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We put forward new general criteria to design successor rules that generatebinary de Bruijn sequences. Prior fast algorithms based on successor rules inthe literature are then shown to be special instances. We implemented thecriteria to join the cycles generated by a number of simple feedback shiftregisters (FSRs) of order $n$. These include the pure cycling register (PCR)and the pure summing register (PSR). For the PCR, we define a transitiverelation on its cycles, based on their weights. We also extend the choices ofconjugate states by using shift operations. For the PSR, we define threedistinct transitive relations on its cycles, namely a run order, a necklaceorder, and a mixed order. Using the new orders, we propose numerous classes ofsuccessor rules. Each class efficiently generates a number, exponential in $n$,of binary de Bruijn sequences. Producing the next bit in each such sequencetakes $O(n)$ memory and $O(n)$ time. We implemented computational routines toconfirm the claims.

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