Additive cyclic codes over finite commutative chain rings

Martı́nez-Moro, Edgar; Otal, Kamil; Özbudak, Ferruh

doi:10.1016/j.disc.2018.03.016

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…Therefore, every linear code is additive but not conversely. After the introduction of additive codes in [9], these codes were studied in both directions of single and mixed alphabets [10,21,22,27]. One important application of such codes in steganography is shown in [23].…”

Section: Introductionmentioning

confidence: 99%

On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

Prakash

Yadav

Islam

et al. 2023

AMC

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}_4 $\end{document}</tex-math></inline-formula> be the ring of integers modulo <inline-formula><tex-math id="M3">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>. This paper studies mixed alphabets <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}_4\mathbb{Z}_4[u^3] $\end{document}</tex-math></inline-formula>-additive cyclic and <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic codes for units <inline-formula><tex-math id="M6">\begin{document}$ \lambda = 1+2u^2,3+2u^2 $\end{document}</tex-math></inline-formula>. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z}_4 $\end{document}</tex-math></inline-formula>-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.</p>

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Section: Introductionmentioning

confidence: 99%

On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

Prakash

Yadav

Islam

et al. 2023

AMC

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“…Since the secret algorithm is not available to average users, it should be focused on the encryption of words, in particular, the audience in question needs protection of a small amount of personal data transmitted over the network. 4. Asymmetric encryption is the third condition.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, the symbol-pair distance of cyclic codes of length p e over Fpm is investigated. The exact symbol-pair distance of all cyclic codes of such length is determined.In this article [4] additive cyclic codes over Galois rings but in a more general ring family finite commutative chain rings are investigated. When focusing on non-Galois finite commutative chain rings, it is necessary to observe two different kinds of additivity.…”

mentioning

confidence: 99%

Development of an encryption method based on cyclic codes

Petrenko¹,

Ryabtsev²,

Pavlov³

et al. 2019

Proceedings of the 21st International Workshop on Computer Science and Information Technologies (CSIT 2019)

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The constant increase in the number of new types of cyber threats actualizes the issues of their information transfer. This article (report) proposes a research on the development of an encryption method based on cyclic codes. The purpose of this article is to develop an open algorithm, even knowing which an attacker will not be able to obtain the source text. The developed software encryption module will solve the problem of protecting information for small companies and private users, and also saves a large amount of resources, since one module solves the problem of ensuring the confidentiality and noise stability of transmitted messages. Also, this module is intended for common operating systems. This module is also intended for common operating systems, and user work is carried out using a friendly user interface. The software module performs RSA (Rivest-Shamir-Adleman) asymmetric encryption and jam resistant coding using the BCH (Bose-Chaudhuri-Hocquenghem) code. It is provided an example of the work of the program module developed on the basis of the proposed method. Based on the collection and analysis of statistical data output software system, a conclusion about the efficiency of the proposed solution and comparison with analogues has been drawn.

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Additive conjucyclic codes over a class of Galois rings

Islam,

Bhunia

2023

J. Appl. Math. Comput.

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As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except $${\mathbb {F}}_{q^2}$$ F q 2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings $$GR(2^r,2)$$ G R ( 2 r , 2 ) , where $$r\ge 2$$ r ≥ 2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over $$GR(2^r,2)$$ G R ( 2 r , 2 ) and the family of linear cyclic codes of length 2n over $${\mathbb {Z}}_{2^r}$$ Z 2 r . This correspondence helps to obtain additive conjucyclic codes over $$GR(2^r,2)$$ G R ( 2 r , 2 ) via known linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ Z 2 r . We prove that the trace dual $${\mathscr {C}}^{Tr}$$ C Tr of an additive conjucyclic code $${\mathscr {C}}$$ C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ Z 2 r contained in additive conjucyclic codes over $$GR(2^r,2)$$ G R ( 2 r , 2 ) . Last but not least, we explicitly derive the generator matrices for these codes.

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Additive cyclic codes over finite commutative chain rings

Cited by 5 publications

References 22 publications

On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

Development of an encryption method based on cyclic codes

Additive conjucyclic codes over a class of Galois rings

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