Mohan S. Shrikhande

Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.

The structure of triangle-free quasi-symmetric designs

Abstract Quasi-symmetric triangle-free designs D with block intersection numbers 0 and y and with no three mutually disjoint blocks are studied. It is shown that the parameters of D are expressible in terms of only two parameters y and m , where m = k / y , k being the block size. Baartmans and Shrikhande proved that 2 ⩽ m ⩽ y + 1 and characterized the extremal values of m . An alternative characterization of the extremal cases and also an alternative proof of the bounds is obtained. It is conjectured that besides the extremal cases, there are only finitely many such designs. It is proved that in such designs if k is a prime power p n , then p = 2 and D is a Hadamard design.

Finiteness questions in quasi-symmetric designs

Abstract Quasi-symmetric designs with block intersection numbers 0 and y⩾2 are considered. It is shown that the number of such designs is finite under any one of the following two restrictions: (1) The block size k is fixed. (2) The integer pair ( e , z ), with the following property is fixed: the number of blocks disjoint from a given block is at most e and the positive block intersection number y is at most z. The connection of these results with a well-known conjecture on symmetric designs is discussed.

A survey of some problems in combinatorial designs— a matrix approach

Abstract In this survey paper matrix theoretic methods dealing with some aspects of the theory of combinatorial designs are discussed. In particular, questions dealing with incidence matrices, quasisymmetric designs, strongly regular graphs, block graphs, special partially balanced designs, partial geometric designs, t 1 2 designs, and resolutions are examined. Some recent work of Haemers on the interlacing of eigenvalues and Wilson on t -designs is also included.

Some characterizations of quasi-symmetric designs with a spread

The designPG2 (4,q) of the points and planes ofPG (4,q) forms a quasi-symmetric 2-design with block intersection numbersx=1 andy=q+1. We give some characterizations of quasi-symmetric designs withx=1 which have a spread through a fixed point. For instance, it is proved that if such a designD is also smooth, thenD≅PG2 (4,q).

On the λ-Design Conjecture

A?-design is a family ofvsubsets (blocks) of av-set such that any two distinct blocks intersect in?points and not all blocks have the same cardinality. Rysers and Woodalls?-design conjecture states that each?-design can be obtained from a symmetric design by complementing with respect to a fixed block. We prove this conjecture forv=p+1, 2p+1, 3p+1, wherepis prime.

Equidistant families of sets

Abstract A family of subsets of the set {1, 2, …, v } is called equidistant if the symmetric difference of any two distinct sets from the family has constant cardinality. Such families can be interpreted as binary equidistant codes of length v . It is known that for v ≡ 3 (mod 4), the maximum size of an equidistant code of length v is v + 1 and existence of a code of length v + 1 is equivalent to existence of a Hadamard matrix of order v + 1. We prove for any v that binary equidistant codes of size v exist if and only if there exist certain combinatorial designs. We specify these designs in case of binary equidistant codes of length v and size v having maximal distance between codewords, thereby extending results of Stinson and van Rees and van Lint.

Quasi-Symmetric Designs with Good Blocks and Intersection Number One

We show that a quasi-symmetric design with intersection numbers 1 and y > 1 and a good block belongs to one of three types: (a) it has the same parameters as PG2(4, q), the design of points and planes in projective 4-space; (b) it is the 2-(23, 7, 21) Witt design; (c) its parameters may be written v = 1 + ((α − 1)λ + 1)(y − 1) and k = 1 + α(y − 1), where α is an integer and α > y ≥ 5, and the design induced on a good block is a 2-(k, y, 1) design. No design of type (c) is known; moreover, for large ranges of the parameters, it cannot exist for arithmetic reasons concerning the parameters. We show also that PG2(4, q) is the only design of type (a) in which all blocks are good.

On classification of two class partially balanced designs

Abstract Let Γ =( V , E ) be a connected strongly regular graph and B a family of k -subsets (blocks) of V . The pair D=(V, B ) is a partially balanced ( v , k , λ , μ )-design on Γ if for any distinct vertices x and y , the number of blocks containing { x , y } is λ if { x , y } is an edge and is μ otherwise. In such a design, the replication number r satisfies r ⩾2 λ − μ . We obtain a complete classification of designs with r =2 λ − μ in case Γ has an eigenvalue −2. We show that there are three infinite families (and their complements) and several sporadic designs. The sporadic designs correspond to the Petersen, Clebsch, and Shrikhande graphs and the triangular graph T (4).

New results for quasi-symmetric designs an application of MACSYMA

Abstract Quasi-symmetric designs with block intersection numbers x and y are investigated. Let Γ be the usual block graph of such a design. Let c be the number of triangles on any edge of Γ , the complement of Γ. It is shown that for fixed values of x, y ⩾ 2 and c ⩾ 0 there are only finitely many such designs. This extends earlier results about quasi-symmetric designs with special properties. We show connections with strongly resolvable designs and also with designs considered by Holliday. The symbolic calculations were carried out using MACSYMA