Partial Cycle Structure of FSRs and Its Applications in Searching De Bruijn Sequences

We propose the concept of partial cycle structure of feedback shift registers, and study its applications in searching the characteristic functions of de Bruijn sequences. We show that, if a function generates de Bruijn sequences then its partial cycle structure does not contain cycles, and conversely, if the partial cycle structure of a function does not contain cycles then it can be extended into a function that generates de Bruijn sequences. By using this property, we analyze the low degree terms in the characteristic functions of de Bruijn sequences, and in particular give a full description of the linear terms in them. We also design an algorithm to search the characteristic functions of de Bruijn sequences which should perform better than the random search algorithm.