We construct linear codes over the finite field $ \mathbb{F}_q $ from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords from a geometric point of view, interpreting them in terms of the combinatorial structure of the associated simplicial complex. This approach allows us to describe the minimum distance of the codes in terms of certain geometric features of the complex. Subsequently, we analyze how various topological operations on simplicial complexes affect the classical parameters of the codes. This study leads to the formulation of geometric criteria that make it possible to explicitly control and manipulate these parameters. Finally, as an application of the obtained results, we construct several families of optimal linear codes over $ \mathbb{F}_2 $ using these geometric methods. Thanks to the previously established geometric properties, we can precisely determine the parameters of these families.
Citation: Antonio Jesús Lorite López, Daniel Camazón Portela, Juan Antonio López Ramos. Linear codes arising from geometrical operations[J]. AIMS Mathematics, 2026, 11(4): 10033-10048. doi: 10.3934/math.2026414
We construct linear codes over the finite field $ \mathbb{F}_q $ from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords from a geometric point of view, interpreting them in terms of the combinatorial structure of the associated simplicial complex. This approach allows us to describe the minimum distance of the codes in terms of certain geometric features of the complex. Subsequently, we analyze how various topological operations on simplicial complexes affect the classical parameters of the codes. This study leads to the formulation of geometric criteria that make it possible to explicitly control and manipulate these parameters. Finally, as an application of the obtained results, we construct several families of optimal linear codes over $ \mathbb{F}_2 $ using these geometric methods. Thanks to the previously established geometric properties, we can precisely determine the parameters of these families.
| [1] |
M. J. Adamaszek, Face numbers of down-sets, Amer. Math. Monthly, 122 (2015), 367–370. https://doi.org/10.4169/amer.math.monthly.122.04.367 doi: 10.4169/amer.math.monthly.122.04.367
|
| [2] |
A. K. Bhagat, R. Sarma, V. Sagar, Subfield codes of $C_D$-codes over $\mathbb{F}_2[x]/\langle x^3 -x\rangle$, Discrete Math., 348 (2024), 114223. https://doi.org/10.1016/j.disc.2024.114223 doi: 10.1016/j.disc.2024.114223
|
| [3] |
S. Chang, J. Y. Hyun, Linear codes from simplicial complexes, Des. Codes Cryptogr., 86 (2018), 2167–2181. https://doi.org/10.1007/s10623-017-0442-5 doi: 10.1007/s10623-017-0442-5
|
| [4] |
Z. Hu, Y. Xu, N. Li, X. Zeng, L. Wang, X. Tang, New constructions of optimal linear codes from simplicial complexes, IEEE Trans. Inf. Theory, 70 (2024), 1823–1835. https://doi.org/10.1109/TIT.2023.3305609 doi: 10.1109/TIT.2023.3305609
|
| [5] |
J. Y. Hyun, H. K. Kim, M. Na, Optimal non-projective linear codes constructed from down-sets, Discrete Appl. Math., 254 (2019), 135–145. https://doi.org/10.1016/j.dam.2018.07.007 doi: 10.1016/j.dam.2018.07.007
|
| [6] |
J. Y. Hyun, J. Y. Lee, Y. J. Lee, Infinite families of optimal linear codes constructed from simplicial complexes, IEEE Trans. Inf. Theory, 66 (2020), 6762–6773. https://doi.org/10.1109/TIT.2020.2993179 doi: 10.1109/TIT.2020.2993179
|
| [7] |
X. Li, Q. Yue, D. Tang, A family of linear codes from constant dimension subspace codes, Des. Codes Cryptogr., 90 (2022), 1–15. https://doi.org/10.1007/s10623-021-00960-x doi: 10.1007/s10623-021-00960-x
|
| [8] |
H. Liu, Z. Yu, Linear codes from simplicial complexes over $\mathbb{F}_n^2$, Des. Codes Cryptogr., 92 (2024), 2993–3016. https://doi.org/10.1007/s10623-024-01424-8 doi: 10.1007/s10623-024-01424-8
|
| [9] |
Y. Pan, Y. Liu, New classes of few-weight ternary codes from simplicial complexes, AIMS Mathematics, 7 (2022), 4315–4325. https://doi.org/10.3934/math.2022239 doi: 10.3934/math.2022239
|
| [10] | V. Sagar, R. Sarma, Linear codes using simplicial complexes, arXiv: 2204.08417, 2022. https://doi.org/10.48550/arXiv.2204.08417 |
| [11] |
V. Sagar, R. Sarma, Certain binary minimal codes constructed using simplicial complexes, Adv. Math. Commun., 18 (2024), 425–454. https://doi.org/10.3934/amc.2023044 doi: 10.3934/amc.2023044
|
| [12] |
Y. Wu, J. Y. Hyun, Few-weight codes over $\mathbb{F}_p+u\mathbb{F}_p$ associated with down sets and their distance optimal gray image, Discrete Appl. Math., 283 (2020), 315–322. https://doi.org/10.1016/j.dam.2020.01.019 doi: 10.1016/j.dam.2020.01.019
|
| [13] |
Y. Wu, X. Zhu, Q. Yue, Optimal few-weight codes from simplicial complexes, IEEE Trans. Inf. Theory, 66 (2019), 3657–3663. https://doi.org/10.1109/TIT.2019.2946840 doi: 10.1109/TIT.2019.2946840
|
Figures(4)
Antonio Jesús Lorite López, Daniel Camazón Portela, Juan Antonio López Ramos. Linear codes arising from geometrical operations[J]. AIMS Mathematics, 2026, 11(4): 10033-10048. doi: 10.3934/math.2026414
DownLoad: