### Abstract

A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) q^{t} distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let K_{λ} = K_{λ}(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the q^{t} possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on K_{λ} that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

Original language | English (US) |
---|---|

Pages (from-to) | 105-117 |

Number of pages | 13 |

Journal | Combinatorics Probability and Computing |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1996 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Combinatorics Probability and Computing*,

*5*(2), 105-117. https://doi.org/10.1017/S0963548300001905

**t-Covering arrays : Upper bounds and poisson approximations.** / Godbole, Anant P.; Skipper, Daphne E.; Sunley, Rachel A.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 5, no. 2, pp. 105-117. https://doi.org/10.1017/S0963548300001905

}

TY - JOUR

T1 - t-Covering arrays

T2 - Upper bounds and poisson approximations

AU - Godbole, Anant P.

AU - Skipper, Daphne E.

AU - Sunley, Rachel A.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

AB - A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

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UR - http://www.scopus.com/inward/citedby.url?scp=0030538878&partnerID=8YFLogxK

U2 - 10.1017/S0963548300001905

DO - 10.1017/S0963548300001905

M3 - Article

AN - SCOPUS:0030538878

VL - 5

SP - 105

EP - 117

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -