t-Covering Arrays: Upper Bounds and Poisson Approximations

A k × n array with entries from the q-letter alphabet {0, 1, . . . , q − 1} is said to be t-covering if each k × t submatrix has (at least one set of) q 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 Kλ = Kλ(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 possible “words” of length t at least λ times. The Lovász lemma yields an upper bound on Kλ 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.