Excess Coverage Arrays and Levenshtein's Conjecture

Abstract

A sequence covering array, denoted \textsf{SCA}$(N;t,v)$, is a set of $N$ permutations of $\{0, \dots, v-1 \}$ such that each sequence of $t$ distinct elements of $\{0, \dots, v-1\}$ reads left to right in at least one permutation. The minimum number of permutations such a sequence covering array can have is $t!$ and Levenshtein conjectured that if a sequence covering array with $t!$ permutations exists, then $v \in \{t,t+1\}$. In this paper, we prove that if an \textsf{SCA}$(7!;7,v)$ exists, then $v \leq 9$. We do this by analysing connections between sequence covering arrays and a special kind of covering array called an excess coverage array. A strength 2 excess coverage array, denoted \textsf{CA}$_{X}(N;2,k,v)$, is an $N \times k$ array with entries from $\{0, \dots, v - 1\}$ such that every ordered pair of distinct symbols appear at least once in each pair of columns and all other pairs appear at least twice. We demonstrate computationally that there is a unique \textsf{CA}$_{X}(42;2,5,6)$, and we prove that this array alone does not satisfy necessary conditions we establish for the existence of an \textsf{SCA}$(7!;7,10)$. Furthermore, we find the maximum possible number of columns for strength 2 excess coverage arrays with symbol sets of sizes between 2 and 6, and more broadly investigate binary excess coverage arrays.