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

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.