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### Abstract

A covering array CA(N; t, k, v) of strength t is an N × k array of symbols from an alphabet of size v such that in every N × t subarray, every t-tuple occurs in at least one row. A covering array is optimal if it has the smallest possible N for given t, k, and v, and uniform if every symbol occurs [N∕v] or [N∕v] times in every column. Before this paper, the only known optimal covering arrays for t = 2 were orthogonal arrays, covering arrays with v = 2 constructed from Sperner's Theorem and the Erdős-Ko-Rado Theorem, and 11 other parameter sets with v > 2 and N > v2. In all these cases, there is a uniform covering array with the optimal size. It has been conjectured that there exists a uniform covering array of optimal size for all parameters. In this paper, a new lower bound as well as structural constraints for small uniform strength-2 covering arrays is given. Moreover, covering arrays with small parameters are studied computationally. The size of an optimal strength-2 covering array with v > 2 and N > v2 is now known for 21 parameter sets. Our constructive results continue to support the conjecture.

Original language | English |
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Pages (from-to) | 5-24 |

Journal | Journal of Combinatorial Designs |

Volume | 28 |

Issue number | 1 |

Early online date | 1 Jan 2019 |

DOIs | |

Publication status | Published - Jan 2020 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- bounds
- computational enumeration
- covering array

## Fingerprint Dive into the research topics of 'On the structure of small strength-2 covering arrays'. Together they form a unique fingerprint.

## Projects

- 1 Finished

## Construction and Classification of Discrete Mathematic Structures

Kokkala, J., Laaksonen, A., Heinlein, D., Ganzhinov, M., Östergård, P. & Szollosi, F.

01/09/2015 → 24/09/2019

Project: Academy of Finland: Other research funding

## Cite this

*Journal of Combinatorial Designs*,

*28*(1), 5-24. https://doi.org/10.1002/jcd.21671