A group-based structure for perfect sequence covering arrays

Abstract

An (n, k)-perfect sequence covering array with multiplicity $\lambda$, denoted $\mathrm{{PSCA}}(n,k,\lambda )$, is a multiset whose elements are permutations of the sequence $(1,2, \dots , n)$ and which collectively contain each ordered length k subsequence exactly $\lambda$ times. The primary objective is to determine for each pair (n, k) the smallest value of $\lambda$, denoted g(n, k), for which a $\mathrm{{PSCA}}(n,k,\lambda )$ exists; and more generally, the complete set of values $\lambda$ for which a $\mathrm{{PSCA}}(n,k,\lambda )$ exists. Yuster recently determined the first known value of g(n, k) greater than 1, namely $g(5,3)=2$, and suggested that finding other such values would be challenging. We show that $g(6,3)=g(7,3)=2$, using a recursive search method inspired by an old algorithm due to Mathon. We then impose a group-based structure on a perfect sequence covering array by restricting it to be a union of distinct cosets of a prescribed nontrivial subgroup of the symmetric group $S_n$. This allows us to determine the new results that $g(7,4)=2$ and $g(7,5) \in \{2,3,4\}$ and $g(8,3) \in \{2,3\}$ and $g(9,3) \in \{2,3,4\}$. We also show that, for each $(n,k) \in \{ (5,3), (6,3), (7,3), (7,4) \}$, there exists a $\mathrm{{PSCA}}(n,k,\lambda )$ if and only if $\lambda \ge 2$; and that there exists a $\mathrm{{PSCA}}(8,3,\lambda )$ if and only if $\lambda \ge g(8,3)$.