A polynomial construction of 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 $\lambda$ 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.