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 implemented the criteria to join the cycles generated by a number of simple feedback shift registers (FSRs) of order $n$. These include the pure cycling register (PCR) and the pure summing register (PSR). For the PCR, we define a transitive relation on its cycles, based on their weights. We also extend the choices of conjugate states by using shift operations. For the PSR, we define three distinct transitive relations on its cycles, namely a run order, a necklace order, and a mixed order. Using the new orders, we propose numerous classes of successor rules. Each class efficiently generates a number, exponential in $n$, of binary de Bruijn sequences. Producing the next bit in each such sequence takes $O(n)$ memory and $O(n)$ time. We implemented computational routines to confirm the claims.