Reversible codes and applications to DNA codes over $ F_{4^{2t}}[u]/(u^2-1) $

Turki Alsuraiheed; Elif Segah Oztas; Shakir Ali; Merve Bulut Yilgor; Turki Alsuraiheed; Elif Segah Oztas; Shakir Ali; Merve Bulut Yilgor

doi:10.3934/math.20231421

AIMS Mathematics

2023, Volume 8, Issue 11: 27762-27774. doi: 10.3934/math.20231421

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Research article

Reversible codes and applications to DNA codes over $ F_{4^{2t}}[u]/(u^2-1) $

1.
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
2.
Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, Turkiye
3.
Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India
4.
Department of Basic Sciences, Altinbas University, Turkiye

Received: 14 August 2023 Revised: 16 September 2023 Accepted: 24 September 2023 Published: 08 October 2023
MSC : 94B05, 94B60, 94B65

Let $ n \geq 1 $ be a fixed integer. Within this study, we present a novel approach for discovering reversible codes over rings, leveraging the concept of $ r $-glifted polynomials. This technique allows us to achieve optimal reversible codes. As we extend our methodology to the domain of DNA codes, we establish a correspondence between $ 4t $-bases of DNA and elements within the ring $ R_{2t} = F_{4^{2t}}[u]/(u^{2}-1) $. By employing a variant of $ r $-glifted polynomials, we successfully address the challenges of reversibility and complementarity in DNA codes over this specific ring. Moreover, we are able to generate reversible and reversible-complement DNA codes that transcend the limitations of being linear cyclic codes generated by a factor of $ x^n-1 $.
- reversible code,
- DNA code,
- $ r $-glifted polynomial
Citation: Turki Alsuraiheed, Elif Segah Oztas, Shakir Ali, Merve Bulut Yilgor. Reversible codes and applications to DNA codes over $ F_{4^{2t}}[u]/(u^2-1) $[J]. AIMS Mathematics, 2023, 8(11): 27762-27774. doi: 10.3934/math.20231421

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Abstract

Let $ n \geq 1 $ be a fixed integer. Within this study, we present a novel approach for discovering reversible codes over rings, leveraging the concept of $ r $-glifted polynomials. This technique allows us to achieve optimal reversible codes. As we extend our methodology to the domain of DNA codes, we establish a correspondence between $ 4t $-bases of DNA and elements within the ring $ R_{2t} = F_{4^{2t}}[u]/(u^{2}-1) $. By employing a variant of $ r $-glifted polynomials, we successfully address the challenges of reversibility and complementarity in DNA codes over this specific ring. Moreover, we are able to generate reversible and reversible-complement DNA codes that transcend the limitations of being linear cyclic codes generated by a factor of $ x^n-1 $.

References

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