Ring-theoretic analysis of error-correcting codes
R. Shantha Mary Joshittarjoshitta@gmail.comDepartment of Computer ScienceJayaraj Annapackiam College for Women (Autonomous)Affiliated with Mother Teresa Women’s UniversityPeriyakulam, Tamil Nadu, 625601, IndiaView full profile → , *Sachin Sakharam GaikwadCorresponding authorsachingaikwad@aitpune.edu.inDepartment of Information TechnologyArmy Institute of TechnologyDighi, Pune, Maharashtra, 411015, IndiaView full profile → , Saumya Goyalsaumyaqsb@quantumeducation.inQuantum University Research CenterQuantum UniversityRoorkee, Uttarakhand, 247667, India0009-0008-4449-6366View full profile → , Narendra Mohannarendra.mohan@gla.ac.inDepartment of Computer Engineering & ApplicationsGLA UniversityMathura, Uttar Pradesh, 281406, IndiaView full profile → , S. Suganyassuganyamba@maher.ac.inDepartment of Management StudiesMeenakshi College of Arts and ScienceMeenakshi Academy of Higher Education & Research (Deemed to be University)Chennai, Tamil Nadu, 600078, IndiaView full profile → , Namrata Somnath Bajarenamrata.bajare@vit.eduDepartment of Computer EngineeringVishwakarma Institute of TechnologyPune, Maharashtra, 411037, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 31 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-3045
- Pages:
- 4213–4224
Abstract
Checking codes play a role in ensuring credibility in delivering information in the noisy communication channels. The paper is a detailed ring-theoretic analysis of error-correcting codes, in the algebraic structure of the linear, cyclic and constacyclic codes of finite rings. In certain applications, the ring-based coding structures are more flexible, have better structural richness and better error-handling capabilities than classical field-based approaches. Algebraic definitions of linear codes (as modules), the construction of cyclic and constacyclic codes in commutative rings, and the geometry of such codes (duality, minimal generating sets and ideal decomposition) are discussed in the paper. In addition, the efficient decoding algorithms which are unique to the ring-based codes are discussed with a focus on their efficiency in computation, and their use in practice. The results indicate that the codes using the ring-theoretic methods are far superior in code performance and flexibility and can thus be used in the existing communication system, e.g., storage, wireless resource and cryptographic protocols.
Keywords
Subject Classifications
References
[1] N. Gassner, M. Greferath, J. Rosenthal, and V. Weger, “Bounds for coding theory over rings,” Entropy, vol. 24, no. 10, Art. no. 1473 (2022), doi: 10.3390/e24101473.
[2] A. N. Alkenani, M. Ashraf, and G. Mohammad, “Quantum Codes from Constacyclic Codes over the Ring Fq[u1,u2]/〈 u 1 2 -u1, u 2 2 -u2,u1u2-u2u1〉,” Mathematics, vol. 8, no. 5, Art. no. 781 (2020), doi: 10.3390/math8050781.
[3] D. Camazón Portela and J. A. López Ramos, “Error-correcting codes on projective bundles over Deligne–Lusztig varieties,” Mathematics, vol. 11, no. 14, Art. no. 3079 (2023), doi: 10.3390/math11143079.
[4] G. G. Jadhav, P. D. Patil, and V. S. Kumbhar, “Signature verification system using machine learning and image processing,” International Journal on Advanced Computer Engineering and Communication Technology, vol. 14, no. 1, pp. 108–114 (2025), doi: 10.65521/ijacect.v14i1.202.
[5] H. Q. Dinh, H. L. Thi, and R. Tansuchat, “γ-dual codes over finite commutative chain rings,” Axioms, vol. 13, no. 10, Art. no. 655 (2024), doi: 10.3390/axioms13100655.
[6] H. Bordbar, “Computational algebra, coding theory, and cryptography: Theory and applications,” Axioms, vol. 13, no. 11, Art. no. 784 (2024), doi: 10.3390/axioms13110784.
[7] H. Kostadinov and N. Manev, “A general construction of integer codes correcting specific errors in binary communication channels,” Mathematics, vol. 11, no. 11, Art. no. 2521 (2023), doi: 10.3390/math11112521.
[8] M. Bras-Amorós, “Ideals of numerical semigroups and error-correcting codes,” Symmetry, vol. 11, no. 11, Art. no. 1406 (2019), doi: 10.3390/sym11111406.
[9] J. Ettayb, “Pseudospectra and condition pseudospectra of multivalued linear operators on non-Archimedean Banach spaces,” Journal of Interdisciplinary Mathematics, vol. 28, no. 7, pp. 2439–2457 (2025), doi: 10.47974/JIM-1900.
[10] P. Khobragade, G. Gandal, A. Somatkar, H. M. Panchabudhe, P. P. Ashok, and B. Madaminov, “Heat transfer modeling using fractional differential equations,” J. Interdiscip. Math., vol. 29, no. 3, pp. 749–756 (2026), doi: 10.47974/JIM-2511.
[11] A. Alahmadi, M. Altaiary, and P. Solé, “Cyclic codes over a non-commutative non-unital ring,” Mathematics, vol. 12, no. 13, Art. no. 2014 (2024), doi: 10.3390/math12132014.
[12] Á. L. Muñoz Castañeda, N. DeCastro-García, and M. V. Carriegos, “On the state approach representations of convolutional codes over rings of modular integers,” Mathematics, vol. 9, no. 22, Art. no. 2962 (2021), doi: 10.3390/math9222962.
[13] M. Ashraf, A. S. Alali, M. Asim, and G. Mohammad, “(β,γ)-skew QC codes with derivation over a semi-local ring,” Symmetry, vol. 15, no. 1, Art. no. 225 (2023), doi: 10.3390/sym15010225.
[14] R. C. Poonia, R. Almakki, A. K. J. Saudagar, A. Altameem, and M. Albathan, “An optimization approach for real-time object detection in IoT devices through edge computing and deep learning,” Journal of Information and Optimization Sciences, vol. 45, no. 5, pp. 1465–1475 (2024), doi: 10.47974/JIOS-1768.
