Finite maximal codes and factorizations of cyclic groups

TL;DR

This paper explores the structure of finite maximal codes, establishing new links with cyclic group factorizations, and proves results related to longstanding conjectures in code theory, including simplified proofs and decidability results.

Contribution

It provides a new, simpler proof of the relationship between finite maximal codes and cyclic group factorizations, and demonstrates decidability of code inclusion related to the factorization conjecture.

Findings

Proves that finite maximal codes containing specific words satisfy the triangle conjecture.

Shows it is decidable whether certain codes are included in finite maximal codes.

Establishes that such codes can be extended to codes satisfying the factorization conjecture.

Abstract

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes, namely the factorization conjecture and the triangle conjecture, proposed by Perrin and Sch\"{u}tzemberger. The latter concerns finite codes $Y$ which are subsets of $a^* B a^*$, where $a$ is a letter and $B$ is an alphabet not containing $a$. A structural property of finite maximal codes has recently been shown by Zhang and Shum. It exhibits a relationship between finite maximal codes and factorizations of cyclic groups. With the aim of highlighting the links between this result and other older ones on maximal and factorizing codes, we give a simpler and a new proof of this result. As a consequence, we prove that for any finite maximal code $X \subseteq (B \cup \{a \})^*$ containing the word $a^{pq}$, where $p,q$ are prime numbers, $X \cap a^* B a^*$ satisfies the triangle conjecture. Let $n$ be a positive integer that is a product of at most two prime numbers. We also prove that it is decidable whether a finite code $Y \cup a^{n} \subseteq a^* B a^* \cup a^*$ is included in a finite maximal code and that, if this holds, $Y \cup a^{n}$ is included in a code that also satisfies the factorization conjecture.