Covering arrays of strength three from extended permutation vectors

Abstract

A covering array $\text{ CA }(N;t,k,v)$ is an $N\times k$ array such that in every $N\times t$ subarray each possible t-tuple over a v-set appears as a row of the subarray at least once. The integers t and v are respectively the strength and the order of the covering array. Let v be a prime power and let ${\mathbb {F}}_v$ denote the finite field with v elements. In this work the original concept of permutation vectors generated by a $(t-1)$-tuple over ${\mathbb {F}}_v$ is extended to vectors generated by a t-tuple over ${\mathbb {F}}_v$. We call these last vectors extended permutation vectors (EPVs). For every prime power v, a covering perfect hash family $\text{ CPHF }(2;v^2-v+3,v^3,3)$ is constructed from EPVs given by subintervals of a linear feedback shift register sequence over ${\mathbb {F}}_v$. When $v\in \{7,9,11,13,16,17,19,23,25\}$ the covering array $\text{ CA }(2v^3-v;3,v^2-v+3,v)$ generated by $\text{ CPHF }(2;v^2-v+3,v^3,3)$ has less rows than the best-known covering array with strength three, $v^2-v+3$ columns, and order v. CPHFs formed by EPVs are also constructed using simulated annealing; in this case the results improve the size of eighteen covering arrays of strength three.