1. bookVolume 30 (2022): Issue 1 (February 2022)
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
access type Open Access

DNA codes over finite local Frobenius non-chain rings of length 5 and nilpotency index 4

Published Online: 12 Mar 2022
Volume & Issue: Volume 30 (2022) - Issue 1 (February 2022)
Page range: 89 - 108
Received: 14 Jun 2021
Accepted: 31 Jul 2021
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
Abstract

A one to one correspondence between the elements of a finite local Frobenius non-chain ring of length 5 and nilpotency index 4, and k-tuples of DNA codewords is established. Using this map the structure of DNA codes over these rings is determined, the length of the code is relatively prime to the characteristic of the residue field of the ring.

Keywords

MSC 2010

[1] T. Abualrub, A. Ghrayeb, X. N. Zeng, Construction of cyclic codes over GF(4) for DNA computing, Journal of the Franklin Institute, 343, 448-457 (2006) Search in Google Scholar

[2] L. Adleman, Molecular computation of solutions to combinatorial problems, Science 266, 1021-1024 (1994)10.1126/science.79736517973651 Search in Google Scholar

[3] C. Álvarez-García, C. A. Castillo-Guillén, DNA codes over finite local Frobenius non-chain rings of length 4, Discrete Mathematics, 341, 112404 (2021)10.1016/j.disc.2021.112404 Search in Google Scholar

[4] C. Álvarez-García, C. A. Castillo-Guillén, 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, (to appear) Search in Google Scholar

[5] A. Bayram, E. S. Oztas, I. Siap, Codes over 𝔽4 + v𝔽4 and some DNA applications, Designs Codes and Cryptography, 80, 379-393 (2016) Search in Google Scholar

[6] C. A. Castillo-Guillén, C. Rentería-Márquez and H. Tapia-Recillas, Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, Finite Fields and Their Aplications, 43, 1-21 (2017) Search in Google Scholar

[7] C. A. Castillo-Guillén, C. Rentería-Márquez and H. Tapia-Recillas, Duals of constacyclic codes over finite local Frobenius non-chain rings of length 4, Discrete Mathematics, 341, 919-933 (2018) Search in Google Scholar

[8] C. A. Castillo-Guillén, C. Rentería-Márquez, E. Sarmiento, R. Villareal and H. Tapia-Recillas, 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, Discrete Mathematics, 342, 2283-2296 (2019) Search in Google Scholar

[9] C. A. Castillo-Guillén, C. Rentería-Márquez, Constacyclic codes over finite local Frobenius non-chain rings of length 5 and nilpotency index 4, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 28(2), 67-91 (2020)10.2478/auom-2020-0020 Search in Google Scholar

[10] H. Q. Dinh, A. K. Singh, S. Pattanayak, S. Sriboonchitta, Cyclic DNA codes over the ring 𝔽2 + u𝔽2 + v𝔽2 + uv𝔽2 + v2𝔽2 + uv2𝔽2, Designs Codes and Cryptography, 86, 1451-1467 (2018) Search in Google Scholar

[11] P. Gaborit, O. D. King, Linear constructions for DNA codes, Theoretical computer science, 334, 99-113 (2005) Search in Google Scholar

[12] K. Guenda, T. A. Gulliver, Construction of cyclic codes over 𝔽2 + u𝔽2 for DNA computing, Appl. Algebra Engrg. Comm. Comput. 24, 445-459 (2013) Search in Google Scholar

[13] J. Kaur, R. Sehmi, S. Dutt, Reversible complement cyclic codes over Galois rings with application to DNA codes, Discrete Applied Mathematics, 280, 162-170 (2020) Search in Google Scholar

[14] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, 10th impression, North-Holland, Amsterdam, 1998 Search in Google Scholar

[15] B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974 Search in Google Scholar

[16] V. Rykov, A. J. Macula, D. Torney, P. White, DNA sequences and quaternary cyclic codes, IEEE International Symposium on Information Theory, Washington, DC, 248-248 (2001) Search in Google Scholar

[17] I. Siap, T. Abualrub, A. Ghrayeb, Cyclic DNA codes over the ring 𝔽2[u]/〈u2 1〉 based on the deletion distance, Journal of the Franklin Institute, 346, 731-740 (2009) Search in Google Scholar

[18] D. C. Tuplan, H. Hoos, A. Condon, Stochastic local search algorithms for DNA word design, Lecture Notes in computer science, Springer, Berlin, 229-241 (2003)10.1007/3-540-36440-4_20 Search in Google Scholar

[19] B. Yildiz, I. Siap, Cyclic codes over 𝔽2[u]/〈u4 1〉 and applications to DNA codes, Computers and Mathematics with applications, 63, 1169-1176 (2012) Search in Google Scholar

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