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

TL;DR

This paper introduces a novel method to determine the minimal polynomials of binary modified de Bruijn sequences by translating the problem into finding Hamiltonian cycles in a specialized graph, utilizing cycle joining techniques.

Contribution

It presents a new general approach based on graph theory and cycle joining methods to compute minimal polynomials of modified de Bruijn sequences.

Findings

Successfully characterizes minimal polynomials using Hamiltonian cycles

Demonstrates the effectiveness of cycle joining methods in the modified context

Provides a computational framework for analyzing binary modified de Bruijn sequences

Abstract

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 on 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. Along the way, we demonstrate the usefullness of computational tools from the cycle joining method in the modified setup.