Factorizations of Cyclic Groups and Bayonet Codes

TL;DR

This paper explores the structure of certain variable-length codes related to cyclic groups, extending existing theory, proving new properties for specific cases, and confirming a longstanding conjecture in those cases.

Contribution

It extends factorizations of cyclic groups theory to Bayonet codes, proving they are prefix and suffix codes for specific n, and confirms the triangle conjecture for those cases.

Findings

Codes are prefix and suffix for n with at most three prime factors or of form pq^k.

Counterexamples exist for other n.

The triangle conjecture holds for these specific n.

Abstract

We study the (variable-length) codes of the form X u {a^n}, where X c a*wa* and |X| = n. We extend various notions and results from factorizations of cyclic groups theory to this type of codes. In particular, when n is the product of at most three primes or has the form pq^k (with p and q prime), we prove that they are composed of prefix and suffix codes. We provide counterexamples for other n. It implies that the long-standing triangle conjecture is true for this type of n. We also prove a conjecture about the size of a potential counterexample to the conjecture.