Ionin Y. J., Shrikhande M. S.-(2s1) Designs Withs Intersection Numbers (1993)

Y U R Y J. I O N I N AND M O H A N S. S H R I K H A N D E *

( 2 s - 1)-DESIGNS WITH s INTERSECTION NUMBERS

ABSTRACT. It follows from Ray-Chaudhuri and Wilson (1975) that a ( 2 s - 1)-(v, k, 2) design has at least s block intersection numbers. The extremal case of such designs having exactly s intersection numbers is studied. Some necessary conditions on the parameters and intersection numbers are obtained. The following characterization results are proved: (i) A (2s-1)-design with exactly s intersection numbers is the Witt 5-(24, 8, 1) design if and only if s >~ 3 and the sum of the intersection numbers is less than or equal to s(s- 1); (ii) A tight 2s-design, that is a 2s-design with exactly s intersection numbers, is the Witt 4-(23, 7, 1) design if and only if s~>2 and the sum of'the intersection numbers is less than or equal to s 2.

O. I N T R O D U C T I O N

Let v, k, 2, and t be positive integers with v > k >~ t. Let # be a v-set and ~ be a collection of k-subsets of # called blocks such that any t-subset of ~ is contained in exactly )~ subsets in ~. Then the pair D = (~, ~) is called a t- design with parameters (v, k, ).). We also refer to D as a t-(v, k, 2) design. For basic definitions and results on t-designs and other related combinatorial configurations, refer to Beth et al. [-2] or Dembowski [-9]. Let B i and B~ be distinct blocks of the design D. The numbers [B~ c~ Bi[ are called the (block) intersection numbers of D. Intersection numbers provide a powerful tool in design theory. For instance, 2-(v, k, 2) designs with exactly one intersection number are precisely symmetric 2-designs. Designs with precisely two block intersection numbers are known as quasi-symmetric designs and are of much current interest [20].

In an important paper Ray-Chaudhuri and Wilson [17] considered t- (v, k, 2) designs D with t-- 2s (s ~> 1). Using linear algebra tools, they showed among other things that if v ~> k + t then (i) the number of blocks >/(~), (ii) D has at least s intersection numbers, (iii) any design with s intersection numbers has at most (~) blocks, so D has exactly (~) blocks if and only if D has precisely s intersection numbers. In [17], a t = 2s-design is calle d tight if it has precisely s intersection numbers. Clearly, a 2-design is tight if and only if it is symmetric. It was shown in [17] that for a tight design the s intersection numbers are roots of a polynomial whose coefficients involve only v, k, and s. These polynomials, in the framework of association schemes and coding theory, were explicitly found by Delsarte [8]. With the Delsarte polynomial of a tight 4-design as the starting point, the combined efforts of Ito [14], Enomoto et al. [10], and Bremner [3] showed that there is exactly one such

*M. S. Shrikhande acknowledges support from a Central Michigan University Summer Fellowship Award No. 42137.

Geometria Dedicata 48: 247-265, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

248 Y U R Y J. I O N I N A N D M O H A N S. S H R I K H A N D E

design (up to complementation), viz. the Witt design with parameters 4- (23, 7, 1). Peterson [16] proved that a tight6-design does not exist. Bannai [1] showed that for each s/> 5, there exist at most finitely many tight 2s-designs.

The situation for t---(2s-1)-designs seems to have received less attention than tight 2s-designs. From [17], it is immediate that a (2s - 1)-design has at least s intersection numbers. This observation is in Cameron [6]. We will be interested in (2s - 1)-designs which have exactly s intersection numbers. We will call such designs extremal. The Delsarte type of polynomials for quasi- symmetric 3-designs occur in Sane and Shrikhande [18] and also in Calderbank [5] and Pawale [15]. See also Shrikhande and Singhi [21]. These polynomials appear in current work of Shrikhande [19] and Ionin and Shrikhande [13] dealing with the following open problem mentioned, for instance, in Hobart [11]: Is the Witt 5-(24, 8, 1)design the only 5-design (up to complementation) with exactly three intersection numbers? Extremal (2s-1)-designs, for s=2,3 , occur in Hobart [11], [12] in the context of certain types of coherent configurations.

In this paper we use elementary linear algebra and basic design theory to investigate extremal (2s - 1)-designs. Using these tools, we find a convenient form for the coefficients of the Delsarte polynomial of such designs. This form allows us to obtain some necessary conditions on the parameters and intersection numbers of extremal designs.

The main results of this paper are the following: If D is an extremal ( 2 s - 1)-(v, k, 2) design with intersection numbers

Xl, x2 , . . . , xs, then:

( s - 1Xk-s)(k-s + 1) s(s- 1) s(k-s)(k-s + 1) (i) v - -2s+2 < ~ x l + x 2 + ' " + x ~ ~ < ~ v-/- 2s+ 1 '

the lower bound is attained if and only if one of the intersection numbers is zero; the upper bound is attained if and only if D is a tight 2s-design;

s(k - 1). (ii) if v/> 2k + 1, then x 1 + x2 + " " + x~ < ~ ,

(iii) v - 2s + 2 divides s(s 2 1.___~) ( k - sXk - s + 1)2(k - s + 2);

S2(S-- 1)3(S-- 2)2. (iv) k - 2s + 2 divides

2

(v) for fixed 2 and s t> 3, there exist at most finitely many extremal (2s - 1)- (v, k, 2) designs;

(2s-1)-DESIGNS WITH S INTERSECTION NUMBERS 249

(vi) x ~ + x2 4-"" 4- Xs ~ S(S - - 1 ) ; if s ~ 3 and xl 4- x2 4 - ' " + xs = s ( s - 1), then D is the Wit t 5-(24, 8, 1) design.

A consequence of (vi) is: a tight 2s-design with intersection numbers x~, XE, . . . , x s is the Wit t 4-(23, 7, 1) design if and only if s > /2 and

X 1 4 - X 2 4 - . . . -.[-:Xs ~ S 2.

1. PRELIMINARIES

Let D be a ( 2 s - 1)-design with usual pa ramete rs (v, k, 2) and having exactly s

block intersection numbers Xl, x2 . . . . . x s. We call such a design an extremal (2s - 1)-design. If D is an ext remal (2s - 1)-(v, k, )0 design, then v >~ k + s.

Indeed, if fB~c~Bjl=x, for two distinct blocks Bi and Bj of D, then IB~ u Bjl = 2 k - x . If x is the smallest intersection number of D, then x ~< k - s ,

so v >>, 2 k - x >1 k+s . It is well known [2] that if D is a t-(v, k, )0 design, then D is also an i-

(v, k, 25) design, where )o~ = ) . ( v - i ) t _ f f ( k - i ) t _~ , for i = 1, 2 . . . . . t - 1. Here we use the no ta t ion (z)q = z ( z - 1 ) . . - ( z - q + 1), (Z)o= 1. For any point p of D, the

derived design Dp is a ( t - 1 ) - (v -1 , k - 1, 2) design whose points are the points of D other than p and whose blocks are the blocks of D passing th rough p.

The residual design D p is a (t - 1)-(v - 1, k, 2 t_ 1 - 2) design whose point set is that of Dp and whose blocks are the blocks of D missing p. If xl , x2 . . . . . xs are all the intersection numbers of D, then any intersection number of D p is conta ined in {Xl, x2 . . . . . xs} and any intersection number of Dp is contained

in { x l - l , x 2 - 1 , . . . , x s - l } . Fo r any t-(v, k, ).) design D, the complementary d e s i g n / ) is obta ined by

replacing each block of D by its c o m p l e m e n t . / ) is a t-(v, v - k, 2) design, where

~=~[=0( - -1 ) i (~ ) ) . i . If xl , x2 . . . . ,x~ are intersection numbers of D, then obviously v - 2k + Xl, v - 2k + x 2 , . . . , v - 2k + x s are the intersection numbers

of D. I t is also clear that for any point p, the designs D p and (b)p are the same.

D E F I N I T I O N 1.1. Given distinct non-negat ive integers x~, x 2 , . . . , let the sequence F~ ~) (s i> 1,0 ~<j ~< s) be defined by: Fro s) = 1, F~ ~) = XlXE...x~, and the

recurrence relat ion F~)= F~ ~- ~) + ( x ~ - s + j)F~Z? ) (s >~ 2, 1 <<, j <~ s).

D E F I N I T I O N 1.2. F o r any distinct non-negat ive integers x~, x2 . . . . . xs and indeterminate z, define the polynomia l P~(z)=H~=I ( z - x i ) .

If x 1, x2 . . . . . x~ are the block intersection numbers of a design D, then P~(z) is usually known as the (Delsarte) annihi la tor po lynomia l of D [8].

R E M A R K 1.3. The numbers F~ ~) arise in several (combinatorial) contexts. Let

2 5 0 Y U R Y J . I O N I N A N D M O H A N S. S H R I K H A N D E

f l (') ¢(~) f,(*) be the (usual) unsigned elementary symmetric functions of , , ]2 , ' ' ' ~

xl, x2, . . . , x, given by

f l (~)= ~ x~, f(s) _~_ ~ X i X j , . . . ' fs(S) = XIX2. . .Xs . i<j i=1

Then, for 1 ~< j ~< s,

j - 1

= E w, - j - - t

i = 0

where the coefficients a(p q) are determined by the expansion of the polynomial (z)q = z q - a(qL ) lz q- x + . . . + ( _ 1)qa~). The numbers a(p q) are the so-called Stirling numbers of the first kind.

The numbers F~ ~) can also be recovered from the polynomial P,(z) by a well-known method:

r~ s) = ( - 1)~P,(0),

r(') = ( - 1) ~- l(ps(1)-- P,(0)) --- (-- 1)*- lAP,(0), S--1

F(S) ( _ 1)s- 2 = A 2 p s ( 0 ) , . . - ' - 2 = 2! ( P , ( 2 ) - 2 P , ( 1 ) + P , ( O ) ) (-1)'-22!

etc.,

where A is the usual difference operator (see [4]). Conversely, the following lemma shows that the polynomial P,(z) can be

expressed in terms of the numbers F) '~.

LEMMA 1.4. The polynomial P,(z) = I-l~.= l ( z - xi) can be expressed as

(1) ~ ( - 1) "-J F,(')_~tz b j = O

Proof. We use induction on s. The relation (1) obviously holds for s = 1. Let s >/2, and suppose the result holds for the polynomial P~_ l(Z). Then

s--1

P,(z) = (z-x,)P,_ 1(z) = (z-x,) ~ ( - I)~F? - l)(z),_j_ I j=O

2 - 1

= ~ (-- 1)~F(p- ')(z - s + j + 1)(z),_j_, j = O

- ~ ( - 1)JF~ "- 1)(X s - S + j + 1)(Z),_j_I j=O

s--1

=E j = O

( - - 1)J F~ " - 1)( z ), _ j + ~ ( - - 1)~ F}L- ( )( x , -- s + j)( z ), _ j j = l

( 2 s = 1 ) - D E S I G N S W I T H S I N T E R S E C T I O N N U M B E R S 251

s - 1 = F ~ - l}(2)s + 2 ( - - 1)J(F} ~- 1)+ F~S-11)(Xs _ s + j ) ) (Z ) s_ j + ( - - 1)SF~ s)

j = l

s - j is) = ( - - 1) F~_gz)j . j=O

We record for future use:

s(s- 1) R E M A R K 1.5. F~S) = Xl + X2 + . . - + x , 2

2. A V A N D E R M O N D E T Y P E D E T E R M I N A N T

One of our a ims in this pape r is to ob ta in cer ta in equa t ions and inequal i t ies

satisfied by the pa rame te r s of ex t remal designs. To this end we first p repa re

some e lemen ta ry results a b o u t cer ta in matr ices and determinants .

Let x l , x 2 , . . . , x s and ao, at , a 2 , . . . , a s be any real numbers . Let

M~ = M , ( x l , x2 . . . . . x~; ao, al , a2 . . . . . a~) be a square mat r ix of o rde r s + 1 with

entries mij , where

ai i f 0 ~ < i ~ < s , j = 0 .

m i j = (xj)i i f 0 ~ i ~ < s , l ~ j ~ < s .

Thus,

a o 1 1 ..- 1

a l x l X 2 • . . X s

M ~ = a2 ( x02 (x2)2 "'" (x~)2

a, (xl) , (Xz)s "" (x,)s

L E M M A 2.1. For s >1 2, let

M~ = M , ( x l , x2 . . . . . x~; ao, a 1, a 2 . . . . . a,)

and

M~_ 1 = M s - I(X1, X 2 . . . . . X s - 1; al - aox, , a2 - a l ( x , - - 1), . . . , a~ - a ,_ l(x~ - s + 1)).

Then

s--1 de t (MD = - det(M~_ 1) H ( x s - xi).

Proof. N o t e tha t for i = 1, 2 , . . . , s; j = 1, 2 . . . . , s,

(x j) i - (x~ - i + 1)(xj)i_ 1 = (x.i-- x~)(xj)i- 1"

252 Y U R Y J . I O N I N A N D M O H A N S. S H R I K H A N D E

For i = s, s - 1 , . . . , 1, add - ( x ~ - i + 1) times the ( i -1) th row of Ms to the ith row of M s and expand the determinant using the last column. []

P R O P O S I T I O N 2.2. The determinant of the matrix Ms(xI, x2,...,xn; ao, al . . . . . as) is equal to

(-- 1) s 1-[ (x j -x i )" ~ ( - 1)Jas_jF} ~,. l <~i<j<~s j = 0

Proof. We use induction on s. For s = 1,

det(Mx(Xl; ao, a l ) ) = d e t [ a° 1 ] = - a l + a o x l = k a l X l ]

Let s/> 2 and assume the result for M~_ r By Lemma 2.1,

s - 1

det(Ms) = - det(Ms_ 1) I-[ (x~-x,), i = I

where

Ms_x = M~- x(xl, x2 . . . . . xs-1; al -aoxs, a 2 - a l ( x s - 1) , . . . , a s - a~_~(xs - s+ 1)).

By the induction hypothesis,

d e t ( M ~ _ l ) = ( - 1) ~ - 1 1-I (xj - x , ) l<~i<j<~s-1

s - 1 • ~ ( - - l y (a~- j - -as - j - l ( x~- - s+j+l ) )F~ ~-1). j=O

We transform the sum in the right-hand side:

s - 1

2 j=O

( -1) i (as_j- -as_j_l(xs--s + j + 1))F~ s-l)

s - 1 s - 1

= Z + + 1))F} j = 0 j = 0

s - 1 = a~F~- l )+ ~ (--1)Ja~-jF}~-x)+ ~ (--1)i(a~-j(x~--s+j))F(~--~)

j = l j = l

$ - 1

= a~F~o ~) + ~ ( _ 1)ia~_ j(F}~- 1) + (x s _ s + j)F}L-~ )) + ( - 1)~a~F~ ~) j = l

= ~ (--1)/a~-jF} ~). j=O

[]

(2S- -1 ) -DESIGNS WITH S INTERSECTION NUMBERS 253

3. A P P L I C A T I O N S TO EXTREMAL DESIGNS

The aim of this section is to obtain convenient explicit expressions for the coefficients of the Delsarte polynomial of an extremal design.

P R O P O S I T I O N 3.1. Let xl , x2 . . . . . xs be any distinct non-negative integers. Let D be an s-(v, k, ).) design whose block intersection numbers are contained

in the set {x 1, x 2 . . . . . x~}. Then

(-1)JF~S)(k)s-j2s-j = Ps(k), j=o

where Ps(k)= ( k - x l X k - x2) . . . ( k - xs) and 2i is the number of blocks containing

any i-tuple points of D. Proof For j = 1, 2 , . . . , s, let mj denote the number of blocks meeting any

fixed block in x s points (if xj is not an intersection number, then mj = 0). Then by the usual method of two-way counting, we obtain, for i = 0, 1 . . . . , s, the equations

By multiplying the ith equation by i! and introducing an artificial variable r a g = - 1, we can consider (2) as a homogeneous linear system in the

unknowns rag, ml . . . . . ms with coefficient matrix Ms = M~(xl, x2 . . . . . xs;

ao, a l , . . . , as), where a i = (k)i(2 i - 1). Thus det(Ms) = 0. Then Proposition 2.2 yields the following equation:

(3) ~ ( - 1)/F~)(k)~_j(2~_;- l) = 0. j=O

Relations (3) and (1) complete the proof of Proposition 3.1. []

We now come to one of our main results.

T H E O R E M 3.2. Let D be an' extremal ( 2 s - 1)-(v, k, 2) design with the

intersection numbers xl , x2, . . . , x s. Then

(4) ~ (--1)JF}S)(k)s_j2s_j = es(k) j=o

and for i = l, 2 , . . . , s - 1,

(5) ~ (-1)JF(S)(k),_j,~s_j+ i = O. j=O

Proof Equation (4) holds by Proposition 3.1. Since D can be considered as

254 Y U R Y J. I O N I N AND M O H A N S. S H R I K H A N D E

an (s+i)-(v, k, As+i) design, for i= 1, 2 , . . . , s- : 1, Equations (5) are implied by the following

ASSERTION. I f E is an (s + i)-(v, k, #) design whose intersection numbers are contained in {x 1, x2, . . . , xs}, then

(-1)JF~s)(k)s_j#s_j+i = 0, j=O

where #i is the number of blocks containing any i-tuple points of E.

We will prove this assertion by induction on i. Let i = 1. Then E can be considered as an s-(v, k, #s) design whose

intersection numbers are contained in {xl, x2,.. •, xs}. Therefore Proposition 3.1 yields the equation

(6) ~ (-- 1)JF)S)(k)~_j#~_j = P~(k). j=O

For any point p in E, the residual design E p is an s - (v - 1, k, #~-p) design whose intersection numbers are contained in {xl, x2 , . . . , xs}. By Proposition 3.1, we obtain the equation

(7) ~ (-1)JF}~)(k)s_j(p~_j-ps_j+l)= P~(k). j=O

Subtracting (7) from (6), we get the assertion for i = 1. Let if> 2, and let E be an (s + i)-(v, k, #) design whose intersection numbers

are contained in {x~, x2,. •., x~}. Then the residual design E p is an (s + i - 1)- ( v - 1 , k, #~+i -~-# ) design whose intersection numbers are contained in {xl, x2 . . . . . xs}. By the induction hypothesis, applied to E p, we obtain

(8) ~ (-- 1)JF}s)(k)~_j(#~__j+~_l-/zs_j+,) = 0. j=o

We also apply the induction hypothesis to the design E as an (s + i - 1 ) - ( v - 1, k, #,+~_ 1) design, obtaining

(9) ~ (--1)JF?)(k)s_jps_j+i = O. j=o

Subtraction of (9) from (8) completes the induction step. []

The proof of the above theorem carries over to the following result.

P R O P O S I T I O N 3.3. Let D be a (t + s)-(v, k, 2~+~) design with the intersection numbers contained in the set {xl, x2 . . . . . x,}. Then (4) and (5), for i = 1, 2 . . . . . t, hold. []

(2s-1)-DESIGNS W I T H S I N T E R S E C T I O N N U M B E R S 255

It is convenient to rewrite Equations (4) and (5) as follows:

(10) ~ ( - - 1)JF~_~(k)j2j = P~(k). j=O

(11) ~, ( - - 1);F~)_i(k)j).i+j = O, i = 1, 2 , . . . , s - 1. j = 0

Using 2i +j = 2 j (k- j ) i / (v-J) i , we can consider (10) and (11) as a system of s linear equations in the s + l unknowns w j = ( - 1 ) ~ - J F ~ ) _ j 2 ) k ! / ( v - j ) ! . The coefficient matrix of the system is A = [ a j , 0 <~ i <~ s - 1, 0 <~ j <~ s, where aij = ( V - - i - - j ) ! / ( k - i - - j ) ! .

Elementary row operations transform the matrix A into the matrix B = [bij], 0 ~ i <~ s - 1, 0 <~ j <~ s, where

I O i f i + j < ~ s - 2

(12) b o = j ! ( v - j - s + l ) ! i f i + j > ~ s - 1 . ( i+ j - s + l ) ! ( k - i - j ) !

The above transformations on A change the r.h.s, of (10). Using the initial row of B, we get the following equation:

( s - 1)!(k),_ 1 s!(k)s = Ps(k) - - ( 1 3 ) --F(l~)2s- 1 (-~-)-~ 1-~-~_ 1 +)'~ ( v - - s ) ~

(k)~_ 1

(v - k) ,_ 1"

Using 2 s = 2(v - s)s- 1/(k - S)s - 1, }~s - 1 = .~(v - s q- 1)s/(k - s + 1)~, Equation (13) can be rewritten as

2(v - k)~ _ 1 [s(k - s + 1) 2 - (v - 2s + 2)F~ s)] = s P~(k).

We record the above as the following

THEOREM 3.4. Le t D be an ex t r ema l ( 2 s - 1)-(v, k, 2) design with intersec-

tion numbers x l , x 2 , . . . , x S. Le t (as above)

s(s- 1) F(1SJ = xl + x2 + "" + x~

2

and

Then

(14)

P s ( k ) = ( k - x l ) ( k - x 2 ) . . . ( k - xs).

2 ( v - k ) s - t [ s ( k - s + 1)2 (v_2s+ 2)F~S)] = ( k - s s + 1) Ps(k) . []


COROLLARY 3.5. Let D be an extremal ( 2 s - 1)-(v, k, 2) design with inter-

section numbers xl , x2 . . . . , x~. Then

(15) ( v -2s+2)F~ ~ < s ( k - s + 1) 2. []

We return to the linear system in unknowns wj whose coefficients are given by (12)• We introduce new unknowns yj, where

j ! ( v - s - j + 1)! YJ = ( k - s + 1)! wi"

These unknowns satisfy a homogeneous linear system obtained by deletion of the initial equation from the previous system. Putting n = k - s + l , the

o o (;) (7) ( : ) o (;) (7) (:) (~)

(7) (~) (~) (:)

"'" ( s n 4 ) ( s ~ 3 ) ( s ~ 2 ) ( s ~ l )

"'" ( s n 3 ) ( s ~ 2 ) ( s ~ l ) ( : )

( s -1 ) x (s+ 1) matrix of this system is

0

0

n O ~ - 0 0

0 (;)

Thus, B o = [b! !)], 1 ~< i ~ s - 1, 0 ~< j < s, where

,j = n if i + j >. s - 1. i + j - - s + l

By elementary row operations, the matrix Bo can be transformed into the following tridiagonal matrix C = [cij], 1 <~ i <~ s - 1 , 0 <~ j <. s, where

f 0 if i + j <~ s - 2 or i + j >~ s + 2

i(i+ 1) if i+ j = s - 1

clj = 12i(n + i - 1) if i + j = s

~ ( n + i - 1 ) ( n + i - 2 ) i f i + j = s + l .

The matrix C yields the following equations for i = 1, 2 , . . . , s - 1:

i(i+ 1)y,-i- 1 + 2 i ( n + i - 1 )y s_ i+(n+i - 1)(n+i-2)y,_~+~ = 0.

(2s-I)-DESIGNS W I T H s I N T E R S E C T I O N N U M B E R S

Since yj = ( - 1) 5-jF~)_j2jj!(k)~_ 1/(v - j)~- l , we obta in

equat ions for i = 1, 2 , . . . , s - 1:

i(i+ l ) ( v - 2 s + i + l ) ( v - 2 s + i + 2) F!S)+t2~_i_l ( v - s + i ) ( v - s + i + l)

_ 2i(s - i)(k - s + i)(v - 2s + i + 1) F~)2~_ v - s + i

the

Putting

257

following

+ ( s - i X s - i + 1 ) ( k - s + i ) ( k - s + i - 1)FI~ 12s_i+ I = O.

( v - s + i ) ( v - s + i + 1) ; t s - i - 1 = 2s-i+1,

( k - s + i X k - s + i+ 1)

v - s + i 2s - i -- k -- s -t-~ "~ - i +1,

we obta in the following result:

T H E O R E M 3.6. Let D be an extremal (2s-1)-design with intersection

numbers x l , x2 . . . . . x~. Then for i = 1, 2 . . . . . s - 1 , the numbers FI ~) (described by Definition 1.1) satisfy the following three-term recurrence relation:

(16) i( i+ 1 ) (v - -2s+i+ 1 ) ( v - 2 s + i + 2 ) F ~ 2 ,

- - 2i(s -- i ) ( k - s + i)(k - s + i + 1)(v - 2s + i + 1)F! ~)

+ (s-- i)(s -- i + 1) (k - s + i - 1)(k - s + i)2(k - s + i + 1)FI~ 1 = 0.

[ ]

C O R O L L A R Y 3.7. Let D be an extremal ( 2 s - O - d e s i g n with intersection

numbers x l , x 2 . . . . . x,. Then for i = 2, 3 . . . . . s,

(17) ( v - 2 s + i + l ) i F l S ~ = ( k - s + O i ( k - s + i - 1 ) i _ 2

[(s--1)(v--2s+z)rl~'~(~' -('-l)(:)(k-s)(k-s+l)l.t x i - 1

Proof Induc t ion on i. [ ]

Equalities (17) express all coefficients FI ~) of the Delsar te po lynomia l (1) in terms of F] s) (and the pa ramete r s v, k, 2, s). In order to obta in an expression for F] ~) in terms of v, k, 2, s, we need the following

L E M M A 3.8. For positive integers m, n, p such that m >1 n + p - 1,

(18) ,~=o ( - - 1 ) i ( p m i ) ( n ; i ) = ( m - - ; - - 1 ) "


Proof. By simple induction, one can prove that

E ("n Therefore, (18) represents the equali ty of the coefficients of z p in the left-

hand and r ight-hand sides of the identity (1 +z ) r"-"- 1 =(1 +z)m(1 +Z ) - " -1 . [ ]

T H E O R E M 3.9. Le t D be an ex tremal (2s -1 ) - (v , k, 2) design with intersec-

tion numbers x l , x 2 . . . . , xs. Le t (as before)

s ( s - 1) F~ ~) = x l + x 2 + . . . + x ~

2

Then

(19) [ ( v - 2s + 2)F? ) - ( s - 1)(k - s ) ( k - s + 1)3

x [ 2 ( s - 1 ) ! ( v - s + 1)s-(k)2 s_ t]

= k ( k - s + 1 ) ( v - s + 1 ) [ 2 ( s - 1 ) t ( v - s )s _ 1 - ( k - 1)2~ - 2 ] .

Proof. Let X = ( v - 2s + 2)F~ ' ) - ( s - 1 ) (k- s ) ( k - s + 1). Then the equalities (17) imply that for i = 2, 3 . . . . . s,

(20) ( v _ 2 s + i + l ) i F l S ) = ( k - s + i ) ! ( k - s + i - 1 ) ! (k-s)!(k-s + 1)!

s-,) Notice that this equality holds also for i = 1 and i = 0 (assuming that

=0, Using Lem ma 1 . 4 , w e now rewrite Equat ion (14) as

2(s - 1)!(v - k)s_ 1 [k(k - s + 1) - X] = (k - s + 1)s k ( - 1)'(k)s _ iF} "). i = 0

We multiply both sides of this equat ion by ( v - s + 1)s and apply (20):

) t (s - 1 ) ! (v - s + 1 ) . ( v - k )~_ 1 [ k ( k - s + 1) - X ]

(-- 1)~(v-- s + 1 ) ! ( k - s + i - 1)! ~ k ! ( k - - s "dl - 1)~

i= ~o (v - 2s + i + 1)!(k - s)!(k - s + 1)!

x s-l)


The coefficient of X in the right-hand side of this equation is equal to

• ~1 l ( v _ s + l ~ ( k _ s + i ) ( s - 1 ) ! ( k ) 2 s - l i = o ( - l ) i + \ s - i - l / \ k - s '

while the sum of the terms free of X is equal to

(k-s+l)(v-s+lXs-1)t(kh~_~ ~o2 ( - 1 ) ~ \ s - i - l ) \ k - s - 1 )"

Applying Lemma 3.8, we get (19). []

4 . F U R T H E R A P P L I C A T I O N S TO E X T R E M A L D E S I G N S

In this s6ction we apply Theorems 3.5 and 3.9 and some general design theory results to obtain certain necessary conditions for extremal designs. We have used these conditions in [13] for investigating extremal 5-designs.

In paper [17], Ray-Chaudhuri and Wilson proved that the number of blocks b of any design D with s intersection numbers satisfies the inequality b ~ (~). Moreover, the equality holds if and only if D is a tight 2s-design (for a proof see [22, Th. 8]). Therefore the number of blocks 20 of an extremal ( 2 s - 1 ) - (v, k, 2) design D satisfies the inequality ~o ~< (~). Using the relation

go =2(v)2s - 1/(k)2s - 1, this inequality can be rewritten as

(21) 2s!(v-s)~_l <. (k)Es-1.

Using Theorem 3.9, we now obtain the following necessary conditions on the sum of the intersection numbers of extremal designs:

T H E O R E M 4.1. Let D be an extremal ( 2 s - 1)-(v, k, ,~) design with intersec-

tion numbers x 1, x 2 . . . . . x s. Then

(s-- 1 ) (k - s ) ( k - s + 1) s ( s - 1) (22) ~< x 1 + x 2 + ." +x~

v - 2 s + 2 2

s ( k - s X k - s + 1) ~< v - 2 s + 1

The lower bound is attained if and only if one of the intersection numbers is zero; the upper bound is attained if and only if D is a tiffht 2s-desiffn.

Proof. Equality (17), for i = s, reads:

(v - s + 1)sf~ ~) = (k)s (k - 1 )s- 2 [(v - 2s + 2)F~ ~} -- (s -- 1)(k - s)(k - s + t)].

Since F~ sJ = x l + x2 + " . + xs-- s(s - 1)/2 and F~ s~ -- xlx2. . .xs , this implies the lower bound in (22). The lower bound is attained if and only if F~ s~ = 0, i.e. one of the intersection numbers is zero.


To obtain the upper bound in (22), we transform (19) into the following equality:

(23) I s ( k - s + 1) 2 - ( v - 2s + 2)F~)] • 2 (s - 1)!(v- s + 1)s

= [ ( k - s + 1)(v- s + 1) + ( s - 1)(k- s ) ( k - s + 1)

- (v - 2 s + 2 ) F ] ' ) ] • ( k ) 2 ~ - i .

Using inequalities (15) and (21), we transform (23) into (22). The upper bound in (22) is attained if and only if the equality in (21) is attained, i.e. D is a tight 2s-design. []

REMARK 4.2. In the case of quasi-symmetric 3-designs with intersection numbers x and y, inequalities (22) read:

(k - 1)(k - 2) 2(k - 1)(k- 2) <~ x + y - 1

v - 2 v - 3

The upper bound for this case (s = 2) was obtained by Calderbank in [5] using linear programming techniques.

COROLLARY 4.3. Let D be an extremal ( 2 s - 1)-(v, k, 2) design with inter-

section numbers x l , x2, . . . , x~, s >~ 2. I f v >1 2k+ 1, then

X I -~- X2"~- " " "~- Xs < - s(k-1)

I f v = 2k or v = 2 k - 1 , then

sk xl + x 2 + " ' + x s < ~ .

Proof. The upper bound in inequality (22) implies immediately that if v >/2k + 1, then x l + x2 + ' " + xs <~ s ( k - 1)/2. The equality holds if and only if v = 2 k + 1 and D is a tight design. If v >~ 2 k - 1 , then (22) implies that Xl + x2 + " " + xs <~ sk/2, the equality holding if and only if v = 2 k - 1 and D is a tight design. By Bannai [1, Prop. 4], tight designs with v= 2k + 1 do not exist. This completes the proof. []

REMARK 4.4. For s=2, Corollary 4.3 strengthens the result of Pawale [15, Th. 3.4].

The following proposition suggests another lower bound for the sum of the intersection numbers of an extremal design.

PROPOSITION 4.5. Let D be an extremal (2s-1)-(v, k, 2) design with

( 2 s - 1 ) - D E S I G N S W I T H S I N T E R S E C T I O N N U M B E R S 261

intersection numbers xl , x2 . . . . . x~ taken in the increasing order. Then for

m-- 1, 2 , . . . , s,

(24) ~ x i >~ m(m-1) . i = 1

The equality holds if and only if xi = 2 i - 2 for i = 1, 2 . . . . . m. Proof. We use induct ion on s. The propos i t ion is obvious for s = I. Let

s >~ 2 and let the propos i t ion hold for ( 2 t - D-designs with t < s. We can

assume that m/> 2. If xi >~ 2i for i = 1, 2 . . . . . m - 1, then (24) holds. Suppose that xi ~< 2 i - 1 for

some i, 1 ~< i ~< m - 1. Let n be the first index such that x, ~< 2 n - 1. Since

2n < k, we can consider the derived design Dx, where X is a set of 2n points in D. The design D x is a ( 2 s - 2 n - 1 ) - d e s i g n whose intersection numbers are

conta ined in the set { x 1 - 2 n , x 2 - 2 n . . . . . x s - 2 n } . Since x i - 2 n < 0 for i ~< n, the design D x has at mos t s - n intersection numbers . Since this is a ( 2 s - 2 n - 1)-design, it has at least s - n intersection numbers . Therefore D x is an ext remal ( 2 s - 2 n - 1 ) - d e s i g n whose intersection numbers are x ,+ 1 - 2 n ,

Xn + 2 - - 2n . . . . , xs-- 2n. By the induct ion hypothesis,

( x l -2n ) >t ( m - n ) ( m - n - 1), i = n + l

which implies

xi >~ re(m- 1) - n ( n - 1), i = n + l

and the equali ty holds if and only if x i = 2 i - 2 for i = n + 1, n + 2 , . . . , m.

By the definition of n, E"~=l x i >~ n ( n - 1 ) + x , . Therefore,

xi >>- re(m-1), i = l

where the equal i ty holds if and only if x i = 2 i - 2, for i = n + 1, n + 2 . . . . . m,

and x , = 0, i.e. xi = 2 i - 2, for i = 1, 2 , . . . , m. [ ]

N o w we obta in the following characterist ic p roper ty of the Wit t 5-(24, 8, 1)

design:

T H E O R E M 4.6. Let D be an extremal ( 2 s - 1)-(v, k, )~) design with intersection numbers xl , x2 , . . . , x s. Then D is the Witt 5-(24, 8, 1)design if and only

if s ~ 3 and E~= a xl <<. s ( s - 1). Proof The necessity is obvious since it is well known that the Wit t 5-

(24, 8, 1) design has intersection numbers 0, 2 and 4 [-2]. Fo r the p roof of the


sufficiency, suppose that X I < X 2 < " ' < X s. Proposition 4.5 implies that X T = l x i = s ( s - 1 ) and x i = 2 i - 2 , for i = 1, 2 , . . . , s. From Definition 1.1, one can now calculate that F]S)=s(s-- l) /2 and F~2S)=s(s+ l)(s--1)(s--2)/8. Since xl =0, Theorem 4.1 yields the equation

2 ( k - s X k - s + 1) (25) v - 2s + 2 =

s

Let P,(z) = z ( z - 2).--(z- 2s + 2) be the Delsarte polynomial of the design D. Since P~(0)= P,(2)= 0, F~')= 0 and F~)_x = F~ )_ 2 (see Remark 1.3). Using these equalities, Equation (16), for i = s - 1 , can be transformed into

( k - 1Xk- 2) (26) v - s =

s - - 1

Subtract (26) from (25) and transform the resulting equation into the equation ( s - 2 ) ( k - s ) ( k - 3 s + 1)=0. Since s >~ 3 and k > s, k = 3 s - 1. Now (26) yields v = 10s-6 . Putting k = 3 s - 1 and v= 1 0 s - 6 in (16) for i = 1, we obtain:

2 (8s - 3)F~ ) - 4 s ( s - 1)(2s + 1)F~ ~) + s 3 ( s - 1)(2s + 1) = 0.

Using expressions for F] ~) and F~z ~), we simplify this equation, obtaining finally the equation s z - 5s + 6 = 0 which yields s = 3. Therefore, k = 8, v = 24. Putting these values in (14), we obtain 2 = 1, i.e. D is the Witt 5-(24, 8, 1)

design. []

P R O P O S I T I O N 4.7. Let D be a tight 2s-design, s >~ 2, with intersection

numbers x 1, Xz . . . . . x~. Then x l + Xz + ... + x~ >>. s2; the equality holds if and

only if D is the Wi t t 4-(23, 7, 1) design.

Proof. For any point p, the derived design Dp is an extremal ( 2 s - 1)-design with intersection numbers x~ - 1, x2 - 1 . . . . . x~ - 1. Applying Proposition 4.5 to Dp, we obtain that x~ + x z + . . . + x , >1 s z. Ifs >i 3, then by Theorem 4.6, the equality x 1 + x 2 + ... + x s = s 2 holds if and only if Dp is the Witt 5-(24, 8, 1) design. In this case, D is a tight 6-design. By Peterson [16], such designs do not exist. Therefore, the equality can hold only for s = 2. Then D is a tight 4- design. By Ito et al. 1-10], [-14], and Bremner [-3], D is the Witt 4-(23, 7, 1) design whose intersection numbers are 1 and 3 or its complement whose intersection numbers are 10 and 12. Therefore, the equality

x l + x 2 + ... + X s = S 2 holds if and only if D is the Witt 4-(23, 7, 1)design.D

LEMMA 4.8. I f D is an extremal (2s-- 1)-(v, k, 2) design without disjoint blocks, then for any point p, the derived design Dv has exact ly s intersection numbers.

Proof. Since Dp is a (2s--2)-design, it has s - 1 or s intersection numbers.


Suppose Dp has exactly s - 1 intersection numbers. Then it is a tight (v - 1, k - 1, 2)-design, and from Ray-Chaudhuri and Wilson [17, Th. 4], the

number of blocks 21 of Dp is equal to ( ~ : ) . Since \ - - - /

21 = 2 ( v - 1)2s_2/(k- 1)2s_ 2, we obtain

2(s - 1)!(v-s)~_ 1 = ( k - 1)~- z.

Equality (19) implies now that

( v - 2s + 2)El ~) = ( s - l ) (k - s)(k- s + 1),

which is the lower bound equality in (22). This implies that one of the intersection numbers of the design D is zero. But this is not the case, because D has no disjoint blocks. Therefore, the derived design Dp has exactly s intersection numbers. []

COROLLARY 4.9. Let D be an extremal ( 2 s - 1)-(v, k, 2) design. Let x 1 be the

smallest intersection number of D. I f v > 2 k - x 1 , then for any point p, the residual design D p has exactly s intersection numbers.

Proof Since v - 2 k + x l is the smallest intersection number of the complementary design D, this design has no disjoint blocks. Therefore Dp has s

intersection numbers. Since D p =Dp, this implies that D p has also s intersection numbers. []

PROPOSITION 4.10. Let D be an extremal (2s-1)-(v, k, 2) design. I f s>~ 3 and v >t 2k, then for any point p, the residual design D ~ has exactly s intersection numbers.

Proof Due to Corollary 4.9, we have only to consider the case v = 2k and x 1 = 0. In this case, D p is a ( 2 s - 2 ) - (2k-1 , k, 2') design. By Bannai [1, Prop. 4], such a design cannot be tight. Therefore it has more than s - 1, i.e. s intersection numbers. []

REMARK 4.11. The above result does not necessarily hold for s = 2. For example, the Hadamard 3-(8, 4, 1) design has intersection numbers 0 and 2, while its residual is a symmetric 2-(7,4, 2) design. For s = 3, Proposition 4.10 was obtained by Hobart [11] by different methods.

THEOREM 4.12. I f D is an extremal (2s - 1)-(v, k, 2) design, then

s ( s - 1) (a) v - 2s + 2 divides ----f--- ( k - s ) (k - s + 1)2(k- s + 2);

s2(s- 1)3(s- 2)2 (b) k - 2s + 2 divides

2

264 YURY J. IONIN AND MOHAN S. S H R I K H A N D E

Proof. Theorem 3.6, for i = 1, immediately implies (a). N o w

v - 2 s + 2 "~2s -- 2 2 . k - 2 s + 2

Hence, k - 2 s + 2 divides [ 2 s ( s - 1 ) / 2 ] ( k - s ) ( k - s + l ) 2 ( k - s + 2 ) . Since k - s + 2 - s m o d ( k - 2 s + 2 ) , k - s + 1 --- s - 1 m o d ( k - 2 s + 2) and k - s - s - 2 m o d ( k - 2s + 2), this implies that k - 2s + 2 divides [s2(s- 1)3(s- 2)2]/2. []

REMARK 4.13. Taking s = 2 in Theorem 4.12, we observe that condition (a) generalizes Pawale's result [15, Cor. 2.7] for quasi-symmetric 3-(v,k, 2) designs, while condition (b) gives no information for such designs. However, as the following two results show, the situation is different if s >~ 3. The first is useful in [ 13].

COROLLARY 4.14. Let D be a 5-(v, k, )~) design with three intersection numbers. Then k - 4 divides 362. []

COROLLARY 4.15. For fixed values of 2 and s >>. 3, there exist at most finitely many extremal (2s - 1)-(v, k, 2) designs.

Proof. From Theorem 4.12(b), for fixed 2 and s, there are finitely many possible k's. Inequality (15) shows that for any fixed k and s, there are finitely many v's. []

REFERENCES

1. Bannai, E., 'On tight designs', Quart. J. Math. (Oxford) 2g (1977), 433-448. 2. Beth, T., Jungnickel, D. and Lenz, H., Design Theory, B.I. Wissenschaftverlag, Mannheim,

1985, Cambridge Univ. Press, Cambridge, 1986. 3. Bremner, A., 'A diophantine equation arising from tight 4-designs', Osaka J. Math. 16 (1979),

353-356. 4. Brualdi, R. A., Introductory Combinatorics, North Holland, New York, Oxford, Amsterdam,

1977. 5. Calderbank, A. R., 'Inequalities for quasi-symmetric designs', J. Combin. Theory Set. A 48(1)

(1988), 53-64. 6. Cameron, P. J., 'Near regularity conditions for designs', Geom. Dedicata 2 (1973), 213-223. 7. Cameron, P. J. and Van Lint, J. H , Graphs, Codes and Desions, London Math. Soc. Lecture

Note Series, vol. 43, Cambridge Univ. Press, Cambridge, 1980. 8. Delsarte, P., 'An algebraic approach to the association schemes of coding theory', Philips

Research Reports Supplements 10 (1973). 9. Dembowski, P., Finite Geometries, Springer, Berlin, Heidelberg, New York, 1968.

10. Enomoto, H., Ito, N. and Noda, R., 'Tight 4-designs', Osaka J. Math. 16 (1979), 39-43. 11. Hobart, S. A., 'Designs of type (2 2),, PhD thesis, Univ. of Michigan, 1987. 12. Hobart, S. A., 'On designs related to coherent configurations of type (2 ~),, Discrete Math. 94

(1991), 103-127.

(2s-1)-DESIGNS W I T H S INTERSECTION NUMBERS 265

13. Ionin, Y. J. and Shrikhande, M. S., '5-designs with three intersection numbers', J. Combin. Theory Ser. A (to appear).

14. Ito, N., 'On tight 4-designs', Osaka J. Math. 12 (1975), 493-522 (corrections and supplements, Osaka J. Math. 15 (1978), 693-697).

15. Pawale, R. M., 'Inequalities and bounds for quasi-symmetric 3-designs', J. Combin. Theory Ser. A, 60 (1992), 159-167.

16. Peterson, C., 'On tight 6-designs', Osaka J. Math. 14 (1977), 417-435. 17. Ray-Chaudhuri, D. K. and Wilson, R. M., 'On t-designs', Osaka d. Math. 12 (1975), 737-744. 18. Sane, S. S. and Shrikhande, M. S., 'Quasi-symmetric 2, 3, 4-designs', Combinatorica 7(3)

(1987), 291-301. 19. Shrikhande, M. S., 'The Delsarte polynomial of 5-(v, k, )0 designs with three intersection

numbers', Journal of Combinatorics, Information and System Sciences, Special volume in honor of Prof. C. R. Rao (to appear).

20. Shrikhande, M. S. and Sane, S. S., Quasi-Symmetric Designs, London Math. Soc. Lecture Note Series~ vol. 164, Cambridge Univ. Press, Cambridge, 1991.

21. Shrikhande, M. S. and Singhi, N. M., 'An elementary derivation of the annihilator polynomial of extremal (2s + D-designs', Discrete Math. 90 (1990), 93-96.

22. Wilson, R. M., 'On the theory of t-designs', in Enumeration and Design (ed. D. M. Jackson and S. A. Vanstone), Academic Press, New York, 1984, pp. 19-49.

Authors' address: Yury J. Ionin and Mohan S. Shrikhande, Department of Mathematics, Central Michigan University, Mt Pleasant, MI 48859, U.S.A.

(Received, June 15, 1992)

Documents

Ionin Y. J., Shrikhande M. S.-(2s1) Designs Withs Intersection Numbers (1993)