Charm bracelets and their application to the construction of periodic Golay pairs

TL;DR

This paper introduces charm bracelets as a mathematical tool and uses an efficient algorithm to generate them, leading to the construction of 29 new periodic Golay pairs of length 68, advancing combinatorial design theory.

Contribution

It presents a novel application of charm bracelets to construct new periodic Golay pairs, utilizing an efficient generation algorithm.

Findings

29 new periodic Golay pairs of length 68 constructed

Efficient $O(n^3)$ algorithm for charm bracelet generation

Application of algebraic structures to combinatorial design

Abstract

A $k$-ary charm bracelet is an equivalence class of length $n$ strings with the action on the indices by the additive group of the ring of integers modulo $n$ extended by the group of units. By applying an $O(n^3)$ amortized time algorithm to generate charm bracelet representatives with a specified content, we construct 29 new periodic Golay pairs of length $68$.