Excess Coverage Arrays and Levenshtein's Conjecture

TL;DR

This paper investigates the existence and limitations of sequence covering arrays related to Levenshtein's conjecture, establishing bounds and uniqueness results for certain array configurations through combinatorial analysis and computational methods.

Contribution

The authors prove that an extsf{SCA}$(7!;7,v)$ cannot have $v > 9$, analyze the uniqueness of specific excess coverage arrays, and determine maximum columns for arrays with small symbol sets.

Findings

An extsf{SCA}$(7!;7,v)$ exists only if $v \\leq 9$

Unique extsf{CA}$_{X}(42;2,5,6)$ array identified

Maximum columns for small symbol sets determined

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.