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, CAN$(t,k,v)$, in a covering array for given values of the parameters $t,k$ and $v$. Asymptotic upper bounds for CAN$(t,k,v)$ have earlier been established using the Stein-Lov\'asz-Johnson strategy and the Lov\'asz local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lov\'asz-Johnson bound is derived. Then using alteration, the Stein-Lov\'asz-Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lov\'asz local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on CAN$(t,k,v)$ are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lov\'asz local lemma and the conditional Lov\'asz local lemma distribution.