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t-Covering Arrays: Upper Bounds and Poisson Approximations

Published online by Cambridge University Press:  12 September 2008

Anant P. Godbole ,
Daphne E. Skipper  and
Rachel A. Sunley
Anant P. Godbole
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Houghton MI 49931, USA
Daphne E. Skipper
Affiliation:
Department of Mathematics, University of Kentucky, Lexington KY 40506, USA
Rachel A. Sunley
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor MI 48109, USA
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Abstract

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) qt 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 qt 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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 2 , June 1996 , pp. 105 - 117
Copyright
Copyright © Cambridge University Press 1996

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