Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform

TL;DR

This paper explores the connections between generalized de Bruijn words, invertible necklaces, and the Burrows-Wheeler transform, revealing their algebraic structures and relationships with finite fields and graph theory.

Contribution

It introduces generalized de Bruijn words and invertible necklaces, establishing their algebraic and combinatorial properties and linking them to finite field bases and graph cycles.

Findings

Generalized de Bruijn words correspond to Hamiltonian cycles in de Bruijn graphs.

Invertible necklaces relate to normal bases of finite fields and form an Abelian group.

A correspondence exists between binary de Bruijn words, necklaces, invertible BWT matrices, and finite field bases.

Abstract

We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles in the generalized de Bruijn graphs introduced in the early '80s in the context of network design. When the size of the alphabet is a prime $p$, we define invertible necklaces as those whose BWT-matrix is non-singular. We show that invertible necklaces of length $n$ correspond to normal bases of the finite field $F_{p^n}$, and that they form an Abelian group isomorphic to the Reutenauer group $RG_p^n$. Using known results in abstract algebra, we can make a bridge between generalized de Bruijn words and invertible necklaces. In particular, we highlight a correspondence between binary de Bruijn words of order $d+1$, binary necklaces of length $2^{d}$ having an odd number of $1$'s, invertible BWT matrices of size $2^{d}\times 2^{d}$, and normal bases of the finite field $F_{2^{2^{d}}}$.