TY - JOUR

T1 - Covering arrays from cyclotomy

AU - Colbourn, Charles

N1 - Funding Information:
Acknowledgments Thanks to Goran Konjevod for very helpful discussions on ε-biased arrays and character theory. Research is supported by DOD grants N00014-08-1-1069 and N00014-08-1-1070.

PY - 2010/5

Y1 - 2010/5

N2 - 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.

AB - 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.

KW - Covering array

KW - Cyclotomic class

KW - Orthogonal array

KW - ε-biased array

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U2 - 10.1007/s10623-009-9333-8

DO - 10.1007/s10623-009-9333-8

M3 - Article

AN - SCOPUS:77951207016

SN - 0925-1022

VL - 55

SP - 201

EP - 219

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

IS - 2-3

ER -