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.
Citation: Fatma Zehra Uzekmek, Elif Segah Oztas, Mehmet Ozen. $ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $[J]. AIMS Mathematics, 2024, 9(10): 27908-27929. doi: 10.3934/math.20241355
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.
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Fatma Zehra Uzekmek, Elif Segah Oztas, Mehmet Ozen. $ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $[J]. AIMS Mathematics, 2024, 9(10): 27908-27929. doi: 10.3934/math.20241355
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