Cyclic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>R</mml:mi></mml:math>-additive codes

Samei, Karim; Mahmoudi, Saadoun

doi:10.1016/j.disc.2016.11.007

Cited by 9 publications

(15 citation statements)

References 12 publications

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“…Throughout this paper R and S are finite commutative rings, where S is a free R-algebra. The class of R-additive codes, as a generalization of F q -linear F q t -codes, was introduced in [8]. In this section, we generalize Shiromoto's results for Radditive codes, and the basic exact sequence for this class of codes is proved.…”

Section: The Basic Exact Sequence For R-additive Codesmentioning

confidence: 74%

“…The theory of F q -linear codes over vector spaces was generalized to R-additive codes in [8]. More precisely let S be an R-algebra.…”

Section: Introductionmentioning

confidence: 99%

“…Linear codes, additive codes and F q -linear codes are special cases of R-additive codes (see Examples 2.2, 2.4 and 2.5 of [9]). In [8] and [9], the structure of cyclic R-additive codes over free R-algebras, where R is a chain ring, have been investigated. In particular, the structure of cyclic additive codes over an arbitrary Galois ring S = GR(p s , m) was given.…”

Section: Introductionmentioning

confidence: 99%

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Singleton bounds for R-additive codes

Samei¹,

Mahmoudi²

2018

Advances in Mathematics of Communications

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Shiromoto (Linear Algebra Applic 295 (1999) 191-200) obtained the basic exact sequence for the Lee and Euclidean weights of linear codes over Z and as an application, he found the Singleton bounds for linear codes over Z with respect to Lee and Euclidean weights. Huffman (Adv. Math. Commun 7 (3) (2013) 349-378) obtained the Singleton bound for Fq-linear F q t-codes with respect to Hamming weight. Recently the theory of Fq-linear F q t-codes were generalized to R-additive codes over R-algebras by Samei and Mahmoudi. In this paper, we generalize Shiromoto's results for linear codes over Z to R-additive codes. As an application, when R is a chain ring, we obtain the Singleton bounds for R-additive codes over free R-algebras. Among other results, the Singleton bounds for additive codes over Galois rings are given.

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Section: The Basic Exact Sequence For R-additive Codesmentioning

confidence: 74%

“…The theory of F q -linear codes over vector spaces was generalized to R-additive codes in [8]. More precisely let S be an R-algebra.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Singleton bounds for R-additive codes

Samei¹,

Mahmoudi²

2018

Advances in Mathematics of Communications

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“…Recently the theory of F q -linear codes was generalized to R-additive codes [18,19]. When R is a finite chain ring, the structure of cyclic R-additive codes over free R-algebras has been given in [18]. In this section, we obtain the structure of cyclic R-additive codes in the case that R is a complete local principal ideal ring.…”

Section: Cyclicr M -Additive Codesmentioning

confidence: 99%

Codes over $ \frak m $-adic completion rings

Mahmoudi¹,

Samei²

2022

AMC

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<p style='text-indent:20px;'>The theory of linear codes over finite rings has been generalized to linear codes over infinite rings in two special cases; the ring of <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-adic integers and formal power series ring. These rings are examples of complete discrete valuation rings (CDVRs). In this paper, we generalize the theory of linear codes over the above two rings to linear codes over complete local principal ideal rings. In particular, we obtain the structure of linear and constacyclic codes over CDVRs. For this generalization, first we study linear codes over <inline-formula><tex-math id="M3">\begin{document}$ \hat{R}_{ \frak m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ R $\end{document}</tex-math></inline-formula> is a commutative Noetherian ring, <inline-formula><tex-math id="M5">\begin{document}$ \frak m = \langle \gamma\rangle $\end{document}</tex-math></inline-formula> is a maximal ideal of <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M7">\begin{document}$ \hat{R}_{ \frak m} $\end{document}</tex-math></inline-formula> denotes the <inline-formula><tex-math id="M8">\begin{document}$ \frak m $\end{document}</tex-math></inline-formula>-adic completion of <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. We call these codes, <inline-formula><tex-math id="M10">\begin{document}$ \frak m $\end{document}</tex-math></inline-formula>-adic codes. Using the structure of <inline-formula><tex-math id="M11">\begin{document}$ \frak m $\end{document}</tex-math></inline-formula>-adic codes, we present the structure of linear and constacyclic codes over complete local principal ideal rings.</p>

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“…Therefore, the generalizations on Galois rings, or moreover finite chain rings are immediate. Some recent works on codes over such rings are [3,7,9,22].…”

Section: Introductionmentioning

confidence: 99%

Additive cyclic codes over finite commutative chain rings

Martı́nez-Moro

Otal

Özbudak

2018

Discrete Mathematics

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Additive cyclic codes over Galois rings were investigated in [3]. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in [3], whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.

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CyclicR-additive codes

Cited by 9 publications

References 12 publications

Singleton bounds for R-additive codes

Singleton bounds for R-additive codes

Codes over $ \frak m $-adic completion rings

Additive cyclic codes over finite commutative chain rings

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