On Perfect Sequence Covering Arrays

TL;DR

This paper investigates perfect sequence covering arrays (PSCAs), determining exact values of the minimal repetition number for certain parameters, improving bounds for many cases, and exploring structural restrictions on symbol distributions.

Contribution

It establishes exact values of g(v, t) for specific parameters and improves upper bounds for many (v, t) pairs using group permutation representations.

Findings

g(6, 3) = 2

g(7, 3) = 2

g(7, 4) = 2","g(8, 3) = 3"],[

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.