Perfect Necklaces

TL;DR

This paper introduces perfect necklaces, a new combinatorial structure related to de Bruijn words, and explores their properties, counting formulas, and statistical test implications.

Contribution

It defines perfect necklaces, proves their existence via arithmetic sequences, provides counting formulas, and analyzes their statistical test properties.

Findings

Every arithmetic sequence with coprime difference induces a perfect necklace.

Concatenation of lexicographically ordered words forms a perfect necklace.

Infinite periodic sequences from perfect necklaces pass all small statistical tests.

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.