Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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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.

On reversible $\mathbb{Z}_2$-double cyclic codes

Nupur Patanker

Indian Institute of Science Education and Research, Pune

Abstract

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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