Bull. Korean Math. Soc. 2023; 60(2): 443-460
Online first article March 21, 2023 Printed March 31, 2023
https://doi.org/10.4134/BKMS.b220184
Copyright © The Korean Mathematical Society.
Nupur Patanker
Indian Institute of Science Education and Research, Pune
A binary linear code is said to be a $\mathbb{Z}_2$-double cyclic code if its coordinates can be partitioned into two subsets such that any simultaneous cyclic shift of the coordinates of the subsets leaves the code invariant. These codes were introduced in [6]. A $\mathbb{Z}_2$-double cyclic code is called reversible if reversing the order of the coordinates of the two subsets leaves the code invariant. In this note, we give necessary and sufficient conditions for a $\mathbb{Z}_2$-double cyclic code to be reversible. We also give a relation between reversible $\mathbb{Z}_2$-double cyclic code and LCD $\mathbb{Z}_2$-double cyclic code for the separable case and we present a few examples to show that such a relation doesn't hold in the non-separable case. Furthermore, we list examples of reversible $\mathbb{Z}_2$-double cyclic codes of length $\leq 10$.
Keywords: $\mathbb{Z}_2$-double cyclic code, reversible $\mathbb{Z}_2$-double cyclic codes, LCD codes
MSC numbers: Primary 94B05; Secondary 94B15, 11T71
Supported by: The author is supported by NBHM, DAE, Govt. of India.
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