Covering arrays from cyclotomy

Abstract

For a prime power ${q \equiv 1 (mod{v})}$ , the q × q cyclotomic matrix, whose entries are the discrete logarithms modulo v of the entries in the addition table of ${\mathbb{F}_q}$ , has been shown using character theoretic arguments to produce an ${\varepsilon}$ -biased array, provided that q is large enough as a function of v and ${\varepsilon}$ . A suitable choice of ${\varepsilon}$ ensures that the array is a covering array of strength t when ${q > t^2 v^{4t}}$ . On the other hand, when v = 2, using a different character-theoretic argument the matrix has been shown to be a covering array of strength t when ${q > t^2 2^{2t-2}}$ . The restrictions on ${\varepsilon}$ -biased arrays are more severe than on covering arrays. This is exploited to prove that for all v ≥ 2, the matrix is a covering array of strength t whenever ${q > t^2 v^{2t}}$ , again using character theory. A number of constructions of covering arrays arise by developing and extending the cyclotomic matrix. For each construction, extensive computations for various choices of t and v are reported that determine the precise set of small primes for which the construction produces a covering array. As a consequence, many covering arrays are found when q is smaller than the bound ${t^2 v^{2t}}$ , and consequences for the existence of covering arrays reported.