On de Bruijn Covering Sequences and Arrays

TL;DR

This paper introduces new bounds and constructions for de Bruijn covering arrays and sequences, utilizing probabilistic methods, coding theory, and innovative folding techniques to improve their size and efficiency.

Contribution

It provides novel probabilistic bounds, a folding construction method, and multiple new constructions for shorter de Bruijn covering sequences and arrays based on coding theory and sequence manipulation.

Findings

Upper bounds on array area using probabilistic techniques

New folding construction method for de Bruijn covering arrays

Several constructions yielding shorter sequences and arrays

Abstract

An $(m,n,R)$-de Bruijn covering array (dBCA) is a doubly periodic $M \times N$ array over an alphabet of size $q$ such that the set of all its $m \times n$ windows form a covering code with radius $R$. An upper bound of the smallest array area of an $(m,n,R)$-dBCA is provided using a probabilistic technique which is similar to the one that was used for an upper bound on the length of a de Bruijn covering sequence. A folding technique to construct a dBCA from a de Bruijn covering sequence or de Bruijn covering sequences code is presented. Several new constructions that yield shorter de Bruijn covering sequences and $(m,n,R)$-dBCAs with smaller areas are also provided. These constructions are mainly based on sequences derived from cyclic codes, self-dual sequences, primitive polynomials, an interleaving technique, folding, and mutual shifts of sequences with the same covering radius. Finally, constructions of de Bruijn covering sequences codes are also discussed.