On LCD codes over $$\mathbb {Z}_4$$ Z 4

Sanjit Bhowmick (), Satya Bagchi () and Ramakrishna Bandi ()
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Sanjit Bhowmick: National Institute of Technology Durgapur
Satya Bagchi: National Institute of Technology Durgapur
Ramakrishna Bandi: Dr. SPM International Institute of Information Technology

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 1200-1210

Abstract: 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 $$\mathbb {Z}_4$$ Z 4 and obtain a necessary and sufficient condition for the existence of LCD codes over $$\mathbb {Z}_4$$ Z 4 . Later, we investigate free LCD cyclic codes of odd lengths over $$\mathbb {Z}_4$$ Z 4 and find a relation between free LCD cyclic codes of odd lengths and reversible codes of odd lengths over $$\mathbb {Z}_4$$ Z 4 . We prove that free LCD cyclic codes of length $$2^k$$ 2 k , a positive integer k, do not exist, and also establish a condition for the existence of free LCD cyclic codes of lengths $$2^km$$ 2 k m for a positive integer m with $$\gcd (2,m)=1$$ gcd ( 2 , m ) = 1 . Finally, we present some good parameters of LCD codes that we obtained.

Keywords: Codes over finite chain rings; LCD codes; Free codes; Reversible codes; Primary 94B05; Secondary 94B15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-024-00563-x

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