Covering arrays from cyclotomy

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34 Scopus citations


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 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 > t2 v4t . 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 22t-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 v2t, 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.

Original languageEnglish (US)
Pages (from-to)201-219
Number of pages19
JournalDesigns, Codes, and Cryptography
Issue number2-3
StatePublished - May 2010


  • Covering array
  • Cyclotomic class
  • Orthogonal array
  • ε-biased array

ASJC Scopus subject areas

  • Computer Science Applications
  • Applied Mathematics


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