Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach

Olufemi Ololade Olaewe; Peter Awonnatemi Agbedemnab; Moses Apambila Agebure

doi:doi:10.11648/j.mcs.20261102.12

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Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach

Olufemi Ololade Olaewe^*

, Peter Awonnatemi Agbedemnab

, Moses Apambila Agebure

Published in Mathematics and Computer Science (Volume 11, Issue 2)

Received: 21 April 2026 Accepted: 30 April 2026 Published: 12 May 2026

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Abstract

This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.

Published in	Mathematics and Computer Science (Volume 11, Issue 2)
DOI	10.11648/j.mcs.20261102.12
Page(s)	33-44
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

Linear Code Equivalence, Linear Code, Graph Theory, Algorithm

References

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[2]	R. M. Tanner, “A Recursive Approach to Low Complexity Codes,” IEEE Trans. Inf. Theory, no. 5, 1981.
[3]	T. J. Richardson and R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 638–656, 2001, https://doi.org/10.1109/18.910579
[4]	H. Dehghani, M. Ahmadi, S. Alikhani, and R. Hasni, “Calculation of Girth of Tanner Graph in LDPC Codes,” Trends Appl. Sci. Res., vol. 7, no. 11, pp. 929–934, Nov. 2012, https://doi.org/10.3923/TASR.2012.929.934
[5]	V. Ninkovic, O. Kundacina, D. Vukobratovic, C. Häger, and A. G. i Amat, “Decoding Quantum LDPC Codes Using Graph Neural Networks,” Aug. 2024, Accessed: Feb. 27, 2025. Available: http://arxiv.org/abs/2408.05170
[6]	T. Etzion, A. Trachtenberg, and A. Vardy, “Which Codes Have Cycle-Free Tanner Graphs?,” IEEE Trans. Inf. Theory, vol. 45, no. 6, p. 2173, 1999. https://doi.org/10.1109/18.782130
[7]	F. J.. MacWilliams and N. J. A.. Sloane, “The theory of error-correcting codes,” p. 762, 20061977, 1977.
[8]	M. Manickam and K. Desikan, “Eigenvalue Interlacing of Bipartite Graphs and Construction of Expander Code using Vertex-split of a Bipartite Graph,” 2023. https://doi.org/10.29020/nybg.ejpam.v17i2.5057
[9]	Z. Zhou and Z. Guo, “Improved Decoding of Tanner Codes,” Jan. 2025, Accessed: May 06, 2025. Available: https://arxiv.org/abs/2501.12293v3
[10]	E. Arıkan, “From sequential decoding to channel polarization and back again,” Aug. 2019, Accessed: May 11, 2025. Available: https://arxiv.org/abs/1908.09594v3
[11]	A. Aliouat and E. Dupraz, “Goal-Oriented Source Coding using LDPC Codes for Compressed-Domain Image Classification,” Mar. 2025, Accessed: May 12, 2025. Available: https://arxiv.org/abs/2503.11954v1
[12]	A. Thomas et al., “Streaming Encoding Algorithms for Scalable Hyperdimensional Computing,” Sep. 2022, Accessed: May 12, 2025. Available: http://arxiv.org/abs/2209.09868
[13]	N. Raviv, “Linear Codes for Hyperdimensional Computing,” Mar. 2024, Accessed: May 12, 2025. Available: https://arxiv.org/abs/2403.03278v1
[14]	M. Grassl, "Computing Extensions of Linear Codes," 2007 IEEE International Symposium on Information Theory (ISIT 2007), Nice, France, 2007, pp. 476-480. https://doi.org/10.48550/arXiv.0704.2596
[15]	L. E. Danielsen and M. G. Parker, “Edge Local Complementation and Equivalence of Binary Linear Codes,” Des. Codes Cryptogr., vol. 49, no. 1–3, pp. 161–170, Oct. 2007, https://doi.org/10.1007/s10623-008-9190-x

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Olaewe, O. O., Agbedemnab, P. A., Agebure, M. A. (2026). Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach. Mathematics and Computer Science, 11(2), 33-44. https://doi.org/10.11648/j.mcs.20261102.12

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ACS Style

Olaewe, O. O.; Agbedemnab, P. A.; Agebure, M. A. Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach. Math. Comput. Sci. 2026, 11(2), 33-44. doi: 10.11648/j.mcs.20261102.12

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AMA Style

Olaewe OO, Agbedemnab PA, Agebure MA. Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach. Math Comput Sci. 2026;11(2):33-44. doi: 10.11648/j.mcs.20261102.12

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@article{10.11648/j.mcs.20261102.12,
  author = {Olufemi Ololade Olaewe and Peter Awonnatemi Agbedemnab and Moses Apambila Agebure},
  title = {Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach},
  journal = {Mathematics and Computer Science},
  volume = {11},
  number = {2},
  pages = {33-44},
  doi = {10.11648/j.mcs.20261102.12},
  url = {https://doi.org/10.11648/j.mcs.20261102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261102.12},
  abstract = {This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.},
 year = {2026}
}

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TY  - JOUR
T1  - Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach
AU  - Olufemi Ololade Olaewe
AU  - Peter Awonnatemi Agbedemnab
AU  - Moses Apambila Agebure
Y1  - 2026/05/12
PY  - 2026
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DO  - 10.11648/j.mcs.20261102.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 33
EP  - 44
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20261102.12
AB  - This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.
VL  - 11
IS  - 2
ER  -

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Author Information

Olufemi Ololade Olaewe

Department of Computer Science, University of Technology and Applied Sciences, Navrongo, Ghana

Contact Email

http://orcid.org/0009-0003-2402-5827
Peter Awonnatemi Agbedemnab

Department of Information Systems and Technology, University of Technology and Applied Sciences, Navrongo, Ghana

Contact Email

http://orcid.org/0000-0001-8904-1009
Moses Apambila Agebure

Department of Computer Science, University of Technology and Applied Sciences, Navrongo, Ghana

Contact Email

http://orcid.org/0000-0003-3555-8349

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APA Style

Olaewe, O. O., Agbedemnab, P. A., Agebure, M. A. (2026). Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach. Mathematics and Computer Science, 11(2), 33-44. https://doi.org/10.11648/j.mcs.20261102.12

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ACS Style

Olaewe, O. O.; Agbedemnab, P. A.; Agebure, M. A. Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach. Math. Comput. Sci. 2026, 11(2), 33-44. doi: 10.11648/j.mcs.20261102.12

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AMA Style

Olaewe OO, Agbedemnab PA, Agebure MA. Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach. Math Comput Sci. 2026;11(2):33-44. doi: 10.11648/j.mcs.20261102.12

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@article{10.11648/j.mcs.20261102.12,
  author = {Olufemi Ololade Olaewe and Peter Awonnatemi Agbedemnab and Moses Apambila Agebure},
  title = {Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach},
  journal = {Mathematics and Computer Science},
  volume = {11},
  number = {2},
  pages = {33-44},
  doi = {10.11648/j.mcs.20261102.12},
  url = {https://doi.org/10.11648/j.mcs.20261102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261102.12},
  abstract = {This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach
AU  - Olufemi Ololade Olaewe
AU  - Peter Awonnatemi Agbedemnab
AU  - Moses Apambila Agebure
Y1  - 2026/05/12
PY  - 2026
N1  - https://doi.org/10.11648/j.mcs.20261102.12
DO  - 10.11648/j.mcs.20261102.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 33
EP  - 44
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20261102.12
AB  - This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.
VL  - 11
IS  - 2
ER  -

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