On de Bruijn Arrays Codes, Part I: Nonlinear Codes

TL;DR

This work defines de Bruijn array codes (DBACs) as sets of $r \times s$ binary doubly-periodic arrays in which every $n \times m$ window appears exactly once, extending de Bruijn concepts to two dimensions. It develops a two-dimensional shift-register framework, including a 2D de Bruijn graph and the $D$-morphism, to construct DBACs directly and recursively from perfect factors. The paper proves necessary existence conditions via PFs, provides explicit $(2^k,2^t;n,m)$-DBAC constructions with sizes like $2^{nm-k-t}$, and introduces recursive and joining techniques (merge-or-split) to obtain broader parameter families, while identifying open questions about sufficiency and parity constraints. Collectively, these results advance 2D sequence-coding theory with potential applications in pattern recognition and sensing, and lay groundwork for future linear pseudo-random array codes.

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.

Figures (2)

• Figure 1: Feedback shift register of order $n$.
• Figure 2: The merge-or-split method.

Theorems & Definitions (40)

• Definition 1
• Definition 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 1
• Conjecture 1