Some results on λ-designs

AbstractA λ-design as introduced by Ryser [3] is a (0, 1)-square matrix with constant column inner products but not all column sums equal. Ryser has shown such a matrix to have two row sums and he constructs an infinite family of λ-designs called H-designs. This paper does three things: (1) generalizes Ryser's H-design construction to an arbitrary (ν, k, λ)-configuration, (2) establishes some additional general properties of λ-designs, and (3) determines all 4-designs

Combinatorial Designs with Two Singular Values I. Uniform Multiplicative Designs

In this and a sequel paper [10] we study combinatorial designs whose incidence matrix has two distinct singular values.These generalize 2-(v, k, É) designs, and include partial geometric designs and uniform multiplicative designs.Here we study the latter, which are precisely the nonsingular designs.We classify all such designs with smallest singular value at most, generalize the Bruck-Ryser-Chowla conditions, and enumerate, partly by computer, all uniform multiplicative designs on at most 30 points.combinatorics;matrices;singularities

Design Lines

The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in $\mathcal{D}$. These lines provide a degree of organization for certain general classes of designs, and the paper is devoted to exploring properties of the lines. Several examples of families of designs that seem naturally to follow the lines are presented.Comment: 16 page

An existing problem for symmetric design: Bruck Ryser Chowla Theorem

Symetric designs are interesting objects of combinatorics, and have some relations with coding theory, difference sets, geometry and finite group theory. They have applications on statistics and design experiments. In the present paper we study an existing problem for symmetric design due to Bruck, Ryser and Chowla and write an algorithm by using their theorem called BRC Theorem

Conditions for Singular Incidence Matrices

Suppose one looks for a square integral matrixN, for which NN has a prescribed form.Then the Hasse-Minkowski invariants and the determinant of NN lead to necessary conditions for existence.The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design.This approach fails when N is singular.In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N are known, or known to be rationally equivalent.This leads for example to non-existence conditions for selfdual generalised polygons, semi-regular square divisible designs and distance-regular graphs.singularities;matrices;graphs