Construction of de Bruijn Sequences from Product of Two Irreducible Polynomials

TL;DR

This paper presents a novel method for constructing de Bruijn sequences using the cycle structure of LFSRs with characteristic polynomials that are products of two irreducible polynomials, including algorithms for cycle analysis and sequence enumeration.

Contribution

It introduces a new approach to construct de Bruijn sequences from LFSRs with composite characteristic polynomials, including cycle analysis and conjugate pair algorithms.

Findings

Derived cycle structure and adjacency graph properties for LFSRs with composite polynomials.

Provided algorithms to identify states in each cycle and find conjugate pairs.

Estimated and, in some cases, exactly counted the number of resulting de Bruijn sequences.

Abstract

We study a class of Linear Feedback Shift Registers (LFSRs) with characteristic polynomial $f(x)=p(x)q(x)$ where $p(x)$ and $q(x)$ are distinct irreducible polynomials in $\F_2[x]$. Important properties of the LFSRs, such as the cycle structure and the adjacency graph, are derived. A method to determine a state belonging to each cycle and a generic algorithm to find all conjugate pairs shared by any pair of cycles are given. The process explicitly determines the edges and their labels in the adjacency graph. The results are then combined with the cycle joining method to efficiently construct a new class of de Bruijn sequences. An estimate of the number of resulting sequences is given. In some cases, using cyclotomic numbers, we can determine the number exactly.