A group-based structure for perfect sequence covering arrays

TL;DR

This paper investigates the minimal multiplicity for perfect sequence covering arrays, introduces a group-based structure to determine new values of g(n,k), and establishes existence conditions for these arrays.

Contribution

It extends the understanding of perfect sequence covering arrays by determining new minimal multiplicities using recursive and group-based methods.

Findings

g(6,3)=2 and g(7,3)=2

g(7,4)=2, g(7,5) in {2,3,4}

g(8,3) in {2,3} and g(9,3) in {2,3,4}

Abstract

An $(n,k)$-perfect sequence covering array with multiplicity $\lambda$, denoted 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 PSCA$(n,k,\lambda)$ exists; and more generally, the complete set of values $\lambda$ for which a 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 PSCA$(n,k,\lambda)$ if and only if $\lambda \ge 2$; and that there exists a PSCA$(8,3,\lambda)$ if and only if $\lambda \ge g(8,3)$.