Cyclic codes over rings of matrices

Hai Quang Dinh; Atul Gaur; Pratyush Kumar; Manoj Kumar Singh; Abhay Kumar Singh

doi:10.3934/amc.2022073

Article Contents

2024, Volume 18, Issue 4: 1100-1122. Doi: 10.3934/amc.2022073

This issue Previous Article Dembowski-Ostrom polynomials and Dickson polynomials Next Article New constructions of large cyclic subspace codes via Sidon spaces

Cyclic codes over rings of matrices

1.
Department of Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, Warren, Ohio 44483, USA
2.
Department of Mathematics, University of Delhi (DU), Delhi 110007, India
3.
Department of Mathematics, School of Chemical Engineering and Physical Science, Lovely Professional University, Jalandhar-144001, India
4.
Department of Mathematics, University of Delhi (DU), Delhi 110007, India
5.
Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India

^*Corresponding author: Manoj Kumar Singh
^*Corresponding author: Manoj Kumar Singh

Author's names are in alphabetical order.
This paper is a part of PhD thesis of Manoj Kumar Singh.

Received: January 2022

Revised: July 2022

Early access: September 2022

Published: August 2024

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Abstract

In this paper, we consider the ring of matrices $ \mathcal{A} $ of order $ 2 $ over the ring $ \mathbb{F}_2 [u] / \langle u^k \rangle $, where $ u $ is an indeterminate with $ u^k = 0 $, i.e. $ \mathcal{A} = M_2 ( \mathbb{F}_2 [u] / \langle u^k \rangle) $. We derive the structure theorem for cyclic codes of odd length $ n $ over the ring $ \mathcal{A} $ with the help of isometry map from $ \mathcal{A} $ to $ \mathbb{F}_4 [u, v] / \langle u^k, v^2, u v - v u \rangle $, where $ v $ is an indeterminate satisfying $ v^2 = 0 $ and $ u v = v u $. We define a map $ \theta $ which takes the linear codes of odd length $ n $ over $ \mathcal{A} $ to linear codes of even length $ 2 k n $ over $ \mathbb{F}_4 $. We also define a weight on the ring $ \mathcal{A} $ which is an extension of the weight defined over the ring $ M_2 ( \mathbb{F}_2) $. An example is also given as applications to construct the linear codes of odd length $ n $ over $ \mathcal{A} $.

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Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 94B60.

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