Spans of Preference Functions for De Bruijn Sequences

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.