Cyclic and negacyclic codes over the Galois ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="normal"><mml:mi>GR</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

Sobhani, R.; Esmaeili, Morteza

doi:10.1016/j.dam.2009.03.001

Cited by 24 publications

(3 citation statements)

References 21 publications

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“…Since the decomposition of polynomials over finite rings is not unique, the structure of repeated-root constacyclic codes over finite rings is more complex. Since 2003, some special classes of repeated-root constacyclic codes over certain finite chain rings have been studied by many authors (see, for example, [6,16,17,27,13,21]).…”

Section: Introductionmentioning

confidence: 99%

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Liu

2022

AMC

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<p style='text-indent:20px;'>In this paper, we generalize the notion of self-orthogonal codes to <inline-formula><tex-math id="M1">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal codes over an arbitrary finite ring. Then, we study the <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonality of constacyclic codes of length <inline-formula><tex-math id="M3">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over the finite commutative chain ring <inline-formula><tex-math id="M4">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a prime, <inline-formula><tex-math id="M6">\begin{document}$ u^2 = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> is an arbitrary ring automorphism of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. We characterize the structure of <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-dual code of a <inline-formula><tex-math id="M10">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code of length <inline-formula><tex-math id="M11">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M12">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. Further, the necessary and sufficient conditions for a <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code to be <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal are provided. In particular, we determine all <inline-formula><tex-math id="M15">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-dual constacyclic codes of length <inline-formula><tex-math id="M16">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M17">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. In the end of this paper, when <inline-formula><tex-math id="M18">\begin{document}$ p $\end{document}</tex-math></inline-formula> is an odd prime, we extend the results to constacyclic codes of length <inline-formula><tex-math id="M19">\begin{document}$ 2 p^s $\end{document}</tex-math></inline-formula>.</p>

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Section: Introductionmentioning

confidence: 99%

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Liu

2022

AMC

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“…En la actualidad los anillos de Galois han adquirido notoriedad en las áreas de la teoría de códigos (cf. [1], [8], [11], [15], [18]) y la criptografía (cf. [3], [14]), entre otras.…”

Section: Introductionunclassified

Imágenes de Gray de códigos consta-cíclicos sobre anillos de Galois R de índice de nilpotencia 3

Ramírez¹,

Andrade²,

Hernández³

2021

Rev Integr temas mat

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Estableceremos condiciones necesarias y suficientes para que la imagen bajo la función de Gray de un R-código consta-cíclico sea un Fpm-código cuasi-cíclico. Estudiamos el anillo de vectores de Witt para obtener una manera de operar las µ-reducciones de las componentes p-ádicas de los elementos de los anillos de Galois de índice de nilpotencia 3, R = GR(p3, m). Analizamos a los anillos de Galois, sus propiedades más relevantes, y en particular la representación p-ádica de sus elementos. Más adelante, examinamos la construcción del anillo de vectores de Witt y sus operaciones, en particular, obtenemos expresiones explícitas para las operaciones de suma y producto de los elementos en el anillo truncado de vectores de Witt de longitud 3, W3(Fpm). Finalmente, utilizamos las operaciones de éstos últimos y un isomorfismo entre GR(p3, m) y W3(Fpm) para operar las µ-reducciones antes descritas.

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“…After that the concepts of cyclic and self-dual codes have been extended and studied over the ring Z 4 (see [1,3,4]). Later on, the study of cyclic and self-dual codes has been generalized to codes over Z p r and Galois rings (see [5,9,10,12,13]).…”

Section: Introductionmentioning

confidence: 99%

On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$

Jitman¹,

Ling²,

Sangwisut³

2016

AMC

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In this paper, cyclic codes over the Galois ring GR(p 2 , s) are studied. The main result is the characterization and enumeration of Hermitian self-dual cyclic codes of length p a over GR(p 2 , s). Combining with some known results and the standard Discrete Fourier Transform decomposition, we arrive at the characterization and enumeration of Euclidean self-dual cyclic codes of any length over GR(p 2 , s).

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Cyclic and negacyclic codes over the Galois ring GR(p2,m)

Cited by 24 publications

References 21 publications

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Imágenes de Gray de códigos consta-cíclicos sobre anillos de Galois R de índice de nilpotencia 3

On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$

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