Locating Patterns in the De Bruijn Torus

TL;DR

This paper advances the construction of de Bruijn torus patterns by introducing new cross-shaped and universal cycle grid methods, enhancing applications in robotics and display technologies.

Contribution

It extends de Bruijn torus constructions beyond cross-shape patterns using half de Bruijn sequences and explores universal cycle grids for practical applications.

Findings

Development of new cross-shaped pattern constructions

Introduction of universal cycle grid results

Potential applications in robotics and displays

Abstract

The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binary matrix in which every possible $j$-by-$k$ submatrix appears exactly once. The existence and construction of these binary matrices was determined in the 70's, with generalizations to $d$-ary matrices in the 80's and 90's. However, these constructions lacked efficient decoding methods, leading to new constructions in the early 2000's. The new constructions develop cross-shaped patterns (rather than rectangular), and rely on a concept known as a half de Bruijn sequence. In this paper, we further advance this construction beyond cross-shape patterns. Furthermore, we show results for universal cycle grids, based off of the one-dimensional universal cycles introduced by Chung, Diaconis, and Graham, in the 90's. These grids have many applications such as robotic vision, location detection, and projective touch-screen displays.