Finite maximal codes and factorizations of cyclic groups

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.