On de Bruijn Rings and Families of Almost Perfect Maps

TL;DR

This paper introduces a novel construction method for generating almost perfect de Bruijn maps, called de Bruijn rings, which are suitable for positional coding and work for arbitrary pattern shapes and non-prime alphabet sizes.

Contribution

The paper presents a new construction technique for almost perfect maps using de Bruijn rings, expanding applicability to arbitrary pattern shapes and non-prime alphabet sizes.

Findings

Successfully generates almost perfect maps for various pattern shapes.

Maps are easily decodable for positional coding.

Method covers cases previously unresolved, such as certain square tori with odd dimensions.

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 element {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 element {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.