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Designs, Codes and Cryptography

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20-01-2020

Do non-free LCD codes over finite commutative Frobenius rings exist?

Authors: Sanjit Bhowmick, Alexandre Fotue-Tabue, Edgar Martínez-Moro, Ramakrishna Bandi, Satya Bagchi

Published in: Designs, Codes and Cryptography | Issue 5/2020

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Abstract

In this paper, we clarify some aspects of LCD codes in the literature. We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD codes over a finite commutative Frobenius ring. We later show that a free constacyclic code over a finite chain ring is an LCD code if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible. We illustrate the minimum Lee distance of LCD codes over some finite commutative chain rings with examples. We found some new optimal cyclic codes over \({\mathbb {Z}}_4\) of different lengths which are LCD codes using computer algebra system MAGMA.

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Title: Do non-free LCD codes over finite commutative Frobenius rings exist?
Authors: Sanjit Bhowmick
Alexandre Fotue-Tabue
Edgar Martínez-Moro
Ramakrishna Bandi
Satya Bagchi
Publication date: 20-01-2020
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 5/2020
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-019-00713-x

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