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Applicable Algebra in Engineering, Communication and Computing

13-07-2024 | Original Paper

DNA codes over \(GR(2^{3},d)[X]/\langle X^{2},2X \rangle\)

Authors: C. Álvarez-García, C. A. Castillo-Guillén, Mohamed Badaoui

Published in: Applicable Algebra in Engineering, Communication and Computing

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Abstract

The main results of this paper are in two directions. First, the family of finite local rings of length 4 whose annihilator of their maximal ideals have length 2 is determined. Second, the structure of constacyclic, reversible and DNA codes over those rings are described, the length of the code is relatively prime to the characteristic of the residue field of the ring.

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Álvarez-García, C., Castillo-Guillén, C.A.: DNA codes over finite local Frobenius non-chain rings of length \(4\). Discret. Math. (2021). https://doi.org/10.1016/j.disc.2021.112404MathSciNetCrossRef

Álvarez-García, C., Castillo-Guillén, C.A.: DNA codes over finite local Frobenius non-chain rings of length \(5\) and nilpotency index \(4\). An. Stiint. Univ. Ovidius Constanta Ser. Mat. 30(1), 89–108 (2022)MathSciNet

Álvarez-García, C., Castillo-Guillén, C.A.: Self dual reversible and complementary duals constacyclic codes over finite local Frobenius non-chain rings of length \(5\) and nilpotency index \(4\). An. Stiint. Univ. Ovidius Constanta Ser. Mat. 29(2), 25–50 (2021)MathSciNet

Abualrub, T., Ghrayeb, A., Zeng, X.N.: Construction of cyclic codes over \(GF(4)\) for DNA computing. J. Frankl. Inst. 343, 448–457 (2006)MathSciNetCrossRef

Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266, 1021–1024 (1994)CrossRef

Bayram, A., Oztas, E.S., Siap, I.: Codes over \(\mathbb{F} _{4}+v\mathbb{F} _{4}\) and some DNA applications. Des. Codes Cryptogr. 80, 379–393 (2016)MathSciNetCrossRef

Castillo-Guillén, C.A., Rentería-Márquez, C., Tapia-Recillas, H.: Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index \(3\). Finite Fields Their Appl. 43, 1–21 (2017)MathSciNetCrossRef

Castillo-Guillén, C.A., Rentería-Márquez, C., Sarmiento, E., Villareal, R., Tapia-Recillas, H.: The dual of a constacyclic code, constacyclic self dual, constacyclic reversible and constacyclic codes with complementary duals over finite local Frobenius non-chain rings of nilpotency index \(3\). Discret. Math. 342, 2283–2296 (2019)CrossRef

Dinh, H.Q., López-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50, 1728–1744 (2004)MathSciNetCrossRef

10.

Dinh, H.Q., Singh, A.K., Pattanayak, S., Sriboonchitta, S.: Cyclic DNA codes over the ring \(\mathbb{F} _{2}+u\mathbb{F} _{2}+v\mathbb{F} _{2}+uv\mathbb{F} _{2}+v^{2}\mathbb{F} _{2}+uv^{2}\mathbb{F} _{2}\). Des. Codes Cryptogr. 86, 1451–1467 (2018)MathSciNetCrossRef

11.

Gaborit, P., King, O.D.: Linear constructions for DNA codes. Theor. Comput. Sci. 334, 99–113 (2005)MathSciNetCrossRef

12.

Guenda, K., Gulliver, T.A.: Construction of cyclic codes over \(\mathbb{F} _{2}+u\mathbb{F} _{2}\) for DNA computing. Appl. Algebra Engrg. Comm. Comput. 24, 445–459 (2013)MathSciNetCrossRef

13.

Head, T.: Formal language theory and DNA: an analysis of the generative capacity of specific recombinant behaviors. Bull. Math. Biol. 49, 737–759 (1987)MathSciNetCrossRef

14.

Kaur, J., Sehmi, R., Dutt, S.: Reversible complement cyclic codes over Galois rings with application to DNA codes. Discret. Appl. Math. 280, 162–170 (2020)MathSciNetCrossRef

15.

MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1998)

16.

Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)

17.

McDonald, B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974)

18.

Siap, I., Abualrub, T., Ghrayeb, A.: Cyclic DNA codes over the ring \(\mathbb{F} _{2}[u]/\langle u^{2}-1\rangle\) based on the deletion distance. J. Frankl. Inst. 346, 731–740 (2009)CrossRef

19.

Yildiz, B., Siap, I.: Cyclic codes over \(\mathbb{F} _{2}[u]/\langle u^{4}-1\rangle\) and applications to DNA codes. Comput. Math. Appl. 63, 1169–1176 (2012)MathSciNetCrossRef

Title: DNA codes over
Authors: C. Álvarez-García
C. A. Castillo-Guillén
Mohamed Badaoui
Publication date: 13-07-2024
Publisher: Springer Berlin Heidelberg
Published in: Applicable Algebra in Engineering, Communication and Computing
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-024-00667-1

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