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arXiv · 2404.13674

On de Bruijn Covering Sequences and Arrays

Yeow Meng Chee·Tuvi Etzion·Hoang Ta·Van Khu Vu
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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.

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Yeow Meng Chee, Tuvi Etzion, Hoang Ta, Van Khu Vu. 2024-04-21. On de Bruijn Covering Sequences and Arrays. https://arxiv.org/abs/2404.13674

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