Cyclic and constacyclic codes over the ring Z4[u]/⟨u3 − u2⟩ and their Gray images

Özen, Mehmet; Uzekmek, Fatma Zehra; Öztaş, Elif Segah

doi:10.3906/mat-2006-112

Cited by 2 publications

(1 citation statement)

References 10 publications

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“…Te structure of a cyclic code of arbitrary length over Z 4 + uZ 4 , u 2 � 0 has been studied by Bandi and Bhaintwal [19]. Cyclic and some constacyclic codes of odd length over the nonchain ring Z 4 + uZ 4 , u 2 � 1 have been studied by Ozen et al [20].…”

Section: Introductionmentioning

confidence: 99%

Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

Jain,

Dutt,

Sehmi

2024

Journal of Mathematics

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The rings Z4+νZ4 have been classified into chain rings and nonchain rings based on the values of ν2∈Z4+νZ4. In this paper, the structure of a cyclic code of arbitrary length over the rings Z4+νZ4 for those values of ν2 for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.

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Section: Introductionmentioning

confidence: 99%

Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

Jain,

Dutt,

Sehmi

2024

Journal of Mathematics

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$ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $

Uzekmek,

Oztas,

Ozen

2024

MATH

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<p>In this paper, three new automorphisms were identified over the ring $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $ where $ u^3 = u^2 $. With the help of these automorphisms, the characteristic structures of the generator polynomials for the $ \theta_i $-cyclic codes and $ (\theta_i, \lambda) $-constacyclic codes of odd length on this ring were investigated. Also, for all the units over the ring, $ \mathbb{Z}_{4} $-images of $ \theta_i $-cyclic and $ (\theta_i, \lambda) $-constacyclic codes were reviewed with the associated codes based on determined transformations. Using these observations, new and optimal codes were obtained and presented in the table. In addition, a new transformation was identified that involved DNA base pairs with the elements of $ \mathbb{Z}_{4} $. Moreover, a unit reverse polynomial was created, and in this way a new generation method has been built to construct reversible DNA codes over this ring. Finally, this article was further enhanced with supporting examples of the DNA as a part of the study.</p>

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Cyclic and constacyclic codes over the ring Z4[u]/⟨u3 − u2⟩ and their Gray images

Cited by 2 publications

References 10 publications

Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

$ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $

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