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A note on the factorization conjecture

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Abstract

We give partial results on the factorization conjecture on codes proposed by Schützenberger. We consider a family of finite maximal codes \(C\) over the alphabet \(A = \{a, b\}\) and we prove that the factorization conjecture holds for these codes. This family contains \((p,4)\)-codes, where a \((p,4)\)-code \(C\) is a finite maximal code over \(A\) such that each word in \(C\) has at most four occurrences of \(b\) and \(a^p \in C\), for a prime number \(p\). We also discuss the structure of these codes. The obtained results once again show relations between factorizations of finite maximal codes and factorizations of finite cyclic groups.

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Notes

  1. Note that in this paper we use the term “positive factorization” with a slightly different meaning with respect to the definition of the same term in [2].

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Acknowledgments

The author thanks the anonymous referee for his/her helpful suggestions and constructive criticism.

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  1. Dipartimento di Informatica, Università degli Studi di Salerno, via Giovanni Paolo II, 132, 84084 , Fisciano, SA, Italy

    Clelia De Felice

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  1. Clelia De Felice
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Correspondence to Clelia De Felice.

Additional information

Partially supported by the \(\textit{FARB}\) Project “Aspetti computazionali e proprietà algebriche degli automi e dei linguaggi formali” (University of Salerno, 2011), the \(\textit{FARB}\) Project “Aspetti algebrici e computazionali nella teoria dei codici e dei linguaggi formali” (University of Salerno, 2012) and the \(\textit{MIUR}\) Project 2010–2011 “Automata and Formal Languages: Mathematical and Applicative Aspects”.

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De Felice, C. A note on the factorization conjecture. Acta Informatica 50, 381–402 (2013). https://doi.org/10.1007/s00236-013-0187-1

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