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On de Bruijn Rings and Families of Almost Perfect Maps

Peer Stelldinger

TL;DR

The paper tackles constructing two-dimensional de Bruijn-like codes that cover local patterns with near-complete efficiency, addressing unresolved existence for certain square tori. It introduces de Bruijn rings as minimal-height sub-perfect maps and formalizes families of almost perfect maps, proving existence for arbitrary $k$, $m$, and $n$ and enabling scalable composition into larger maps. The approach yields decodable, near-complete pattern coverage with linear-time construction and decoding, making it practical for positional coding and optical localization. The work demonstrates substantial pattern coverage gains, provides explicit constructions (including square shapes and prime-alphabet considerations), and outlines pathways to higher dimensions and rotated-pattern variants.

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.

On de Bruijn Rings and Families of Almost Perfect Maps

TL;DR

The paper tackles constructing two-dimensional de Bruijn-like codes that cover local patterns with near-complete efficiency, addressing unresolved existence for certain square tori. It introduces de Bruijn rings as minimal-height sub-perfect maps and formalizes families of almost perfect maps, proving existence for arbitrary , , and and enabling scalable composition into larger maps. The approach yields decodable, near-complete pattern coverage with linear-time construction and decoding, making it practical for positional coding and optical localization. The work demonstrates substantial pattern coverage gains, provides explicit constructions (including square shapes and prime-alphabet considerations), and outlines pathways to higher dimensions and rotated-pattern variants.

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.
Paper Structure (8 sections, 12 theorems, 33 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 8 sections, 12 theorems, 33 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic Definitions and Related Work for the 1D Case
  3. Basic Definitions and Related Work for the 2D Case
  4. De Bruijn Rings
  5. From Rings to larger Sub-Perfect Maps
  6. Application for Optical Localization
  7. Conclusion and Future Work
  8. Additional Tables and Figures

Key Result

lemma 1

Let $k,m\geq2$ be natural numbers. Then $\mathrm{M}(k,m)$ is smaller than $\frac{1}{m} k^m$ and larger than $\frac{1}{m}\left(k^m-k^{\lfloor m/2\rfloor+1}\right)$.

Figures (7)

  • Figure 1: The $(2,2)_2$-ring graph.
  • Figure 2: The $(3,2)_2$-ring graph.
  • Figure 3: Size comparison of square shaped sub-perfect maps according to theorem \ref{['thm:sqrRingsPrime']} and (not guaranteed to exist) de Bruijn tori of the same type. When $N$ is the side length of a sub-perfect map, $\tilde{N}$ denotes the side length of the corresponding de Bruijn torus. The plot shows the percentage of $N^2/\tilde{N}^2$, i.e. the ratio of covered to all $(n,n)_{k^2}$-patterns for different values of $k$.
  • Figure 4: The $(2,2)_3$-ring graph. If no label is given at an edge, the label is equal to the target node label. A bidirectional edge represents two edges in opposing directions (with not necessarily equal labels).
  • Figure 5: The $(2,3)_2$-ring graph. If no label is given at an edge, the label follows uniquely from the rules given in definition \ref{['def:ringgraph']}.
  • ...and 2 more figures

Theorems & Definitions (25)

  • lemma 1
  • proof
  • corollary 1
  • definition 1: De Bruijn ring
  • theorem 1
  • definition 2: Ring graph
  • lemma 2
  • proof
  • lemma 3
  • proof
  • ...and 15 more