The Cross-Joint Method and Generation of De Bruijn Sequences

This manuscript studies the construction of nonlinear feedback shift registers that generate binary de Bruijn sequences by means of the cross-join pairs method. We explicitly state the original contribution of our study as follows: We combine the classical theorem of Mykkeltveit and Szmidt with constructive algorithms, implementation details for small orders, and an interpretation in terms of Fryers’ polynomials. We also clarify the relation to earlier methods, discuss computational complexity and practical limitations, formalize the definition of the weighted adjacency matrix, and expand the conclusion with applications and directions for future work. Numerical results for small orders demonstrate the feasibility of the approach.