### Abstract

Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, CAN(t; k; v), in a covering array for given values of the parameters t; k, and v. Asymptotic upper bounds for CAN(t; k; v) have been established using the Stein-Lovász-Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lovász-Johnson bound is derived. Then using alteration, the Stein-Lovász-Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovász local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on CAN(t; k; v) are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lovász local lemma and the conditional Lovász local lemma distribution.

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

Pages (from-to) | 1277-1293 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

State | Published - 2017 |

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### Keywords

- Conditional expectation algorithm
- Covering array
- Lovász local lemma

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(2), 1277-1293. https://doi.org/10.1137/16M1067767

**Upper bounds on the size of covering arrays.** / Sarkar, Kaushik; Colbourn, Charles.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 31, no. 2, pp. 1277-1293. https://doi.org/10.1137/16M1067767

}

TY - JOUR

T1 - Upper bounds on the size of covering arrays

AU - Sarkar, Kaushik

AU - Colbourn, Charles

PY - 2017

Y1 - 2017

N2 - Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, CAN(t; k; v), in a covering array for given values of the parameters t; k, and v. Asymptotic upper bounds for CAN(t; k; v) have been established using the Stein-Lovász-Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lovász-Johnson bound is derived. Then using alteration, the Stein-Lovász-Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovász local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on CAN(t; k; v) are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lovász local lemma and the conditional Lovász local lemma distribution.

AB - Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, CAN(t; k; v), in a covering array for given values of the parameters t; k, and v. Asymptotic upper bounds for CAN(t; k; v) have been established using the Stein-Lovász-Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lovász-Johnson bound is derived. Then using alteration, the Stein-Lovász-Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovász local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on CAN(t; k; v) are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lovász local lemma and the conditional Lovász local lemma distribution.

KW - Conditional expectation algorithm

KW - Covering array

KW - Lovász local lemma

UR - http://www.scopus.com/inward/record.url?scp=85021883944&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021883944&partnerID=8YFLogxK

U2 - 10.1137/16M1067767

DO - 10.1137/16M1067767

M3 - Article

VL - 31

SP - 1277

EP - 1293

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -