On de Bruijn rings and families of almost perfect maps

Abstract

De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape $(m,n)$ appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd $m=n\in \{3,5,7\}$ and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd $m=n\in \{3,5,7\}$ as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications.