Constructing de Bruijn Sequences Based on a New Necessary Condition

Abstract

In this paper, we propose a new necessary condition for feedback functions of de Bruijn sequences and discuss its application in constructing de Bruijn sequences. It is shown that a large number of de Bruijn sequences could be easily constructed by precomputing an [Formula: see text]-stage nonlinear feedback shift register (NFSR) with a special cycle structure—that is, if a state [Formula: see text] is on a cycle generated by this NFSR, then all the states with the same Hamming weight as [Formula: see text] are also on this cycle. Moreover, if there are [Formula: see text] different cycles in the state graph of the precomputed NFSR, then we can construct [Formula: see text] de Bruijn sequences by the different choices of conjugate state pairs, where [Formula: see text].