Upper Bounds on the Size of Covering Arrays

Abstract

Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, $\mathsf{CAN}(t,k,v)$, in a covering array for given values of the parameters $t,k$, and $v$. Asymptotic upper bounds for $\mathsf{CAN}(t,k,v)$ have been established using the Stein--Lovasz--Johnson strategy and the Lovasz local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein--Lovasz--Johnson bound is derived. Then using alteration, the Stein--Lovasz--Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovasz local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on $\mathsf{CAN}(t,k,v)$ are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lovasz local lemma and the conditional Lovasz local lemma ...