Universal Partial Words over Non-Binary Alphabets

TL;DR

This paper investigates universal partial words over non-binary alphabets, establishing their structural properties, number-theoretic existence conditions, and providing explicit constructions for certain alphabet sizes.

Contribution

It proves structural and existence results for universal partial words over non-binary alphabets, including periodicity, cyclicity, and explicit constructions for even-sized alphabets.

Findings

Universal partial words have periodic $ullet$ structure.

They are cyclic over non-binary alphabets.

Explicit constructions exist for even-sized alphabets.

Abstract

Chen, Kitaev, M\"{u}tze, and Sun recently introduced the notion of universal partial words, a generalization of universal words and de Bruijn sequences. Universal partial words allow for a wild-card character $\diamond$, which is a placeholder for any letter in the alphabet. We settle and strengthen conjectures posed in the same paper where this notion was introduced. For non-binary alphabets, we show that universal partial words have periodic $\diamond$ structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for a family of universal partial words over alphabets of even size.