Spans of Preference Functions for De Bruijn Sequences

TL;DR

This paper characterizes preference functions that generate de Bruijn sequences and shows that such functions can generate sequences of all higher orders, introducing the concept of preference function complexity.

Contribution

It provides a characterization of preference functions for de Bruijn sequences and introduces the concept of preference function complexity.

Findings

Preference functions that generate de Bruijn sequences are characterized.

Any such preference function also generates sequences of higher order.

Introduces the concept of preference function complexity for de Bruijn sequences.

Abstract

A nonbinary Ford sequence is a de Bruijn sequence generated by simple rules that determine the priorities of what symbols are to be tried first, given an initial word of size $n$ which is the order of the sequence being generated. This set of rules is generalized by the concept of a preference function of span $n-1$, which gives the priorities of what symbols to appear after a substring of size $n-1$ is encountered. In this paper we characterize preference functions that generate full de Bruijn sequences. More significantly, We establish that any preference function that generates a de Bruijn sequence of order $n$ also generates de Bruijn sequences of all orders higher than $n$, thus making the Ford sequence no special case. Consequently, we define the preference function complexity of a de Bruijn sequence to be the least possible span of a preference function that generates this de Bruijn sequence.