On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials

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 Ω(f(x)) that contains all sequences produced by a specific LFSR on distinct inputs and provide an efficient way to find a state of each cycle. Our structural results lead to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach provides a practical method to generate a large class of de Bruijn sequences. Many recently-proposed constructions of de Bruijn sequences are shown to be special cases of our construction.