An efficiently generated family of binary de Bruijn sequences

Abstract

We study how to generate binary de Bruijn sequences efficiently from the class of simple linear feedback shift registers with feedback function f (x0,x1, ...xn-1) = x0 +x1 +xn-1 for n >_ 3, using the cycle joining method. Based on the properties of this class of LFSRs, we propose two new generic successor rules, each of which produces at least 2n-3 de Bruijn sequences. These two classes build upon a framework proposed by Gabric, Sawada,Williams andWong in Discrete Mathematics vol. 341, no. 11, pp. 2977–2987, November 2018. Here we introduce new useful choices for the uniquely determined state in each cycle to devise valid successor rules. In each class, the next bit costs O(n) time and O(n) space for a fixed n.