Agenda

Speaker:
Prof. Gabriele Fici, University of Palermo

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_{pⁿ}, and that they form an Abelian group isomorphic to the Reutenauer group RG_pⁿ. 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}× 2^{d}, and normal bases of the finite field F_{2^{2^{d}}}.