A note on the factorization conjecture

TL;DR

This paper investigates the factorization conjecture for certain finite maximal codes over a binary alphabet, providing partial results and characterizations that relate positive factorizations to group cyclic structures.

Contribution

It offers new partial results on the factorization conjecture, characterizes positive factorizations for specific codes, and links code factorizations to cyclic group factorizations.

Findings

Positive factorizations have coefficients 0 or 1.

Codes with words having at most 4 'b's' have positive factorizations.

Structural characterization of these codes and their relation to cyclic groups.

Abstract

We give partial results on the factorization conjecture on codes proposed by Schutzenberger. We consider finite maximal codes C over the alphabet A = {a, b} with C \cap a^* = a^p, for a prime number p. Let P, S in Z <A>, with S = S_0 + S_1, supp(S_0) \subset a^* and supp(S_1) \subset a^*b supp(S_0). We prove that if (P,S) is a factorization for C then (P,S) is positive, that is P,S have coefficients 0,1, and we characterize the structure of these codes. As a consequence, we prove that if C is a finite maximal code such that each word in C has at most 4 occurrences of b's and a^p is in C, then each factorization for C is a positive factorization. We also discuss the structure of these codes. The obtained results show once again relations between (positive) factorizations and factorizations of cyclic groups.