Paper 2014/686

The Adjacency Graph of Some LFSRs

Ming Li and Dongdai Lin

Abstract

In this paper, we discuss the adjacency graph of feedback shift registers (FSRs) whose characteristic polynomial can be written as $g=(x_0+x_1)*f$ for some linear function $f$. For $f$ contains an odd number of terms, we present a method to calculate the adjacency graph of FSR$_{(x_0+x_1)*f}$ from the adjacency graph of FSR$_f$. The parity of the weight of cycles in FSR$_{(x_0+x_1)*f}$ can also be determined easily. For $f$ contains an even number of terms, the theory is not so much complete. We need more information than the adjacency graph of FSR$_f$ to determine the adjacency graph of FSR$_{(x_0+x_1)*f}$. Besides, some properties about the cycle structure of linear feedback shift registers (LFSR) are presented.