Block designs with near-minimal number of blocks

Abstract

For our purposes, we can begin by saying that a combinatorial design is a binary matrix on which we have imposed some conditions; this is a simple and easy way to introduce this concept. The mathematical study of these designs was promoted by problems in the design and analysis of experiments in statistics, especially since the 1930s. For the mathematical aspects, two classic references are the books by Hall [2] and by Ryser [8]; as examples of more recent references, we have the books by Ionin and Shrikhande [3], and by Van Lint and Wilson [4]. For the statistical aspects, a classic reference is the book by Box, Hunter, and Hunter [1]. Binary matrices and vectors are of interest in combinatorics, in the theories of coding and of information, and in cryptology; some applications and several mathematical results are discussed in [7]. The central problems are the existence and the construction of such designs. In general, these problems remain unsolved, but there are partial results [2, 3, 4, 8]. Also important are problems concerned with minimal or maximal properties of combinatorial designs. A classic example is Fisher’s inequality [2, p. 129]. More recently, Marrero [5] obtained a maximal property for binary nonsingular matrices in which each row has the same number of ones. Also, Marrero and Pasles [6] have found the best possible bounds