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 nn 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−1n−1, which gives the priorities of what symbols to appear after a substring of size n−1n−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 nn also generates de Bruijn sequences of all orders higher than nn, 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.