A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes

Akram Saleh; Mohammad Reza Soleymani

doi:doi:10.11648/j.ajist.20261001.13

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A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes

Akram Saleh^*

, Mohammad Reza Soleymani

Published in American Journal of Information Science and Technology (Volume 10, Issue 1)

Received: 15 December 2025 Accepted: 26 December 2025 Published: 20 January 2026

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Abstract

This paper investigates a construction algorithm for two specific families of quasi-cyclic codes defined over a finite commutative chain ring. First, by employing the Generalized Discrete Fourier Transform, we develop an efficient and systematic algorithm for constructing the generator matrix of repeated-root quasi-cyclic codes under specific structural constraints on the code length and the underlying ring. The proposed method avoids the need for exhaustive enumeration of constituent subcodes and instead operates directly on their generator matrices, leading to improved computational performance. Building on this result, using the Discrete Fourier Transform, we further specialize the proposed framework to derive an algorithm for obtaining the generator matrix of a particular class of simple-root quasi-cyclic codes over a restricted and well-defined category of rings. This specialization demonstrates the performance of the proposed approach and highlights its applicability to different quasi-cyclic code structures in a unified algebraic setting. The proposed construction methods offer significant improvements in computational efficiency when compared to existing approaches that rely on code construction techniques based on constituent subcodes. To evaluate the practical benefits of the proposed algorithms, we present a detailed performance comparison in terms of encoding time per iteration and memory consumption. The comparative results, illustrated through histograms, clearly indicate that the proposed methods achieve faster encoding, lower memory usage, highlighting their superior performance compared to conventional construction techniques as the code length increases.

Published in	American Journal of Information Science and Technology (Volume 10, Issue 1)
DOI	10.11648/j.ajist.20261001.13
Page(s)	15-24
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Quasi-cyclic Codes, Repeated-root Codes, Simple-root Codes, Chain Rings, Discrete Fourier Transform

References

[1]	N. Aydin and I. Siap, “New quasi-cyclic codes over F5,” Applied Mathematics Letters, vol. 15, no. 7, pp. 833–836, Oct. 2002. https://doi.org/10.1016/S0893-9659(02)00050-2
[2]	Z. Chen, “Six new binary quasi-cyclic codes,” IEEE Transactions on Information Theory, vol. 40, no. 5, pp. 1666–1667, 1994. https://doi.org/10.1109/18.333888
[3]	E. Z. Chen, “New quasi-cyclic codes from simplex codes,” IEEE Transactions on Information Theory, vol. 53, no. 3, pp. 1193–1196, 2007. https://doi.org/10.1109/TIT.2006.890727
[4]	R. Daskalov, P. Hristov, and E. Metodieva, “New minimum distance bounds for linear codes over GF (5),” Discrete Mathematics, vol. 275, no. 1–3, pp. 97–110, 2004. https://doi.org/10.1016/S0012-365X(03)00126-2
[5]	A. Saleh and M. R. Soleymani, “A novel framework for relating quasi-cyclic codes and quasi-twisted codes,” in 2023 Biennial Symposium on Communications (BSC), pp. 38–41, IEEE, Jul. 2023. https://doi.org/10.1109/BSC57238.2023.10201835
[6]	A. Saleh and M. Soleymani, “A novel construction technique for some classes of quasi-cyclic codes,” in 2023 IEEE International Symposium on Information Theory (ISIT), pp. 591–595, IEEE, Jun. 2023. https://doi.org/10.1109/ISIT54713.2023.10206961
[7]	M. N. Danish, S. A. Pasha, and A. J. Hashmi, “Quasi-cyclic LDPC codes for short block-lengths,” in 2021 IEEE Aerospace Conference, pp. 1–8, IEEE, 2021. https://doi.org/10.1109/AERO50100.2021.9438238
[8]	M. I. Hidayat, Irwansyah, and I. G. A. W. Wardhana, “A construction of generalized quasi-cyclic codes over finite field using Gray map,” AIP Conference Proceedings, vol. 2641, no. 1, 2022. https://doi.org/10.1063/5.0114988
[9]	S. Benjwal, M. Bhaintwal, and R. Kumar, “On quantum codes derived from quasi-cyclic codes over a non-chain ring,” Quantum Information Processing, vol. 23, no. 9, art. no. 309, 2024. https://doi.org/10.1007/S11128-024-04514-7
[10]	S. Ling and P. Solé, “On the algebraic structure of quasi-cyclic codes. I: Finite fields,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 2751–2760, 2001. https://doi.org/10.1109/18.959257
[11]	S. Ling and P. Solé, “On the algebraic structure of quasi-cyclic codes II: Chain rings,” Designs, Codes and Cryptography, vol. 30, pp. 113–130, 2003. https://doi.org/10.1023/A:1024715527805
[12]	S. Ling, P. Solé, and H. Niederreiter, “On the algebraic structure of quasi-cyclic codes IV: Repeated roots,” Designs, Codes and Cryptography, vol. 38, no. 3, pp. 337–361, 2006. https://doi.org/10.1007/S10623-005-1431-7
[13]	A. Saleh and M. Esmaeili, “Some classes of quasi-twisted codes over finite chain rings,” Journal of Applied Mathematics and Computing, vol. 57, no. 1, pp. 629–646, 2018. https://doi.org/10.1007/S12190-017-1125-0
[14]	M. F. Ezerman, M. Grassl, S. Ling, F. Ozbudak, andB. Ozkaya, “Characterization of nearly self-orthogonal quasi-twisted codes and related quantum codes,” IEEE Transactions on Information Theory, vol. 71, no. 1, pp. 499–517, 2024. https://doi.org/10.1109/TIT.2024.3503420
[15]	B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.

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Saleh, A., Soleymani, M. R. (2026). A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes. American Journal of Information Science and Technology, 10(1), 15-24. https://doi.org/10.11648/j.ajist.20261001.13

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ACS Style

Saleh, A.; Soleymani, M. R. A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes. Am. J. Inf. Sci. Technol. 2026, 10(1), 15-24. doi: 10.11648/j.ajist.20261001.13

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AMA Style

Saleh A, Soleymani MR. A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes. Am J Inf Sci Technol. 2026;10(1):15-24. doi: 10.11648/j.ajist.20261001.13

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@article{10.11648/j.ajist.20261001.13,
  author = {Akram Saleh and Mohammad Reza Soleymani},
  title = {A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes
},
  journal = {American Journal of Information Science and Technology},
  volume = {10},
  number = {1},
  pages = {15-24},
  doi = {10.11648/j.ajist.20261001.13},
  url = {https://doi.org/10.11648/j.ajist.20261001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajist.20261001.13},
  abstract = {This paper investigates a construction algorithm for two specific families of quasi-cyclic codes defined over a finite commutative chain ring. First, by employing the Generalized Discrete Fourier Transform, we develop an efficient and systematic algorithm for constructing the generator matrix of repeated-root quasi-cyclic codes under specific structural constraints on the code length and the underlying ring. The proposed method avoids the need for exhaustive enumeration of constituent subcodes and instead operates directly on their generator matrices, leading to improved computational performance. Building on this result, using the Discrete Fourier Transform, we further specialize the proposed framework to derive an algorithm for obtaining the generator matrix of a particular class of simple-root quasi-cyclic codes over a restricted and well-defined category of rings. This specialization demonstrates the performance of the proposed approach and highlights its applicability to different quasi-cyclic code structures in a unified algebraic setting. The proposed construction methods offer significant improvements in computational efficiency when compared to existing approaches that rely on code construction techniques based on constituent subcodes. To evaluate the practical benefits of the proposed algorithms, we present a detailed performance comparison in terms of encoding time per iteration and memory consumption. The comparative results, illustrated through histograms, clearly indicate that the proposed methods achieve faster encoding, lower memory usage, highlighting their superior performance compared to conventional construction techniques as the code length increases.
},
 year = {2026}
}

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TY  - JOUR
T1  - A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes

AU  - Akram Saleh
AU  - Mohammad Reza Soleymani
Y1  - 2026/01/20
PY  - 2026
N1  - https://doi.org/10.11648/j.ajist.20261001.13
DO  - 10.11648/j.ajist.20261001.13
T2  - American Journal of Information Science and Technology
JF  - American Journal of Information Science and Technology
JO  - American Journal of Information Science and Technology
SP  - 15
EP  - 24
PB  - Science Publishing Group
SN  - 2640-0588
UR  - https://doi.org/10.11648/j.ajist.20261001.13
AB  - This paper investigates a construction algorithm for two specific families of quasi-cyclic codes defined over a finite commutative chain ring. First, by employing the Generalized Discrete Fourier Transform, we develop an efficient and systematic algorithm for constructing the generator matrix of repeated-root quasi-cyclic codes under specific structural constraints on the code length and the underlying ring. The proposed method avoids the need for exhaustive enumeration of constituent subcodes and instead operates directly on their generator matrices, leading to improved computational performance. Building on this result, using the Discrete Fourier Transform, we further specialize the proposed framework to derive an algorithm for obtaining the generator matrix of a particular class of simple-root quasi-cyclic codes over a restricted and well-defined category of rings. This specialization demonstrates the performance of the proposed approach and highlights its applicability to different quasi-cyclic code structures in a unified algebraic setting. The proposed construction methods offer significant improvements in computational efficiency when compared to existing approaches that rely on code construction techniques based on constituent subcodes. To evaluate the practical benefits of the proposed algorithms, we present a detailed performance comparison in terms of encoding time per iteration and memory consumption. The comparative results, illustrated through histograms, clearly indicate that the proposed methods achieve faster encoding, lower memory usage, highlighting their superior performance compared to conventional construction techniques as the code length increases.

VL  - 10
IS  - 1
ER  -

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Author Information

Akram Saleh

Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada

Contact Email

http://orcid.org/0009-0004-3959-6929
Mohammad Reza Soleymani

Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada

Contact Email

http://orcid.org/0009-0005-6805-4341

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APA Style

Saleh, A., Soleymani, M. R. (2026). A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes. American Journal of Information Science and Technology, 10(1), 15-24. https://doi.org/10.11648/j.ajist.20261001.13

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ACS Style

Saleh, A.; Soleymani, M. R. A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes. Am. J. Inf. Sci. Technol. 2026, 10(1), 15-24. doi: 10.11648/j.ajist.20261001.13

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AMA Style

Saleh A, Soleymani MR. A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes. Am J Inf Sci Technol. 2026;10(1):15-24. doi: 10.11648/j.ajist.20261001.13

Copy | Download

@article{10.11648/j.ajist.20261001.13,
  author = {Akram Saleh and Mohammad Reza Soleymani},
  title = {A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes
},
  journal = {American Journal of Information Science and Technology},
  volume = {10},
  number = {1},
  pages = {15-24},
  doi = {10.11648/j.ajist.20261001.13},
  url = {https://doi.org/10.11648/j.ajist.20261001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajist.20261001.13},
  abstract = {This paper investigates a construction algorithm for two specific families of quasi-cyclic codes defined over a finite commutative chain ring. First, by employing the Generalized Discrete Fourier Transform, we develop an efficient and systematic algorithm for constructing the generator matrix of repeated-root quasi-cyclic codes under specific structural constraints on the code length and the underlying ring. The proposed method avoids the need for exhaustive enumeration of constituent subcodes and instead operates directly on their generator matrices, leading to improved computational performance. Building on this result, using the Discrete Fourier Transform, we further specialize the proposed framework to derive an algorithm for obtaining the generator matrix of a particular class of simple-root quasi-cyclic codes over a restricted and well-defined category of rings. This specialization demonstrates the performance of the proposed approach and highlights its applicability to different quasi-cyclic code structures in a unified algebraic setting. The proposed construction methods offer significant improvements in computational efficiency when compared to existing approaches that rely on code construction techniques based on constituent subcodes. To evaluate the practical benefits of the proposed algorithms, we present a detailed performance comparison in terms of encoding time per iteration and memory consumption. The comparative results, illustrated through histograms, clearly indicate that the proposed methods achieve faster encoding, lower memory usage, highlighting their superior performance compared to conventional construction techniques as the code length increases.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - A Novel Approach to Constructing Certain Classes of Quasi-Cyclic Codes

AU  - Akram Saleh
AU  - Mohammad Reza Soleymani
Y1  - 2026/01/20
PY  - 2026
N1  - https://doi.org/10.11648/j.ajist.20261001.13
DO  - 10.11648/j.ajist.20261001.13
T2  - American Journal of Information Science and Technology
JF  - American Journal of Information Science and Technology
JO  - American Journal of Information Science and Technology
SP  - 15
EP  - 24
PB  - Science Publishing Group
SN  - 2640-0588
UR  - https://doi.org/10.11648/j.ajist.20261001.13
AB  - This paper investigates a construction algorithm for two specific families of quasi-cyclic codes defined over a finite commutative chain ring. First, by employing the Generalized Discrete Fourier Transform, we develop an efficient and systematic algorithm for constructing the generator matrix of repeated-root quasi-cyclic codes under specific structural constraints on the code length and the underlying ring. The proposed method avoids the need for exhaustive enumeration of constituent subcodes and instead operates directly on their generator matrices, leading to improved computational performance. Building on this result, using the Discrete Fourier Transform, we further specialize the proposed framework to derive an algorithm for obtaining the generator matrix of a particular class of simple-root quasi-cyclic codes over a restricted and well-defined category of rings. This specialization demonstrates the performance of the proposed approach and highlights its applicability to different quasi-cyclic code structures in a unified algebraic setting. The proposed construction methods offer significant improvements in computational efficiency when compared to existing approaches that rely on code construction techniques based on constituent subcodes. To evaluate the practical benefits of the proposed algorithms, we present a detailed performance comparison in terms of encoding time per iteration and memory consumption. The comparative results, illustrated through histograms, clearly indicate that the proposed methods achieve faster encoding, lower memory usage, highlighting their superior performance compared to conventional construction techniques as the code length increases.

VL  - 10
IS  - 1
ER  -

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