### Abstract

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 > 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 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 t^{2} v ^{2t}, and consequences for the existence of covering arrays reported.

Original language | English (US) |
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Pages (from-to) | 201-219 |

Number of pages | 19 |

Journal | Designs, Codes, and Cryptography |

Volume | 55 |

Issue number | 2-3 |

DOIs | |

State | Published - May 1 2010 |

### Keywords

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

### ASJC Scopus subject areas

- Computer Science Applications
- Applied Mathematics

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## Cite this

*Designs, Codes, and Cryptography*,

*55*(2-3), 201-219. https://doi.org/10.1007/s10623-009-9333-8