On de Bruijn Covering Sequences and Arrays
Yeow Meng Chee, Tuvi Etzion, Hoang Ta, Van Khu Vu · 2024
An$(m, n, R)-\mathbf{de}$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$(m, n, R)-\mathbf{dBCA}$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$(m, n, R)-\mathbf{dBCAs}$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.