A polynomial construction of perfect sequence covering arrays

TL;DR

This paper introduces a new explicit construction method for perfect sequence covering arrays (PSCAs) using projective space representations, establishing bounds on the minimal repetition number for fixed parameters.

Contribution

It provides an explicit construction proving bounds on the minimal repetition number for PSCAs using projective geometry, a novel approach in the field.

Findings

Established that g(v,t) = O(v^{t(t-2)}) for fixed t ≥ 4.

Constructed PSCAs using permutation representations of projective space groups.

Showed coverage of most 4-sequences in Desarguesian projective planes.

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 $\lambda$ permutations. For $v \geq t$, let $g(v, t)$ be the smallest positive integer $\lambda$ such that a PSCA$(v, t, \lambda)$ exists. We present an explicit construction that proves $g(v,t) = O(v^{t(t-2)})$ for fixed $t \geq 4$. The method of construction involves taking a permutation representation of the group of projectivities of a suitable projective space of dimension $t - 2$ and deleting all but a certain number of symbols from each permutation. In the case that this space is a Desarguesian projective plane, we also show that there exists a permutation representation of the group of projectivities of the plane that covers the vast majority of 4-sequences of its points a fixed number of times.