Application-Oriented Study of Next-Generation Alternant Codes over Gaussian Integers for Secure and Efficient Communication

Muhammad Sajjad () and Nawaf A. Alqwaifly ()
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Muhammad Sajjad: NUTECH School of Applied Science and Humanities, National University of Technology, Islamabad 44000, Pakistan
Nawaf A. Alqwaifly: Department of Electrical Engineering, College of Engineering, Qassim University, Buraydah 52571, Saudi Arabia

Mathematics, 2025, vol. 13, issue 14, 1-13

Abstract: This paper presents the construction and analysis of a novel class of alternant codes over Gaussian integers, aimed at enhancing error correction capabilities in high-reliability communication systems. These codes are constructed using parity-check matrices derived from finite commutative local rings with unity, specifically Z n [ i ] , where i 2 = − 1 . A detailed algebraic investigation of the polynomial x n − 1 over these rings is conducted to facilitate the systematic construction of such codes. The proposed alternant codes extend the principles of classical BCH and Goppa codes to complex integer domains, enabling richer algebraic structures and greater error-correction potential. We evaluate the performance of these codes in terms of error correction capability, and redundancy. Numerical results show that the proposed codes outperform classical short-length codes in scenarios requiring moderate block lengths, such as those applicable in certain segments of 5G and IoT networks. Unlike conventional codes, these constructions allow enhanced structural flexibility that can be tuned for various application-specific parameters. While the potential relevance to quantum-safe communication is acknowledged, it is not the primary focus of this study. This work demonstrates how extending classical coding techniques into non-traditional algebraic domains opens up new directions for designing robust and efficient communication codes.

Keywords: alternant codes; Gaussian integers; finite commutative local rings; secure communication; wireless communications; digital storage; source codes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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