Cyclic codes over the ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>2</m

Kewat, Pramod Kumar; Ghosh, Bappaditya; Pattanayak, Sukhamoy

doi:10.1016/j.ffa.2015.01.005

Cited by 16 publications

(3 citation statements)

References 21 publications

(38 reference statements)

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“…Theorem 4.1 gives the generators of cyclic codes of length n over R, see [10]. Now, extending this result for λ-constacyclic codes of length n over R, we present the generators for λ-constacyclic codes of length n over R, where gcd(n, p) = 1.…”

Section: Generators Of (1 + (P − 2)u)-constacyclic Codesmentioning

confidence: 75%

A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/hu 2 − u, v2 − v, uv − vui

Bag¹,

Islam²,

Prakash³

et al. 2019

Journal of Algebra Combinatorics Discrete Structures and Applications

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Section: Generators Of (1 + (P − 2)u)-constacyclic Codesmentioning

confidence: 75%

A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/hu 2 − u, v2 − v, uv − vui

Bag¹,

Islam²,

Prakash³

et al. 2019

Journal of Algebra Combinatorics Discrete Structures and Applications

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“…Cengellenmis et al [2] gave the structure of codes over finite non-chain rings and studied their Gray map's images. In [7], constacyclic codes over a non-chain ring were introduced by Karadeniz et al In 2015, cyclic codes over the non-chain ring Z p [u, v]/⟨u 2 , v 2 , uv − vu⟩ [8] were considered, a generator set for these codes was found and proved to be unique. Furthermore, the authors studied the rank of the generator matrix and obtained some optimal codes over this ring as well.…”

mentioning

confidence: 99%

“…These properties also applied to LCD codes over finite local rings, because of local rings can be regarded as a direct product of some fields. Recently, there was some research on linear codes over finite rings, such as [8,10,18]. In 2003, Ling et al [10] investigated quasi-cyclic (QC) codes and double circulant codes over a chain ring.…”

mentioning

confidence: 99%

Linear complementary dual codes and double circulant codes over a semi-local ring

Cheng

Cao

Qian

2024

AMC

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>

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Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$ F 2 + u F 2

Dinh

Singh

Pattanayak

et al. 2017

Des. Codes Cryptogr.

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Cyclic codes over the ring Zp[u,v]/〈u2

Cited by 16 publications

References 21 publications

A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/hu 2 − u, v2 − v, uv − vui

A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/hu 2 − u, v2 − v, uv − vui

Linear complementary dual codes and double circulant codes over a semi-local ring

Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$ F 2 + u F 2

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