On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials

TL;DR

This paper introduces a new method for constructing de Bruijn sequences using cycle joining from LFSRs with any characteristic polynomial, providing an efficient algorithm and unifying many previous approaches.

Contribution

It presents a novel cycle joining construction for de Bruijn sequences from LFSRs with arbitrary characteristic polynomials, along with an efficient algorithm for cycle analysis.

Findings

Efficient algorithm to find all conjugate pairs between cycles.

Practical generation of de Bruijn sequences up to order 20.

Many existing constructions are special cases of this method.

Abstract

We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial $f(x)$. We study in detail the cycle structure of the set $\Omega(f(x))$ that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order $n \approx 20$. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.