3)(u - 4)F 3 =

k(k

-

1)2(k

-

2)[(v

-

4)F 1 -

2(k

-

2)(k

-

3)1.

(7)

This i m m e d i a t e l y gives the desired conclusion. LEMMA 2.6. I f D is a 5-(v, k, A) design with exactly three intersection numbers, then u > k + 6.

Proof.

W e have to rule out u = k + 3 , v=k+4, and u = k + 5 . S u p p o s e v = k + 3. T h e n , for any two distinct blocks B 1 a n d B z of the design D, IB 1 U B21 < k + 3, a n d t h e r e f o r e IB 1 C3 B2[ >__ k - 3. T h u s the i n t e r s e c t i o n n u m b e r s of the design D are k - 3, k - 2, a n d k - 1. This implies F 1 = 3(k - 3) a n d P ( k ) = 6. P u t t i n g these values in (2), o b t a i n A = (k - 2)(k - 3)(k - 4 ) / 6 . T h e r e f o r e , Ao =

7 (;)=( )io

ev°

oto .o,nt

case since the design D is n o t complete.

is a b , o c k

is not the


41

S u p p o s e u = k + 4. T h e n L e m m a 2.1 i m p l i e s t h a t k divides 36 a n d L e m m a 2.3 i m p l i e s t h a t k + 1 divides 216. N o k > 6 satisfies b o t h divisibility c o n d i t i o n s . S u p p o s e v = k + 5. T h e n L e m m a 2.1 i m p l i e s t h a t k + 1 divides 216 a n d L e m m a 2.3 i m p l i e s t h a t k + 2 d i v i d e s 1080. T h e o n l y p o s s i b l e v a l u e o f k is k = 7. T h e n u = 12 a n d (4) yields 1 5 F 3 - 7 F 2 + 7 0 F l = 0. T h e r e f o r e 7 d i v i d e s F 3. T h i s is i m p o s s i b l e since F 3 is t h e p r o d u c t of t h e i n t e r s e c t i o n n u m b e r s of t h e d e s i g n D a n d t h e i n t e r s e c t i o n n u m b e r s are less t h a n k=7. T h e p r o o f of L e m m a 2.6 is n o w c o m p l e t e . T h e n e x t r e s u l t i m p r o v e s S h r i k h a n d e [15, C o r o l l a r y B]. THEOREM 2.7. L e t D be a 5-(u, k, A) design with exactly three intersection n u m b e r s . Then 2A(v - 3)(u - 4) > ( k - 1)(k - 2)(k - 3)(k - 4) with equality i f a n d only i f one o f the intersection n u m b e r s is zero. Proof. P u t e x p r e s s i o n s for F 2 a n d F 3 f r o m (3) a n d (4) i n t o Eq. (2). U s i n g P ( k ) = k ( k - 1)(k - 2) - F l k ( k - 1) + F z k - F3, we o b t a i n [(v - 4)F 1 - 2(k - 2)(k -

3)1 [ 2 A ( u -k(k

= k(k

- 2 ) ( u - 3 ) ( v - 4) - 1 ) ( k - 2 ) ( k - 3 ) ( k - 4)]

- 2 ) ( u - 2 ) [ 2 A ( u - 3 ) ( u - 4) -(k

- 1)(k - 2)(k - 3)(k - 4)].

(8)

S u p p o s e t h a t 2A(u - 3)(u - 4) < ( k - 1)(k - 2 ) ( k - 3)(k - 4). T h e n (8) a n d L e m m a 2.4 i m p l y t h a t 2A(u - 2)(u - 3)(v - 4) < k ( k - 1)(k 2 ) ( k - 3 ) ( k - 4). S i n c e k < u - 2, this i m p l i e s k(k

- 1 ) ( k - 2 ) ( k - 3 ) ( k - 4) - 2 a ( v - 2 ) ( u - 3 ) ( u - 4) < (v - 2)[(k

- 1 ) ( k - 2 ) ( k - 3 ) ( k - 4) - 2 A ( u - 3 ) ( u - 4 ) ] .

T h e r e f o r e , t h e e q u a l i t y (8) i m p l i e s t h a t (u - 4 ) F 1 - 2 ( k - 2 ) ( k - 3) > k ( k - 2). T h i s f o r c e s (u - 4 ) F 1 > 3 ( k - 2) 2, w h i c h c o n t r a d i c t s L e m m a 2.4. Thus, 2A(u- 3)(u-4)>_ (k1 ) ( k - 2 ) ( k - 3 ) ( k - 4). W e o b s e r v e f r o m (8) t h a t e q u a l i t y h o l d s if a n d o n l y if (v - 4 ) F 1 = 2 ( k - 2)(k - 3). U s i n g L e m m a 2.5, we n o t e t h a t 2A(u - 3)(u - 4) = ( k - 1)(k - 2)(k 3 ) ( k - 4) h o l d s if a n d o n l y if o n e o f t h e i n t e r s e c t i o n n u m b e r s is zero. T h i s c o m p l e t e s t h e p r o o f o f T h e o r e m 2.7.

42


The next result follows from Ionin and Shrikhande [9, Theorem 4.1]. LEMMA 2.8. bers. Then

Let D be a 5-(~, k, A) design with three intersection num-

(u - 5)F~ < 3 ( k - 2 ) ( k - 3).

(9)

In [14] Ray-Chaudhuri and Wilson proved that the number of blocks b of any design D with s intersection numbers satisfies the inequality b < ( : ) . Moreover, b = ( : ) holds if and only if D i s a tight 2s-design f l o r a proof, see [17, T h e o r e m 8]). Applying this result to 5-(v, k, A) designs with three intersection numbers, we obtain the inequality A0 < (~/" The equality never holds, since by Peterson [13] there is no tight 6-design. \--/

Remark 2.9. T h e o r e m 2.7 and the inequality A0 < (~) imply the following bounds for the parameter A of a 5-(u, k, A) design with three intersection numbers:

( k - 1 ) ( k - 2 ) ( k - 3 ) ( k - 4) 2(v - 3 ) ( v - 4) k ( k - 1 ) ( k - 2 ) ( k - 3 ) ( k - 4)

_ 3(k - 2) 2. This c o n t r a dicts L e m m a 2.4. S u p p o s e t h a t x I = 1, x 2 = 2, a n d x 3 = 4. T h e n F l = 4 a n d F 2 = 8, so (6) yields t h e e q u a t i o n 8(u - 3 ) ( k - 4) = ( k - 1)(k - 2)(5k - 26), which gives t h e following i n t e g e r s o l u t i o n s satisfying t h e c o n d i t i o n u _> 2 k : k = 10, v = 3 9 and k=22, v=248. H o w e v e r , b o t h of t h e s e solutions v i o l a t e L e m m a 2.1. S u p p o s e t h a t x I = 1, x 2 = 3, a n d x 3 = 4. T h e n F l = 5 F 2 = 12, so (6) yields t h e e q u a t i o n 3(u - 3)(k 2 - l l k + 22) = 2 ( k - 1)(k - 2)(k - 3) ( k - 7). T h e following i n t e g e r solutions of this e q u a t i o n satisfy t h e c o n d i tion u > 2 : k = 8 , v = 7 0 , k = 10, u = 8 7 , a n d k = 13, u = 113. N o n e o f t h e s e solutions satisfies L e m m a 2.1. If x l = 2 , x 2 = 3 , a n d x 3 = 4 , t h e n F 1 = 6 , F 2 = 18, a n d F 3 = 2 4 , so (5) yields the e q u a t i o n 2 ( u - 3 ) = ( k - 1 ) ( k - 2), which c o n t r a d i c t s L e m m a 2.4. Thus, we have p r o v e d the t h e o r e m for t h e case u > 2 k . If u < 2 k , t h e n c o n s i d e r t h e c o m p l e m e n t a r y d e s i g n D. It has i n t e r s e c t i o n n u m b e r s u 2k+x i 1 or x 3 > 7. Thus, we have to rule o u t t h e case xl = 0, x 2 = 5, a n d x 3 = 6. In this case, F 1 = 8, F 2 = 20, a n d F 3 = 0. T h e n L e m m a 2.5 a n d (5) imply t h e system o f e q u a t i o n s 4(u - 4) = ( k - 2)(k - 3) a n d 5(v - 3) = ( k - 1)(k - 2). Solving we get k = 6, u = 7 or k = 7, v = 9. This implies v < k + 3, a c o n t r a d i c t i o n .

COROLLARY 5.3. I f D is n o t the Witt 5-(24, 8, 1) design or its c o m p l e m e n t , then k >_ 11 a n d v >>_k + 11. 582a/69/1-4

46


Proof. Since, by L e m m a 2.6, u _> k + 6, the c o m p l e m e n t D of the design D is a 5 - ( v , v - k , ~ ) design. Hence, if, for a > 6, one of the inequalities k > a or ~ _> k + a holds for any 5-(v, k, 3`) design D other than the Witt 5-(24, 8, 1) design or its complement, then the o t h e r inequality also holds for any such design. W e wish to prove the inequality k > 11, so we have to rule out 6 < k < 10. By Corollary 5.2, F 1 > 9, so (9) gives 3(u - 5) < ( k - 2 ) ( k - 3).

(11)

If k = 6, then (11) yields u _< 8; if k = 7, then (11) yields u < 11 and L e m m a 2.6 rules out these possibilities. Thus k > 8 and therefore v _> k+8. If k = 8, then (11) yields v < 14, which violates v _> k + 8. T h u s k >_ 9 and therefore v > k + 9. If k = 9, t h e n (11) implies v < 18. Since v -> k + 9, then u = 18. But k = 9, u = 18 violate L e m m a 2.3. Thus k _> 10 and v > k + 10. If k = 10, then (11) gives v < 23. Since u > k + 10 = 20, we have to rule out 2 0 < ~ < 2 3 . But k = 10, v = 2 3 and k = 10, u = 2 1 violate L e m m a 2.1; k = 10, v = 22 and k = 10, u = 20 violate L e m m a 2.3. This completes the p r o o f of Corollary 5.3.

6. FINITENESS RESULTS T h e o r e m 3.2 asserts that the Witt 5-(24, 8, 1) design is the only 5-(v, k, 3`) design with 3` = 1 and having exactly three intersection numbers. T h e next two r e s u l t s show that there is no 5-(u, k, 3`) design with 2 < 3` < 4 and having exactly three intersection numbers. THEOREM 6.1. There is no 5-(u, k, 3`) design with 3` = 2 and having exactly three intersection numbers.

Proof. Suppose that there exists a 5-(v, k , 2 ) design D with three intersection numbers. T h e n using T h e o r e m 2.7, 4 0 , - 3 ) ( v - 4)>_ ( k - 1 ) ( k - 2 ) ( k - 3 ) ( k - 4). This can be simplified to ( 2 u - 7)2_> ( k 2 - 5 k + 5) 2, which implies that v - 5 _> ( k 2 - 5 k + 2 ) / 2 . By Corollary 5.2, F~ > 9, so L e m m a 2.8 implies that v - 5 < (k 2 - 5k + 6 ) / 3 . Therefore, (k 2 - 5k + 6 ) / 3 > (k 2 - 5k + 2 ) / 2 , which gives k < 5, a contradiction. Modifying this reasoning, we obtain THEOREM 6.2. There is no 5-(u, k, 3.) design with 3` = 3 or 4 and having exactly three intersection numbers.

5-DESIGNS WITH

3

INTERSECTION SIZES

47

Proof. Let D be a 5-(u, k, A) design with A > 3 and having exactly three intersection numbers. We rewrite the inequality given by T h e o r e m 2.7 as h ~(2u-

7) 2 > ( k 2 -

h 5k + 5 ) 2 + ~ - 1.

Since 5A -- 1 > 0, this implies 2 u - 7 < V ~ / h ( k 2 5 k + 5 ) and so u - 5 > (k 2 5k "~- 5 ) / 2}/~2" By Corollary 5.2, F 1 > 9, and by L e m m a 2.8, (k - 2)(k - 3 ) / 3 > (k 2 g. 5k+5)/2d2 3 This last inequality can be transformed into --

3

-

-

11 (3 - 2 f ~ - ) ( k 2 - 5k + 5) < -~- 2 ~ .

(12)

If a = 3, then (12) implies k _< 7. This contradicts Corollary 5.3. Suppose A = 4. Then (12) implies k < 12. By Corollary 5.3, we must rule out k = 11 and k = 12. Note that, by Corollary 2.2, k - 4 must divide 36A = 144, so k ~ 11. If k = 12, then, by T h e o r e m 2.7, v > 35, while (9) implies v < 34. This completes the proof of T h e o r e m 6.2. THEOREM 6.3. For a fixed A, there exist at most finitely many 5-@, k, A) designs having exactly three intersection numbers.

Proof. Corollary 2.2 implies that, for a fixed A, there exist finitely many ks and L e m m a 2.1 implies that, for a fixed k, there exist finitely many vs. T h e o r e m 6.3 and R e m a r k 2.9 imply THEOREM 6.4. For a fixed k, there exist at most finitely many 5-(v, k, A) designs having exactly three intersection numbers.

Remark 6.5. Using MAPLE, we verified that, for 5 < h < 50 or k _< 500 (and h va 1), there is no 5-(v, k, A) design having exactly three intersection numbers.

7. P R O O F OF T H E O R E M

5.1

First note that if T h e o r e m 5.1 holds for a 5-(u, k, A) design D with u _> 2k, it also holds for its complement. This allows us to assume from now on that u > 2k. We assume also that D is not the Witt 5-(24, 8, 1) design, so x 1 =g 0, A > 2, x) > 5. Thus, we have to rule out the possibilities 1 < x I < x 2 < 4.

48

IONIN

AND

SHRIKHANDE

Case 7.1. x 1 = 1, x 2 ~--- 2, x 3 = x > 5. I n this case, F 1 = x , F 2 2x. T h e r e f o r e , we can apply L e m m a 1.1 (iii), which yields k < 5.

= F 3 =

Case 7.2. x a = 1, x 2 = 3 , x 3 = x > 5 . Let p a n d q be two distinct p o i n t s of the design D. T h e second derived design (Dp)q is a quasi-symmetric 3 - ( v - 2, k - 2, A) design with the i n t e r s e c t i o n n u m b e r s 1 a n d x - 2. By C a l d e r b a n k a n d M o r t o n [4] or also by Pawale a n d S a n e [12], (Dp)q is either 3-(23, 7, 5) or 3-(22, 7, 4) design. So either u = 25, k = 9, and A = 5 o r v = 2 4 ,

k=9,

a n d A = 4 . I n either case A0 = A { ~ j / { ~ } is

n o t integer.

Case 7.3. r e w r i t t e n as

x 1 = 1, x 2

=

4, x 3 = x > 5. I n this case Eq. (6) c a n b e

2(~, - 3 ) [ ( x - 6 ) k 2 + ( 5 x + 1 8 ) k - ( 1 8 x + 12)] = (k - 1 ) ( k - 2 ) ( k - 3 ) [ ( x - 7)k + (2x + 22)].

(13)

F o r x > 7, L e m m a 2.10 implies the i n e q u a l i t y nx 2 < (11n + 36)(x - 2), w h e r e n = k 2 - k - 6. F o r x > 10, x 2 > 12(x - 2), so n < 36, k < 7 which is n o t the case, b e c a u s e k > x + 1 > 11. Thus, we have to rule out 5 < x _< 9. U s i n g (13), we apply L e m m a 2.10 for x = 5 a n d x = 9 a n d L e m m a 2.5 for x = 6 a n d x = 7 to rule o u t x = 9 a n d to o b t a i n the following b o u n d s in the o t h e r cases: 6 < k < 15 (for x = 5), 7 _< k __< 13 (for x = 6), 8 _< k _< 29 (for x = 7). P u t t i n g these values of x a n d k in (13), we o b t a i n only o n e v a l u e of v satisfying L e m m a 2.1: v = 8 (for x = 5, k = 6). But it violates the i n e q u a l i t y v > 2k. F o r x = 8, we use (13) a n d c o n g r u e n c e s k 2 + 2 9 k - 7 8 - = - 4 8 m o d ( k - 1) a n d k 2 + 29k - 78 = - 1 6 m o d ( k - 2) to show that k 2 + 29k - 78 divides 48(k - 3)(k + 38) a n d t h e r e f o r e divides 288(k - 6). It is easy to see that the ratio 288(k - 6 ) / ( k 2 + 29k - 78) is less t h a n 5, for any k, a n d n o t equal to 1, 2, 3, or 4, for i n t e g e r k.

Case 7.4. x 1 = 2 , x 2 = 3 , x 3 = x > 5 . 2, F 3 = 6x, so Eq. (5) can be r e w r i t t e n as

Then

F l =x+2,

F 2=4x+

( u - 3 ) [ ( x 2 + 10x - 2 ) k 2 - ( 1 9 x 2 + 46x - 2 ) k - 5 4 x ( x + 1)] = -(k

- 1)(k - 2)[(X e - llx +~l)k 2 -(x 2-

65x + 1 ) k -

9x(x+

11)].

(14)

L e m m a 1.1 (iii) yields the i n e q u a l i t y k < 1 8 x ( x + 2 ) / ( x 2 + 10x - 2), which implies k < x + 4.

5-DESIGNS WITH

3

49

INTERSECTION SIZES

If x + l < k 7, t h e n t h e l e f t - h a n d side of (14) is positive, while t h e r i g h t - h a n d side is negative. F o r x = 5, 6, this e q u a t i o n d o e s n o t yield i n t e g e r values o f u.

Case 7.5. x 1 = 2 , x 2 = 4 , x 3 = x > 5. T h e n 3, F 3 = 8 x , so Eq. (5) can b e r e w r i t t e n as 2(v

-

z:) = ( k

-

-

F~ = x + 3 ,

2)t

(x,

F 2=5x+

(15)

H e r e a(x, k) = (x 2 - 4 2 x + 9)k 2 + (47x 2 + 186x - 9)k - 2 4 x ( 7 x + 9), a n d / 3 ( x , k ) = ( 5 x e - 5 4 x + 9 ) k 2-(5x 2-342x-9)k-48x(x+ 12). If x > 11, t h e n k > 12 a n d t h e r e f o r e /3(x, k ) > 0. A p p l y i n g L e m m a 2.10, we o b t a i n t h e i n e q u a l i t y 3a(x, k ) > 7/3(x, k), which c a n n o t h o l d for x > 11 a n d k___ 12. If 5 < x < 10, t h e n 3FaF3 - F 2 > 0, so L e m m a 1.1 (iii) yields k < 4 8 x ( x + 3 ) / ( - x 2 + 4 2 x - 9). F o r t h e v a l u e s o f x a n d k satisfying t h e s e inequalities, o n e can c a l c u l a t e f r o m (15) to o b t a i n t h e following t h r e e p a i r s of i n t e g e r s ( k , u) satisfying t h e c o n d i t i o n v > 2 k : k = 8, v = 18, k = 10, v = 39, a n d k = 12, u = 38. N o n e of t h e s e p a i r s satisfies L e m m a 2.1.

Case 7.6. r e w r i t t e n as

xl=3,

x 2=4,x

3=x_>5.

12(v - 3 ) 7 ( x , k ) = - ( k

In this case, Eq. (5) can b e - 1)(k - 2)6(x,k).

(16)

Here y(x,k) = (2x1)k 2 - ( 2 x 2 + 1 0 x - 1)k + 2 x ( 4 x + 7) a n d 6 ( x , k ) = ( x 2 - 13x + 4)k 2 - ( x 2 - 85x + 4)k - 1 2 x ( x + 13). L e m m a 1.1 (iii) yields t h e i n e q u a l i t y k < 2x(x + 4 ) / ( 2 x - 1), which i m p l i e s t h a t k < x + 5, so k - 4 _< x _< k - 1. It follows t h a t 7 ( x , k ) > 0, so (16) i m p l i e s t h a t 3 ( x , k ) < 0. But 6(x, k ) > 0 if x _> 8 a n d x + 2 < k 13 a n d k = x + 1. T h e r e m a i n i n g possibilities are: (i) x = 5 , 6 < k < 9 , ( i i ) x = 6 , 7 < k < 10, (iii) x = 7 , 8 < k _ < 11, a n d (iv) 8 _ < x _ < 12, k = x + 1. P u t t i n g all t h e s e values o f k a n d x in (16), we o b t a i n t h e following values of v satisfying the c o n d i t i o n v >_ 2 k : u = 839 (for k = 13 a n d x = 12), u = 5 5 3 (for k = 12 a n d x = 11), u = 3 4 8 (for k = 11 a n d x = 10), v = 207 (for k = 10 a n d x = 9 ) , v = 115 (for k = 9 a n d x = 8 ) , v=59(for k=Sand x=7),u=28(for k=7and x = 6 ) , v = 17 (for k = 8 a n d x = 5), a n d also any ~,>_ 12 if k = 6 a n d x = 5 . If k = 6 a n d x = 5, t h e n F 1 = 9, F 2 = 36, a n d F 3 = 60, so Eq. (3) yields u = 8 o r v = 9. T h e c o n d i t i o n u > 2 k is violated. If k = 7, x = 6, a n d v = 28, t h e n F 1 = 10, F 2 = 42, a n d F 3 = 72. T h e s e values do not satisfy Eq. (3). A l l o t h e r possibilities a r e r u l e d out by L e m m a 2.1. T h e p r o o f o f T h e o r e m 5.1 is now c o m p l e t e .

50

IONIN AND SHRIKHANDE REFERENCES

1. E. BANNAI, personal communications. 2. T. BETH, D. JUNGNICKEL, AND H. LENZ, "Design Theory," B. I. Wissenschaftverlag, Mannheim, 1985, Cambridge Univ. Press, London/New York, 1986. 3. A. BREMNER, A diophantine equation arising from tight 4-designs, Osaka J. Math. 16 (1979), 353-356. 4. A. R. CALDERBANK AND P. MORTON, Quasi-symmetric 3-designs and elliptic curves, SIAM J. Discrete Math. 3 (1990), 178-196. 5. P. DELSARTE, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Supp. 10 (1973). 6. H. ENOMOTO, N. ITO, AND R. NODA, Tight 4-designs, Osaka J. Math. 16 (1979), 39-43. 7. B. H. GROSS, Intersection triangles and block intersection numbers of Steiner systems, Math. Z. 139 (1974), 87-104. 8. S. A. HOBART, "Designs of Type (242)," Ph.D. thesis, University of Michigan, Ann Arbor, MI, 1987. 9. Y. J. IONIN AND M. S. SHRIK~ANDZ, (2S -- D-designs with s intersection numbers, Geom. Dedicata, 48 (1993), 247-265. 10. N. ITO, On tight 4-designs, Osaka J. Math. 12 (1975), 493-522; corrections and supplements, Osaka J. Math. 15 (1978), 693-697. 11. R. NODA, Steiner systems which admit block transitive automorphism groups of small rank, Math. Z. 125 (1972), 113-121. 12. R. M. PAWALE AND S. S. SANE, A short proof of a conjecture on quasi-symmetric 3-designs, Discrete Math. 96 (1991), 71-74. 13. C. PETERSON, On tight 6-designs, Osaka J. Math. 14 (1977), 417-435. 14. D. K. RAY-CHAUDHURI AND R. M. WILSON, On t-designs, Osaka, J. Math. 12 (1975), 737-744. 15. M. S. SHmKHANDE, The Delsarte polynomial of 5-(u, k, 3.) designs with three intersection numbers, J. Combin. Inform. System Sci. 18, Nos. 1-2 (1993), 53-60. 16. M. S. SHRW,Z~aNDE AND S. S. SANE, "Quasi-Symmetric Designs," London Math. Soc. Lecture Notes Series 164, Cambridge Univ. Press, London/New York, 1991. 17. R. M. WiLson, On the theory of t-designs, in "Enumeration and Design" (D. M. Jackson and S. A. Vanstone, Eds.), Academic Press, New York, 1984, pp. 19-49.