Consecutive Covering Arrays and a New Randomness Test

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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 = qt for an orthogonal array to exist and n ≥ qt 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 = Wk, 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.