Nonbinary Counterparts of the Prefer-Same and Prefer-Opposite de Bruijn Sequences

TL;DR

This paper generalizes binary prefer-same and prefer-opposite de Bruijn sequences to q-ary sequences using preference functions, exploring their properties, relationships, and discrepancy profiles.

Contribution

It introduces q-ary sequences based on preference functions, extending known binary sequences and analyzing their fundamental properties and relationships.

Findings

Sequences share properties with binary versions

Prefer-higher sequence derived from homomorphic image

Discrepancy profiles similar to binary case

Abstract

The well known prefer-one, prefer-opposite, and prefer-same binary de Bruijn sequences are all constructed using simple preference rules. We apply the technique of preference functions of span one to define q-ary sequences that generalize the prefer-opposite and prefer-same sequences and we present some of their basic properties that are shared with their binary versions. In particular, we show that the prefer-higher sequence (the nonbinary counter-part of the prefer-one sequence) is obtained from a homomorphic image of the proposed prefer-opposite, when repetitions are cleaned up. This mirrors a known relationship between the binary versions. We also perform calculations that demonstrate that the discrepancy profile of the proposed sequences is similar to that of the binary case.