Covering Arrays for Equivalence Classes of Words

TL;DR

This paper investigates covering arrays for words over a finite alphabet, considering equivalence classes of words based on induced partitions or weight, and provides logarithmic bounds on their minimal sizes for various parameters.

Contribution

It introduces new bounds for covering arrays under word equivalence relations, extending classical results to more general and practical scenarios.

Findings

Logarithmic upper bounds on covering array sizes for equivalence classes

Results for specific cases t=2,3,4 and general t

Comparison of two equivalence schemes for words

Abstract

Covering arrays for words of length $t$ over a $d$ letter alphabet are $k \times n$ arrays with entries from the alphabet so that for each choice of $t$ columns, each of the $d^t$ $t$-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case words are equivalent if they induce the same partition of a $t$ element set. In the second case, words of the same weight are equivalent. In both cases we produce logarithmic upper bounds on the minimum size $k=k(n)$ of a covering array. Definitive results for $t=2,3,4$, as well as general results, are provided.