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arXiv · 1901.10836

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

Sanjit Bhowmick·Alexandre Fotue-Tabue·Edgar Martínez-Moro·Ramakrishna Bandi·Satya Bagchi
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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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Sanjit Bhowmick, Alexandre Fotue-Tabue, Edgar Martínez-Moro, Ramakrishna Bandi, Satya Bagchi. 2019-01-30. Do non-free LCD codes over finite commutative Frobenius rings exist?. https://arxiv.org/abs/1901.10836

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