A Semi-Random Construction of Small Covering Arrays

Given a set S of v ≥ 2 symbols, and integers k ≥ t ≥ 2 and N ≥ 1, an N × k array A ∈ SN×k is an (N ; t, k, v)-covering array if all sequences in S appear as rows in every N × t subarray of A. These arrays have a wide variety of applications, driving the search for small covering arrays. The covering array number, CAN(t, k, v), is the smallest N for which an (N ; t, k, v)-covering array exists. In this paper, we combine probabilistic and linear algebraic constructions to improve the upper bounds on CAN(t, k, v) by a factor of ln v, showing that for prime powers v, CAN(t, k, v) ≤ (1 + o(1)) ( (t− 1)v/(2 log2 v − log2(v + 1)) ) log2 k, which also offers improvements for large v that are not prime powers. Our main tool, which may be of independent interest, is a construction of an array with v rows that covers the maximum possible number of subsets of size t.