Covering arrays from cyclotomy

For a prime power q ≡ 1 (mod{v}), the q × q cyclotomic matrix, whose entries are the discrete logarithms modulo v of the entries in the addition table of F_q, has been shown using character theoretic arguments to produce an ε-biased array, provided that q is large enough as a function of v and ε . A suitable choice of ε ensures that the array is a covering array of strength t when {q > t² 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 ε -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 t² v ^2t, and consequences for the existence of covering arrays reported.