Investigation into the structure of finite Abelian groups and their applications in linear coding theory

Articles in Press

Document Type : Research Paper

Author
Ebrahim Zangooiezadeh
Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
10.22075/ijnaa.2026.38039.5489
Abstract
This article presents a comprehensive analysis of the application of finite abelian groups in linear coding theory. In the last few decades, algebraic coding theory has found widespread applications in various disciplines due to its rich, fascinating mathematical structure. We examine the theoretical foundations, structural properties, and practical implementations of group codes, with particular emphasis on their capacity characteristics and error-correcting capabilities. We show how such techniques can be generalized to study systems and codes defined over Abelian groups. These findings underscore their growing relevance in modern fields such as post-quantum cryptography and quantum error correction. The analysis includes detailed mathematical formulations, theorems governing group code behavior, and concrete examples demonstrating their effectiveness in various communication scenarios. The capacity of finite Abelian group codes over symmetric memoryless channels is determined.
Keywords
Finite abelian groups
linear codes
group algebras
coding theory
error correction
[1] T.G. Alabi, Direct product of finite Abelian group, Int. J. Appl. Sci. Res. 5 (2021), no. 5, 166–173.
[2] G.N. Alfarano, D. Napp, A. Neri, and V. Requena, Weighted Reed‑Solomon convolutional codes, arXiv:2012.11417, 2020, https://doi.org/10.48550/arXiv.2012.11417.
[3] K. Amin, Constructing single‑error‑correcting codes using factorization of finite Abelian groups, Int. J. Algebra 8 (2014), no. 7, 311–315.
[4] S. Benson, Students ask the darnedest things: A result in elementary group theory, Math. Mag. 70 (1997), no. 3, 207–211.
[5] S. Bhowmicka, A. Fotue Tabueb, and J. Pal, On the ℓℓ‑DLIPs of codes over finite commutative rings, Discrete Math. 347 (2020), no. 4, 12–23.
[6] J. Chen, R.M. Tanner, J. Zhang, and P.C. Marc, Construction of irregular LDPC codes by quasi‑cyclic extension, IEEE Trans. Inform. Theory 53 (2007), no. 4, 1479–1483.
[7] K.K. Cheung and M. Mosca, Decomposing finite Abelian groups, arXiv:cs.DS/0101004, 2001.
[8] G. Como and F. Fagnani, The capacity of finite abelian group codes over symmetric memoryless channels, IEEE Trans. Inf. Theory 55 (2009), no. 5, 2037–2054.
[9] T. Crilly, A first course in group theory, by Cyril F. Gardiner, Math. Gaz. 65 (1981), no. 434, 314–315.
[10] M. Devos, The structure of critical product sets, arXiv:1301.0096, 2013.
[11] F. Fagnani and S. Zampieri, Dynamical systems and convolutional codes over finite abelian groups, IEEE Trans. Inf. Theory 42 (1996), no. 6, 1892–1912.
[12] L. Fuchs, Abelian Groups, 1st ed., CRC Press, 1993.
[13] J.T. Griesmer, Semicontinuity of structure for small sumsets in compact abelian groups, Discrete Anal. 2019 (2019), Paper No. 18, 46.
[14] J.I. Hall, Notes on Coding Theory, Chapter 5, Generalized Reed‑Solomon Codes, 2015.
[15] J.N. Henry, Groups Satisfying the Converse to Lagrange's Theorem, Master's Thesis, Missouri State University, 2019.
[16] L. Honold, Linear codes over finite chain rings, Electron. J. Combin. 7 (2000), no. 2, 1–22.
[17] S. Jitman, Complementary dual abelian codes in group algebras of some finite abelian groups, Proc. 19th IMT‑GT Int. Conf. Math. Stat. Appl. (ICMSA), 2024.
[18] J. Justesen, New convolutional code constructions and a class of asymptotically good time‑varying codes, IEEE Trans. Inform. Theory 19 (1973), no. 2, 220–225.
[19] N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), no. 10, 203–209.
[20] L.E. Laurent, A classification of groups satisfying the converse of Lagrange's theorem, University of Nevada, Las Vegas, 1995.
[21] Y. Li and J.Y. Li, Quantum coding via quasi‑cyclic block matrix, Entropy 3 (2023), no. 6, 537–550.
[22] B. Lynn, Group Theory, Lagrange's Theorem, Applied Crypto Group, Stanford University, 2024.
[23] J. Massey, Minimal Codewords and Secret Sharing, Proc. 6th Joint Swedish‑Russian Int. Workshop Inf. Theory, 1993.
[24] S. Mesnager, A. Sinak, and A. Yayla, Minimal linear codes with few weights and their secret sharing, Int. J. Inf. Secur. Sci. 8 (2019), no. 4, 77–87.
[25] A. Ovchinnikov, A. Veresova, and A. Fominykh, Decoding of linear codes for single error bursts correction based on the determination of certain events, Inf. Control Syst. 6 (2022), no. 11, 41–52.
[26] J. Petrillo, Counting subgroups in a direct product of finite cyclic groups, College Math. J. 42 (2011), no. 3, 215–222.
[27] A. Piro, The Fundamental Theorem for Finite Abelian Groups: A Brief History and Proof, Georgia College & State University, Department of Mathematics, Milledgeville, GA 31061, United States, 2015.
[28] S. Pumpluen, Using cyclic (f,δ)(f,δ)‑codes over finite chain rings to construct ZpZp​‑ and FqFq​‑lattices, arXiv:2501.10838, 2025.
[29] R. Raja, Codes and Orbit Covers of Finite Abelian Groups, arXiv:2306.15387, 2022, 1–12.
[30] A. Riaz, S. Kousar, N. Kausar, D. Pamucar, and G.M. Addis, An analysis of algebraic codes over lattice valued intuitionistic fuzzy type‑submodules, Comput. Intell. Neurosci. 2022 (2022), no. 1, 8148284.
[31] D.M. Russinoff, A formalization of finite group theory: Part II, Electron. Proc. Theor. Comput. Sci. 393 (2023), no. 12, 16–32.
[32] V. Sagar and R. Sarma, Codes Over the Non‑Unital Non‑Commutative Ring E Using Simplicial Complexes, arXiv:2304.06758, 2023.
[33] T.N. Shankar and G. Sahoo, Cryptography with elliptic curves, Int. J. Comput. Sci. Appl. 2 (2009), no. 1, 38–42.
[34] S.A. Spence, Introduction to Algebraic Coding Theory, Supplementary material for Math 336, Cornell University, 2018, 1–19.
[35] D.B. Surowski, The uniqueness aspect of the fundamental theorem of finite Abelian groups, Amer. Math. Monthly 102 (1995), no. 2, 162–163.
[36] J.A. Tolar, Classification of finite quantum kinematics, J. Phys. Conf. Ser. 538 (2014), 012020.
[37] Wikipedia, Lagrange's theorem (group theory), 2025, https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory).
[38] K. Yasunaga and T. Fujiwara, On correctable errors of binary linear codes, IEEE Trans. Inf. Theory 56 (2010), no. 6, 2537–2548.
[39] C. Yu, S. Zhu, H. Chen, Y. Li, and X. Zhang, New Record‑Breaking Binary Linear Codes Constructed from Group Codes, arXiv:2412.15551, 2022.
[40] W. Zhao, Study on Cauchy theorem as a special case of Sylow theorem, Highlights Sci. Eng. Technol. 128 (2025), 91–96.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 05 October 2026

  • Receive Date 09 June 2025
  • Accept Date 11 August 2026