The Cycle Structure of LFSR with Arbitrary Characteristic Polynomial over Finite Fields

We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.Comment: An extended abstract containing preliminary results was presented at SETA 201

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 $\Omega(f(x))$ that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order $n \approx 20$. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction