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On LCD codes over \(\mathbb {Z}_4\)

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Abstract

Linear Complementary Dual (LCD) code is a linear code with a trivial intersection with its dual. In this paper, we prove that non-free LCD codes do not exist over \(\mathbb {Z}_4\) and obtain a necessary and sufficient condition for the existence of LCD codes over \(\mathbb {Z}_4\). Later, we investigate free LCD cyclic codes of odd lengths over \(\mathbb {Z}_4\) and find a relation between free LCD cyclic codes of odd lengths and reversible codes of odd lengths over \(\mathbb {Z}_4\). We prove that free LCD cyclic codes of length \(2^k\), a positive integer k, do not exist, and also establish a condition for the existence of free LCD cyclic codes of lengths \(2^km\) for a positive integer m with \(\gcd (2,m)=1\). Finally, we present some good parameters of LCD codes that we obtained.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Durgapur, Durgapur, West Bengal, 713209, India

    Sanjit Bhowmick & Satya Bagchi

  2. Department of science and applied mathematics, Dr. SPM International Institute of Information Technology, Atal Nagar-Nava Raipur, India

    Ramakrishna Bandi

Authors
  1. Sanjit Bhowmick
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  2. Satya Bagchi
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  3. Ramakrishna Bandi
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Corresponding author

Correspondence to Satya Bagchi.

Additional information

Communicated by Sudhir R Ghorpade.

The first author of the paper would like to thank the Ministry of Human Resource and Development India for supporting financially to carry out this work. The second author would like to acknowledge the support of the National Board of Higher Mathematics (NBHM), Government of India (Grant No. 02011/2/2019 NBHM(R.P)/R &D II/1092).

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Bhowmick, S., Bagchi, S. & Bandi, R. On LCD codes over \(\mathbb {Z}_4\). Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00563-x

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  • DOI: https://doi.org/10.1007/s13226-024-00563-x

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