The Adjacency Graphs of LFSRs With Primitive-Like Characteristic Polynomials

Abstract

We consider the adjacency graphs of the linear feedback shift registers (LFSRs) with characteristic polynomials of the form $l(x)p(x)$ , where $l(x)$ is a polynomial of small degree and $p(x)$ is a primitive polynomial. It is shown that their adjacency graphs are closely related to the association graph of $l(x)$ and the cyclotomic numbers over finite fields. By using this connection, we give a unified method to determine their adjacency graphs. As an application of the method, we explicitly calculate the adjacency graphs of LFSRs with the characteristic polynomials of the form $(1+x+x^{3}+x^{4})p(x)$ , and construct a large class of De Bruijn sequences from them.