Sign-balanced covering matrices

Abstract

A q × n array with entries from 0, 1,..., q − 1 is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of [0, 1,... q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1] t , a λq × n array will be said to form a ( t, q, λ, Δ) sign-balanced matrix if for each choice C 1, C 2,..., C t of t columns and for each choice ɛ = ( ɛ 1,..., ɛ t ) ∈ Δ of signs, the linear combination ∑ j = 1 t ε j C j contains (mod q) each entry of [0, 1,..., q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k( n, t, q, λ, δ) of rows have to be so that for each choice of t columns and for each choice ( ɛ 1, ..., ɛ t ) of signs in δ, the linear combination ∑ j = 1 t ε j C j contains each entry of [0, 1,..., q − 1] at least λ times? We use probabilistic methods, in particular the Lovász local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k( n, t, q, λ, δ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described.