Preliminary results on the minimal polynomial of modified de Bruijn sequences

Modified de Bruijn sequences are created by removing a single zero from the longest run of zeros of de Bruijn sequences. There are few theoretical results on the minimal polynomial and linear complexity of modified de Bruijn sequences. Some preliminary results are presented in this paper. It shows that for the minimal polynomial of a modified de Bruijn sequence of order n there exists at least one irreducible factor of degree n. An equivalent condition on which a polynomial is the minimal polynomial of some modified de Bruijn sequence is derived, using the tool of rational fraction representation of periodic sequences. Based on the equivalent condition, the impossible linear complexity of modified de Bruijn sequences can be discussed.