Constructing de Bruijn sequences by concatenating smaller universal cycles

TL;DR

This paper establishes conditions for concatenating universal cycles to construct de Bruijn sequences, generalizing existing methods and introducing three new constructions for these sequences.

Contribution

It provides a unified framework for concatenating universal cycles to generate de Bruijn sequences, extending prior methods and proposing three novel constructions.

Findings

Derived sufficient conditions for concatenating universal cycles.

Generalized two known de Bruijn sequence constructions.

Developed three new de Bruijn sequence constructions.

Abstract

We present sufficient conditions for when an ordering of universal cycles $\alpha_1, \alpha_2, \ldots, \alpha_m$ for disjoint sets $\mathbf{S}_1, \mathbf{S}_2, \ldots , \mathbf{S}_m$ can be concatenated together to obtain a universal cycle for $\mathbf{S} = \mathbf{S}_1 \cup \mathbf{S}_2 \cup \cdots \cup \mathbf{S}_m$. When $\mathbf{S}$ is the set of all $k$-ary strings of length $n$, the result of such a successful construction is a de Bruijn sequence. Our conditions are applied to generalize two previously known de Bruijn sequence constructions and then they are applied to develop three new de Bruijn sequence constructions.