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

January 30, 2019 · Declared Dead · 🏛 Designs, Codes and Cryptography

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Authors Sanjit Bhowmick, Alexandre Fotue-Tabue, Edgar Martínez-Moro, Ramakrishna Bandi, Satya Bagchi arXiv ID 1901.10836 Category cs.IT: Information Theory Citations 39 Venue Designs, Codes and Cryptography Last Checked 3 months ago
Abstract
In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal $\mathbb{Z}_4$ codes of different lengths {which are} cyclic LCD codes over $\mathbb{Z}_4$.
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