Tower of covering arrays

Abstract

Covering arrays are combinatorial objects that have several practical applications, specially in the design of experiments for software and hardware testing. A covering array of strength t and order v is an N×k array over Zv with the property that every N×t subarray covers all members of Zvt at least once. In this work we explore the construction of a Tower of Covering Arrays (TCA) as a way to produce covering arrays that improve or match some current upper bounds. A TCA of height h is a succession of h+1 covering arrays C0,C1,…,Ch in which for i=1,2,…,h the covering array Ci is one unit greater in the number of factors and the strength of the covering array Ci−1; this way, if the covering array C0 is of strength t and has k factors then the covering arrays C1,…,Ch are of strength t+1,…,t+h and have k+1,…,k+h factors respectively. We note that the ratio between the number of rows of the last covering array Ch in a TCA of height h and the number of rows of the best known covering array for the same values of t, k, and v as for Ch is reduced as h grows. Therefore, we search for TCAs with the greatest height possible. The relevant results are the improvement of nineteen current upper bounds for v=2 and t∈{7,8,9,10,11}, and the construction of twenty-one covering arrays that matched current upper bounds.