Bir zincir olmayan halka üzerinde DNA kodlar

Öz

$R=F_2[u,v,w]/\langle u^2=v^2,\, uv=0,\, w^2=w\rangle$ halkası üzerinde oluşturulan DNA kodlarının cebirsel yapısını inceliyoruz. Bu halka, değişmeli yerel Frobenius zincirsiz halkadır. $R$ üzerinde bir Gray haritası tanımlıyoruz ve Gray haritasının görüntülerini kullanarak DNA kodları oluşturuyoruz. Halka üzerinde tersinir DNA kodları ve tersinir tamamlayıcı DNA kodları tanımlıyoruz.

Anahtar Kelimeler

• Linear codes
• DNA codes

DNA Codes over a Non-chain Ring

Öz

We study the algebraic structure of DNA codes constructed over the ring R=F_2 [u,v,w] ⟨u^2=v^2,uv=0,w^2=w⟩, which is a commutative local Frobenius non-chain ring. We define a gray map over R and generate DNA codes using the images of the gray map. We define reversible DNA codes and reversible complement DNA codes over the ring.

Anahtar Kelimeler

• Linear codes
• DNA codes
• algebraic codes

Kaynakça

1. [1] Abulraub, T., Ghrayeb, A., Nian Zeng, X., “Construction of cyclic codes over GF(4) for DNA computing”, J. Franklin Inst. 343 (2006) : 448-457.
2. [2] Adleman, L. “Molecular computation of solutions to combinatorial problems”, Science 266(5187) (1994) : 1021-1024.
3. [3] Adleman, L., Rothemund, P.W.K., Roweis, S., Winfree, E., “On applying molecular computation to the Data Encryption Standard” J. Comput. Biol. 6 (1) (1999) : 53-63.
4. [4] Alsuraiheed, T., Oztas, E. S., Ali, S., Yilgor, M. B., “Reversible codes and applications to DNA codes over F2t4 [u]/(u2 − 1)”, AIMS Math. 8(11) (2023) : 27762-27774.
5. [5] Bayram, A., Oztas, E.S., Siap, I. “Codes over F4 +vF4 and some DNA applications”, Des. Codes Cryptogr. 80(2) (2015) : 379-393.
6. [6] Brand˜ao, M. M., Spoladore, L., Faria, L. C., Rocha, A. S., Silva–Filho, M. C., Palazzo, R. “Ancient DNA sequence revealed by error-correcting codes”, Scientific reports. 5 (2015) : 12051.
7. [7] Darehmiraki, M. “A semi–general method to solve the combinatorial optimization problems based on nanocomputing”, Int. J. Nanosci. 9(5) (2010) : 391-398.
8. [8] Faria, L. C., Rocha, A. S., Kleinschmidt, J. H., Silva-Filho, M. C., Bim, E., Herai, R. H., Yamagishi, M. E., Palazzo, R. Jr. “Is a genome a codeword of an error– correcting code?” PloS one. 7(5), e36644 (2012).

9. [9] Guozhen, X. , Mingxin L., Lei, Q., Xuejia L. “New field of cryptography: DNA Cryptography”, Chinese Sci. Bull. 51 (2006) : 1413-1420.
10. [10] Hesketh, E. E., Sayir, J., Goldman, N. “Improving communication for interdisciplinary teams working on storage of digital information in DNA”.F1000Research.7, 39 (2018).
11. [11] Liebovitch, L. S., Tao, Y., Todorov, A. T., Levine, L. “Is there an error correcting code in the base sequence in DNA?” Biophys J. 71(3) (1996) : 1539-1544.
12. [12] Oztas, E.S., Siap., I. “Lifted polynomials over F16 and their applications to DNA Codes”, Filomat. 27(3) (2013) : 459-466.
13. [13] Oztas, E.S., Siap, I. “On a generalization of lifted polynomials over finite fields and their applications to DNA codes”, Int. J. Comput. Math. 92(9) (2015) : 1976-1988.
14. [14] Oztas, E. S., Yildiz, B., Siap, I. “On DNA codes from a family of chain rings”, J. Algebra Comb. Discrete Struct. Appl. 4(1) (2017) : 93-102.
15. [15] Siap, I., Abulraub, T., Ghrayeb, A. “Cyclic DNA codes over the ring F2[u]/(u2 −1) based on the deletion distance”, J. Franklin Inst. 346(8) (2009) : 731-740.
16. [16] Ubaidur Rahman, N. H., Balamurugan, C., Mariappan, R.: A Novel DNA Computing Based Encryption and Decryption Algorithm. Procedia Comput. Sci. 46 (2015) : 463-475.
17. [17] Wang, X., Bao, Z., Hu, J., Wang, S., Zhan, A.: Solving the SAT problem using a DNA computing algorithm based on ligase chain reaction. BioSystems. 91(1) (2008) : 117-125.
18. [18] Yildiz, B., Siap, I. “Cyclic codes over F2[u]/(u4 − 1) and applications to DNA codes” Comput. Math. Appl. 63(7) (2012) : 1169-1176.
19. [19] Yilgor, M. B., Gursoy, F. , Oztas,E. S., Demirkale, F., “Cyclic codes over F2 + uF2 +vF2 + v2F2 with respect to the homogeneous weight and their applications to DNA codes.” AAECC, 32 (2021), 621-636.
20. [20] Merve B. Yilgor, “Cyclic codes over F2[u, v, w]/(u2 = v2, uv = 0, w2 =w) and its applications”, AIMS Mathematics, 2025, 10(12): 28396- 28406.
21. [21] Massey J.L. “Reversible codes”, Inf. Control 7 (1964) : 369-380.