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Entropy
Volume 27
Issue 12
10.3390/e27121187
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

On Double Cyclic Codes over Finite Chain Rings for DNA Computing

by
Shakir Ali
1,2,*,
Amal S. Alali
3,
Mohd Azeem
1,
Atif Ahmad Khan
1 and
Kok Bin Wong
2
1
Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India
2
Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia
3
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
*
Author to whom correspondence should be addressed.
Entropy 2025, 27(12), 1187; https://doi.org/10.3390/e27121187 (registering DOI)
Submission received: 18 October 2025 / Revised: 20 November 2025 / Accepted: 20 November 2025 / Published: 22 November 2025
(This article belongs to the Section Information Theory, Probability and Statistics)
Versions Notes

Abstract

Let e be a fixed positive integer and n1,n2 be odd positive integers. The main objective of this article is to investigate the algebraic structure of double cyclic codes of length (n1,n2) over the finite chain ring eR=F4e+vF4e, where v2=0. Building upon this structural framework, we further demonstrate the construction of DNA codes derived from these double cyclic codes over eR. In addition, we provide the necessary and sufficient criteria showing that these codes possess reversibility and reverse-complement properties over eR. Furthermore, we introduce a generalized Gray map that extends the classical Gray map from the ring F2+vF2 with v2=0 to the ring eR, showing a direct correspondence between elements of eR and DNA sequences over S={A,T,G,C} utilizing double cyclic codes. To illustrate the applicability of our results, we present some examples demonstrating the effectiveness of the mapping in generating reversible and reverse-complement DNA codes from algebraic structures over the ring eR.
Keywords: double cyclic code; chain ring; DNA code; reversible code; reversible-complement code double cyclic code; chain ring; DNA code; reversible code; reversible-complement code

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MDPI and ACS Style

Ali, S.; Alali, A.S.; Azeem, M.; Khan, A.A.; Wong, K.B. On Double Cyclic Codes over Finite Chain Rings for DNA Computing. Entropy 2025, 27, 1187. https://doi.org/10.3390/e27121187

AMA Style

Ali S, Alali AS, Azeem M, Khan AA, Wong KB. On Double Cyclic Codes over Finite Chain Rings for DNA Computing. Entropy. 2025; 27(12):1187. https://doi.org/10.3390/e27121187

Chicago/Turabian Style

Ali, Shakir, Amal S. Alali, Mohd Azeem, Atif Ahmad Khan, and Kok Bin Wong. 2025. "On Double Cyclic Codes over Finite Chain Rings for DNA Computing" Entropy 27, no. 12: 1187. https://doi.org/10.3390/e27121187

APA Style

Ali, S., Alali, A. S., Azeem, M., Khan, A. A., & Wong, K. B. (2025). On Double Cyclic Codes over Finite Chain Rings for DNA Computing. Entropy, 27(12), 1187. https://doi.org/10.3390/e27121187

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