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Universal partial tori

William D. Carey, Matthew David Kearney, Rachel Kirsch, Stefan Popescu

TL;DR

This paper develops two-dimensional generalizations of De Bruijn structures by introducing uptorus and upmatrix, incorporating a wildcard symbol to compress coverage. It provides constructive frameworks that yield infinitely many uptori and upmatrices, based on one-dimensional universal cycles (including a new universal partial family variant) and on two-dimensional analogues built from upcycles. The authors supply computationally identified small instances, prove key structural lemmas about bottom rows and rotations, and introduce lifting techniques via perfect necklaces to create uptori with variable diamondicity. The work surveys existing existence results, extends them to new objects, and proposes open problems and directions toward higher-dimensional analogues.

Abstract

A De Bruijn cycle is a cyclic sequence in which every word of length $n$ over an alphabet $\mathcal{A}$ appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words, which shorten De Bruijn cycles using a wildcard character, we introduce universal partial tori and matrices. We find them computationally and construct infinitely many of them using one-dimensional variants of universal cycles, including a new variant called a universal partial family.

Universal partial tori

TL;DR

This paper develops two-dimensional generalizations of De Bruijn structures by introducing uptorus and upmatrix, incorporating a wildcard symbol to compress coverage. It provides constructive frameworks that yield infinitely many uptori and upmatrices, based on one-dimensional universal cycles (including a new universal partial family variant) and on two-dimensional analogues built from upcycles. The authors supply computationally identified small instances, prove key structural lemmas about bottom rows and rotations, and introduce lifting techniques via perfect necklaces to create uptori with variable diamondicity. The work surveys existing existence results, extends them to new objects, and proposes open problems and directions toward higher-dimensional analogues.

Abstract

A De Bruijn cycle is a cyclic sequence in which every word of length over an alphabet appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words, which shorten De Bruijn cycles using a wildcard character, we introduce universal partial tori and matrices. We find them computationally and construct infinitely many of them using one-dimensional variants of universal cycles, including a new variant called a universal partial family.
Paper Structure (11 sections, 15 theorems, 32 equations, 1 figure)

This paper contains 11 sections, 15 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computationally Identified Upmatrices
  4. Upmatrices Constructed from Upwords
  5. Computationally Identified Uptori
  6. Uptori Constructed from Upcycles
  7. Uptori Constructed from Universal Partial Families
  8. Universal Partial Quasi-Families
  9. Uptori Constructed from Lifted Upfamilies
  10. Open Problems
  11. Acknowledgments

Key Result

Lemma 2.5

Every nontrivial uptorus for $\mathcal{A}^{w \times \ell}$ has at least $w+1$ rows and at least $\ell+1$ columns.

Figures (1)

  • Figure 1: The transpose of matrix $m(u,s)$

Theorems & Definitions (54)

  • Definition 2.1: Subarray
  • Example 2.2
  • Definition 2.3: Universal Partial Matrix
  • Definition 2.4: Universal Partial Torus
  • Lemma 2.5
  • proof
  • Definition 2.6: Alternating Word
  • Definition 2.7: Alternating De Bruijn Cycle
  • Definition 2.8: Unrolled Alternating De Bruijn Cycle
  • Remark 2.9
  • ...and 44 more