Constructive bijection between binary de Bruijn words and equivalent algebraic objects

Construct an explicit and efficient bijection between binary de Bruijn words of length 2^{d+1}, binary necklaces of length 2^d with an odd number of 1s, normal bases of the finite field GF(2^{2^d}), and binary polynomials of degree less than 2^d that are coprime with X^{2^d}−1, in order to enable uniform generation of binary de Bruijn words.

Background

The paper establishes a bijection among four classes of objects in the special binary case: binary de Bruijn words of length 2{d+1}; necklaces of length 2d having an odd number of 1s; normal bases of GF(2{2d}); and binary polynomials of degree less than 2d coprime with X{2d}−1.

The unresolved issue is not the existence of the abstract correspondence, but the construction of a constructive bijection that would permit effective interchange among these representations. The authors note that such a bijection could be used to generate every binary de Bruijn word with uniform probability; an efficient positive-probability generation algorithm was known, but the uniform constructive correspondence remained open.

References

One could for example use such a bijection to generate every binary de Bruijn word with uniform probability, a problem that is still open --- in the authors presented an efficient algorithm to generate every binary de Bruijn word with positive probability.

— Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform  (2502.12844 - Fici et al., 18 Feb 2025) in Remark following the final theorem in Section 5, “Generalized de Bruijn words and invertible necklaces”