Universal partial tori

TL;DR

This paper introduces universal partial tori and matrices, extending De Bruijn cycles to two dimensions with wildcard characters, and constructs infinitely many using novel one-dimensional variants.

Contribution

It defines universal partial tori and matrices, and develops new methods to construct infinitely many, advancing the theory of universal cycles with wildcards.

Findings

Constructed infinitely many universal partial tori and matrices.

Developed a new variant called universal partial family.

Extended the concept of universal cycles to two-dimensional structures.

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.