Covering arrays from cyclotomy

For a prime power q ≡ 1 (mod v), the q × q cyclotomic matrix, whose entries are the discrete logarithmsmodulo v of the entries in the addition table of Fq , 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 > t2v4t . 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 > t222t−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 > t2v2t , 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 t2v2t , and consequences for the existence of covering arrays reported.