A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. The graph is isomorphic to the modified de Bruijn–Good graph. Along the way, we demonstrate the usefulness of some computational tools from the cycle joining method in the modified setup