Design and Analysis of Shortened Bose-Chaudhuri-Hocquenghem Codes for Efficient Data Transmission over the Eisenstein Fields

Authors

  • Muhammad Sajjad National University of Technology (NUTECH), Islamabad, Pakistan https://orcid.org/0000-0003-0006-1156
  • Rida Asghar Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
  • Mushtaq K. Abdalrahem College of Pharmacy, University of Al-Ameed
  • Adnan Burhan Rajab Department of Computer Engineering, College of Engineering, Knowledge University, Erbil 12 44001, Iraq
  • Mahesha Narayana Department of Mathematics, The University of the West Indies, Kingston 7, Jamaica
  • Moin-ud-Din Junjua School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China
  • Salman A. AlQahtani New Emerging Technologies and 5G Network and Beyond Research Chair, Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i3.6275

Keywords:

Eisenstein field; Shortened BCH codes; Error correcting codes; Modified BMA

Abstract

This article focuses on the constructing and decoding for Shortened Bose-Chaudhuri-Hocquenghem (BCH) codes over the Eisenstein fields, based on the Berlekamp-Massey Algorithm (BMA) with improved decoding performance. Thus, Eisenstein fields, being a natural generalization of Gaussian fields, form a well-devised algebraic structure for error-correcting codes and have better parameters for transmission of information in noisy channels. The work starts with the definition of shortened BCH codes over the Eisenstein fields with the mention of generator polynomials and the study of certain algebraic characteristics. These codes are designed to provide the best balance between the words length and the ability to cope with error detection and correction in variety of high-efficiency applications. The modified BMA is presented as a decoding mechanism that is equipped for the translation of Eisenstein field arithmetic which is different in nature. The modification concerns an improved approach to the residual terms defined by the classes of higher level, which leads to the efficiency of convergence and the reduction of numerical complexity in contrast to the BMA. The imitations show that the proposed codes offer a higher error correcting capability and faster computation compared to the traditional BCH codes over finite fields when reliability and low latency are desired.

Author Biography

  • Muhammad Sajjad, National University of Technology (NUTECH), Islamabad, Pakistan

    Dr. Muhammad Sajjad is a dedicated mathematician from Pakistan, currently serving as an Assistant Professor at the National University of Technology (NUTECH), Islamabad. He earned his PhD from Quaid-i-Azam University Islamabad, Pakistan, in 2024 and is an esteemed member of the American Mathematical Society (AMS). Recognised for his outstanding contributions to cybersecurity and cryptography, Dr. Sajjad has received several prestigious accolades, including two international research awards in cybersecurity and cryptography, the Computer Scientist Award from China, and the National Cryptology Award from NCCS Pakistan. His research spans multiple disciplines, including mathematics, electrical engineering, and computer science, with a strong emphasis on vector algebras, non-commutative and non-associative algebras, number theory, coding theory, channel coding, cryptography, cryptology, post-quantum cryptography, graph theory, and elliptic curves. His work is particularly focused on designing efficient error-correcting codes, advancing data security, and exploring complex algebraic structures. Dr. Sajjad is an active contributor to the global academic community, having delivered talks at prestigious international conferences. His research collaborations extend to leading universities in South America, Denmark, and Saudi Arabia, further strengthening his impact on the field. He has undergone professional training in various aspects of academic management and possesses strong programming skills in Python, MATLAB, C++, and more. Over his academic career, he has taught 21 bachelor and master's courses and published 13 research papers. Through his dedication to teaching, research, and innovation, Dr. Muhammad Sajjad continues to shape the future of mathematical sciences and information security.

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Published

2025-08-01

Issue

Vol. 18 No. 3: (July 2025)

Section

Coding Theory and Cryptology

License

Copyright (c) 2025 Muhammad Sajjad, Rida Asghar, Mushtaq K. Abdalrahem, Adnan Burhan Rajab, Mahesha Narayana, Moin-ud-Din Junjua, Salman A. AlQahtani

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How to Cite

Design and Analysis of Shortened Bose-Chaudhuri-Hocquenghem Codes for Efficient Data Transmission over the Eisenstein Fields. (2025). European Journal of Pure and Applied Mathematics, 18(3), 6275. https://doi.org/10.29020/nybg.ejpam.v18i3.6275