Sharad S. Sane

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.

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).

A short proof of a conjecture on quasi-symmetric 3-designs

It was conjectured by Sane and M.S. Shrikhande that the only nontrivial quasi-symmetric 3-design with the smaller block intersection number one is either the Witt 4-(23, 7, 1) design or its residual. Calderbank and Morton recently proved this conjecture using sophisticated number theoretic arguments. A short and elementary proof of this conjecture is presented in this paper.

Maximal arcs in designs

An (α,n)-arc in a 2-design is a set ofn points of the design such that any block intersects it in at most α points. For such an arc,n is bounded by 1+(r(α−1)/λ), with equality if and only if every block meets the arc in either 0 or α points. An (α,n) arc with equality in above is said to be maximal.A maximal block arc can be dually defined. This generalizes the notion of an oval (α=2) in a symmetric design due to Asmus and van Lint. The aim of this paper is to study the infinite family of possibly extendable symmetric designs other than the Hadamard design family and their related designs using maximal arcs. It is shown that the extendability corresponds to the existence of a proper family of maximal arcs. A natural duality between point and block arcs is established, which among other things implies a result of Cameron and van Lint that extendability of a given design in this family is equivalent to extendability of its dual. Similar results are proved for other related designs.

Maximal intersecting families of finite sets and

The following theorem is proved. The collection of lines of an n-uniform projective Hjelmslev plane is maximal when considered as a collectiion of mutually intersecting sets of equal cardinality.

n

Abstract A stronger version of a theorem of Mavron, Mullin and Rosa on the strong index of a generalised quasi-residual design is obtained. Some examples of non-isomosphic generalised quasi-residual designs are given.

-uniform Hjelmslev planes

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

On generalised quasi-residual designs

Using the triangular graph T(6) it is shown that up to isomorphism any (16,6,2)-design is uniquely determined by the number of ovals in it, which is 12, 28 or 60. It is shown that the (16,6,2)-design with precisely 60 ovals is embeddable in a 3-(22,6,1)-design. As a consequence the uniqueness of the 3-(22,6,1)-design is obtained.