On de Bruijn Arrays Codes, Part I: Nonlinear Codes

TL;DR

This paper explores the existence, construction, and theoretical framework of de Bruijn array codes, which are two-dimensional binary arrays that contain every possible submatrix exactly once, extending the concept of de Bruijn sequences.

Contribution

It introduces necessary conditions, provides multiple constructions, and proposes a theoretical framework for two-dimensional feedback shift registers for de Bruijn array codes.

Findings

Necessary conditions for existence are identified.

Several direct and recursive constructions are presented.

A new framework for two-dimensional feedback shift registers is proposed.

Abstract

A de Bruijn array code is a set of $r \times s$ binary doubly-periodic arrays such that each binary $n \times m$ matrix is contained exactly once as a window in one of the arrays. Such a set of arrays can be viewed as a two-dimensional generalization of a perfect factor in the de Bruijn graph. Necessary conditions for the existence of such codes are given. Several direct constructions and recursive constructions for such arrays are given. A framework for a theory of two-dimensional feedback shift registers which is akin to (one-dimensional) feedback shift registers is suggested in the process.