Cycle Structure and Adjacency Graphs of a Class of LFSRs and a New Family of De Bruijn Cycles

Feedback shift registers can be applied to the fields of communications, stream ciphers, computers, and design theory. The linear feedback shift registers are often used in the construction of De Bruijn sequences. For any given linear shift register, its cycle structure and adjacency graphs are features that must be investigated in the construction of the De Bruijn sequences by using the cycle-joining method. A class of linear feedback shift registers is discussed in this paper. The cycle structure of some linear feedback shift registers is derived. And the adjacency graphs are divided into two categories to analyze their structure in detail. Based on this kind of linear feedback shift registers combined with the cycle-joining method, a novel family of De Bruijn cycles is obtained. The number of the corresponding De Bruijn cycles produced is also proposed exactly.