Consecutive covering arrays and a new randomness test

A k × n array with entries from an “alphabet” A = { 0 , 1 , … , q − 1 } of size q is said to form a t-covering array (resp. orthogonal array) if each t × n submatrix of the array contains, among its columns, at least one (resp. exactly one) occurrence of each t-letter word from A (we must thus have n = q t for an orthogonal array to exist and n ≥ q t for a t -covering array). In this paper, we continue the agenda laid down in Godbole et al. (2009) in which the notion of consecutive covering arrays was defined and motivated; a detailed study of these arrays for the special case q = 2 , has also carried out by the same authors. In the present article we use first a Markov chain embedding method to exhibit, for general values of q, the probability distribution function of the random variable W = W k , n , t defined as the number of sets of t consecutive rows for which the submatrix in question is missing at least one word. We then use the Chen–Stein method (Arratia et al., 1989, 1990) to provide upper bounds on the total variation error incurred while approximating L ( W ) by a Poisson distribution Po ( λ ) with the same mean as W. Last but not least, the Poisson approximation is used as the basis of a new statistical test to detect run-based discrepancies in an array of q-ary data.