Designing preference functions for de Bruijn sequences with forbidden words

Abstract

A preference function provides a method to build periodic sequences by specifying a set of rules that determine which symbols are to be attempted before others, when the sequence is constructed one symbol at a time. The well-known prefer-one, prefer-opposite, and prefer-same binary de Bruijn sequences are all constructed using appropriate preference functions. In this article we provide some fairly general results that give conditions for a pair of an initial word and a preference function on a q-ary alphabet to produce sequences that include every pattern of given size n≥ 1 –except possibly some specified set of patterns. We provide several old and new constructions that showcase the flexibility of the results. Specifically, we give a construction for square-free and general separative de Bruijn sequences. The existence of these sequences was established more than a decade ago but nonconstructively. An important special case of these separative sequences produces universal cycles for permutations. We also build a preference function for binary de Bruijn sequences of patterns with a maximum density of ones. As for full de Bruijn sequences, the main result helps furnish a recursive construction from arbitrary cyclic permutations of q symbols. Finally, we build a preference function that extends a full de Bruijn sequence of order n into one of order n+ 1. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.