On skew cyclic codes over $ M_{2}(\mathbb{F}_{2}) $

Xuesong Si; Chuanze Niu; Xuesong Si; Chuanze Niu

doi:10.3934/math.20231246

AIMS Mathematics

2023, Volume 8, Issue 10: 24434-24445. doi: 10.3934/math.20231246

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On skew cyclic codes over $ M_{2}(\mathbb{F}_{2}) $

Xuesong Si ,
Chuanze Niu ^,

School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China

Received: 06 June 2023 Revised: 22 July 2023 Accepted: 06 August 2023 Published: 17 August 2023
MSC : 11T71, 94B05, 94B15

The algebraic structure of skew cyclic codes over $ M_{2} $($ \mathbb{F}_2 $), using the $ \mathbb{F}_4 $-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree $ d(x) $ is a free code generated by $ d(x) $. According to our findings, skew cyclic codes of odd and even lengths are cyclic and $ 2 $-quasi-cyclic over $ M_{2}(\mathbb{F}_{2}) $, respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over $ M_{2}(\mathbb{F}_{2}) $ is considered in this paper.
- skew cyclic code,
- finite chain ring,
- skew polynomial rings,
- Gray map
Citation: Xuesong Si, Chuanze Niu. On skew cyclic codes over $ M_{2}(\mathbb{F}_{2}) $[J]. AIMS Mathematics, 2023, 8(10): 24434-24445. doi: 10.3934/math.20231246

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Abstract

The algebraic structure of skew cyclic codes over $ M_{2} $($ \mathbb{F}_2 $), using the $ \mathbb{F}_4 $-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree $ d(x) $ is a free code generated by $ d(x) $. According to our findings, skew cyclic codes of odd and even lengths are cyclic and $ 2 $-quasi-cyclic over $ M_{2}(\mathbb{F}_{2}) $, respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over $ M_{2}(\mathbb{F}_{2}) $ is considered in this paper.

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