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.

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Journal of Combinatorial Designs
School of Mathematics and Statistics

Kokkala, J.I. (Janne I.), Meagher, K. (Karen), Naserasr, R. (Reza), Nurmela, K.J. (Kari J.), Östergård, P.R.J. (Patric R. J.), & Stevens, B. (2019). On the structure of small strength-2 covering arrays. Journal of Combinatorial Designs. doi:10.1002/jcd.21671