On Perfect Sequence Covering Arrays

Abstract

A PSCA$(v, t, \lambda)$ is a multiset of permutations of the $v$-element alphabet $\{0, \dots, v-1\}$ such that every sequence of $t$ distinct elements of the alphabet appears in the specified order in exactly $\lambda$ of the permutations. For $v \geq t \geq 2$, we define $g(v, t)$ to be the smallest positive integer $\lambda$ such that a PSCA$(v, t, \lambda)$ exists. We show that $g(6, 3) = g(7, 3) = g(7, 4) = 2$ and $g(8, 3) = 3$. Using suitable permutation representations of groups we make improvements to the upper bounds on $g(v, t)$ for many values of $v \leq 32$ and $3\le t\le 6$. We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.