Noncyclic BCH and Srivastava codes over Eisenstein integers toward next-generation error-correcting codes

Muhammad Sajjad; Nawaf A. Alqwaifly; Muhammad Sajjad; Nawaf A. Alqwaifly

doi:10.3934/math.2026015

AIMS Mathematics

2026, Volume 11, Issue 1: 353-365. doi: 10.3934/math.2026015

Previous Article Next Article

Research article Special Issues

Noncyclic BCH and Srivastava codes over Eisenstein integers toward next-generation error-correcting codes

Muhammad Sajjad ^{1
,
,},
Nawaf A. Alqwaifly ^{2
,
,}

1.
NUTECH School of Applied Science and Humanities, National University of Technology, Islamabad, 44000, Pakistan
2.
Department of Electrical Engineering, College of Engineering, Qassim University, Buraydah, Saudi Arabia

Received: 08 August 2025 Revised: 22 November 2025 Accepted: 12 December 2025 Published: 05 January 2026
MSC : 11T71, 68P30, 94A24, 97G70

This article investigates noncyclic BCH and Srivastava codes over Eisenstein integer fields $ \mathbb{Z}_p[\omega] $, where $ p \equiv 2 \pmod{3} $. By leveraging the algebraic structure of Eisenstein fields, we construct parity-check matrices with an alternant structure such that all maximal-order determinants satisfy the required algebraic conditions. These constructions provide explicit lower bounds on the minimum distance and enhance the error-correcting performance of the codes. The study further generalizes noncyclic forms of BCH and Srivastava codes, enabling higher code rates and larger minimum distances than their cyclic counterparts. Numerical examples illustrate the feasibility and effectiveness of the proposed constructions for reliable data transmission.
- Eisenstein integer codes,
- alternant codes,
- noncyclic BCH codes,
- Srivastava codes,
- minimum distance,
- signal processing,
- cryptography,
- wireless communications
Citation: Muhammad Sajjad, Nawaf A. Alqwaifly. Noncyclic BCH and Srivastava codes over Eisenstein integers toward next-generation error-correcting codes[J]. AIMS Mathematics, 2026, 11(1): 353-365. doi: 10.3934/math.2026015

Related Papers:

Abstract

This article investigates noncyclic BCH and Srivastava codes over Eisenstein integer fields $ \mathbb{Z}_p[\omega] $, where $ p \equiv 2 \pmod{3} $. By leveraging the algebraic structure of Eisenstein fields, we construct parity-check matrices with an alternant structure such that all maximal-order determinants satisfy the required algebraic conditions. These constructions provide explicit lower bounds on the minimum distance and enhance the error-correcting performance of the codes. The study further generalizes noncyclic forms of BCH and Srivastava codes, enabling higher code rates and larger minimum distances than their cyclic counterparts. Numerical examples illustrate the feasibility and effectiveness of the proposed constructions for reliable data transmission.

References

[1]	R. C. Bose, D. K. Ray-Chaudhuri, On a class of error correcting binary group codes, Inf. Control, 3 (1960), 68–79. https://doi.org/10.1016/S0019-9958(60)90287-4 doi: 10.1016/S0019-9958(60)90287-4
[2]	H. J. Helgert, Noncyclic generalizations of BCH and Srivastava codes, Inf. Control, 21 (1972), 280–290. https://doi.org/10.1016/S0019-9958(72)80007-X doi: 10.1016/S0019-9958(72)80007-X
[3]	E. Spiegel, Codes over $\mathbb{Z}_m$, Inf. Control, 35 (1977), 48–51. https://doi.org/10.1016/S0019-9958(77)90526-5 doi: 10.1016/S0019-9958(77)90526-5
[4]	E. Spiegel, Codes over $\mathbb{Z}_m$, revisited, Inf. Control, 37 (1978), 100–104. https://doi.org/10.1016/S0019-9958(78)90461-8 doi: 10.1016/S0019-9958(78)90461-8
[5]	E. R. Berlekamp, Algebraic coding theory, World Scientific Publishing Co., Inc., 2015.
[6]	T. Muir, W. H. Metzler, A treatise on the theory of determinants, New York: Dover Publications, 1960.
[7]	M. Sajjad, T. Shah, M. M. Hazzazi, A. R. Alharbi, I. Hussain, Quaternion integers based higher length cyclic codes and their decoding algorithm, CMC Comput. Mater. Continua, 73 (2022), 1177–1194. https://doi.org/10.32604/cmc.2022.025245 doi: 10.32604/cmc.2022.025245
[8]	M. Sajjad, T. Shah, M. Alammari, H. Alsaud, Construction and decoding of BCH-codes over the Gaussian field, IEEE Access, 11 (2023), 71972–71980. https://doi.org/10.1109/ACCESS.2023.3293007 doi: 10.1109/ACCESS.2023.3293007
[9]	M. Sajjad, T. Shah, Q. Xin, B. Almutairi, Eisenstein field BCH codes construction and decoding, AIMS Mathematics, 8 (2023), 29453–29473. https://doi.org/10.3934/math.20231508 doi: 10.3934/math.20231508
[10]	Y. Lei, C. Li, Y. Wu, P. Zeng, More results on hulls of some primitive binary and ternary BCH codes, Finite Fields Appl., 82 (2022), 102066. https://doi.org/10.1016/j.ffa.2022.102066 doi: 10.1016/j.ffa.2022.102066
[11]	G. Xu, G. Luo, X. Cao, H. Xu, Hulls of linear codes from simplex codes, Des. Codes Cryptogr., 92 (2024), 1095–1112. https://doi.org/10.1007/s10623-023-01331-4 doi: 10.1007/s10623-023-01331-4
[12]	Z. Sun, X. Liu, S. Zhu, Y. Tang, Negacyclic BCH codes of length $\frac{q^{2m-1}}{q+1}$ and their duals, Des. Codes Cryptogr., 92 (2024), 2085–2101. https://doi.org/10.1007/s10623-024-01380-3 doi: 10.1007/s10623-024-01380-3
[13]	F. Zullo, Multi-orbit cyclic subspace codes and linear sets, Finite Fields Appl., 87 (2023), 102153. https://doi.org/10.1016/j.ffa.2022.102153 doi: 10.1016/j.ffa.2022.102153
[14]	M. Sajjad, T. Shah, M. Abbas, M. Alammari, R. J. Serna, The impact of alternant codes over Eisenstein integers on modern technology, Comp. Appl. Math., 44 (2025), 95. https://doi.org/10.1007/s40314-024-03057-y doi: 10.1007/s40314-024-03057-y

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)