t-Covering arrays: Upper bounds and poisson approximations

A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) q^t distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let K_λ = K_λ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the q^t possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on K_λ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.