Perfect Necklaces
Abstract
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under rotations. For positive integers k and n, we call a necklace (k,n)-perfect if each word of length k occurs exactly n times at positions which are different modulo n for any convention on the starting point. We call a necklace perfect if it is (k,k)-perfect for some k. We prove that every arithmetic sequence with difference coprime with the alphabet size induces a perfect necklace. In particular, the concatenation of all words of the same length in lexicographic order yields a perfect necklace. For each k and n, we give a closed formula for the number of (k,n)-perfect necklaces. Finally, we prove that every infinite periodic sequence whose period coincides with some (k,n)-perfect necklace for any n, passes all statistical tests of size up to k, but not all larger tests. This last theorem motivated this work.
Keywords
Cite
@article{arxiv.1601.07975,
title = {Perfect Necklaces},
author = {Nicolás Álvarez and Verónica Becher and Pablo A. Ferrari and Sergio A. Yuhjtman},
journal= {arXiv preprint arXiv:1601.07975},
year = {2016}
}