Pseduo-Random and de Bruijn Array Codes
Tuvi Etzion
TL;DR
The paper addresses extending one-dimensional sequence/window properties to two-dimensional binary arrays by defining pseudo-random array codes (PRAC) and de Bruijn array codes (DBAC) as two-dimensional analogs of zero and perfect factors in the de Bruijn framework. It develops folding-based constructions of PRACs from M-sequences and from irreducible polynomials (and their products) and analyzes minimum distance via the smallest-weight sequence, providing concrete examples. It then presents direct and recursive constructions for DBACs through concatenation of PF sequences and derivative morphisms, linking DBAC existence to PF parameters and offering weight-variant and recursive expansion techniques, with several open questions for future research. Together, these results contribute systematic 2D array code designs with precise window-coverage properties and potential applications in coding theory and array-based signaling.
Abstract
Pseudo-random arrays and perfect maps are the two-dimensional analogs of M-sequences and de Bruijn sequences, respectively. We modify the definitions to be applied to codes. These codes are also the two-dimensional analogs of certain factors in the de Bruijn graph. These factors are called zero factors and perfect factors in the de Bruijn graph. We apply a folding technique to construct pseudo-random array codes and examine the minimum distance of the constructed codes. The folding is applied on sequences generated from irreducible polynomials or a product of irreducible polynomials with the same degree and the same exponent. Direct and recursive constructions for de Bruijn array codes are presented and discussed.