New Successor Rules to Efficiently Produce Exponentially Many Binary de Bruijn Sequences

Abstract

We put forward new general criteria to design successor rules that generate binary de Bruijn sequences. Prior fast algorithms based on successor rules in the literature are then shown to be special instances. We implement the criteria to join the cycles generated by a number of simple feedback shift registers (FSRs). These include the pure cycling register (PCR) and the pure summing register (PSR). For PCR, we define a preorder on its cycles, based on their weights. For PSR, we define two orders, namely a necklace order on its cycles and a mixed order on the cycles based on both the weight and the necklace orders. Using the new orders, we propose numerous classes of successor rules that can efficiently generate binary de Bruijn sequences. Producing the next bit takes no more than O(n) memory and between O(n) and O(n log n) time. We implemented computational routines to confirm the claims.