Large Order Binary de Bruijn Sequences via Zech's Logarithms

This work shows how to efficiently construct binary de Bruijn sequences, even those with large orders, using the cycle joining method. The cycles are generated by an LFSR with a chosen period e whose irreducible characteristic polynomial can be derived from any primitive polynomial of degree n satisfying e = 2 n −1 t by t-decimation. The crux is our proof that determining Zech’s logarithms is equivalent to identifying conjugate pairs shared by any pair of cycles. The approach quickly finds enough number of conjugate pairs between any two cycles to ensure the existence of trees containing all vertices in the adjacency graph of the LFSR. When the characteristic polynomial f(x) is a product of distinct irreducible polynomials, we combine the approach via Zech’s logarithms and a recently proposed method to determine the conjugate pairs. This allows us to efficiently generate de Bruijn sequences with larger orders. Along the way, we establish new properties of Zech’s logarithms.