Upper Bounds on the Size of Covering Arrays
Kaushik Sarkar, Charles J. Colbourn · SIAM Journal on Discrete Mathematics · 2017
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--Lovász--Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein--Lovász--Johnson bound is derived. Then using alteration, the Stein--Lovász--Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovász 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 Lovász local lemma and the conditional Lovász local lemma distribution.