Incidence Matrix Based Construction of Linear Error-Correcting Codes from PG(9,q) and PG(10,q)

Section: Research Article
Issue
Vol. 35 No. 3 (2026): Volume 35, Issue 3
Pages
40-49

Keywords:

Finite Projective Space, Incidence Matrix, Linear Code, PG(9,q), PG(10,q)

Abstract

Error-correcting codes over finite projective geometries are of significant value in ensuring data reliability. However, because higher-dimensional spaces are far too computationally complex to tinker with, most research has been concentrated on lower dimensions. To help bridge this gap, this paper constructs new projective linear codes directly out of the high-dimensional spaces  and . We generated the companion matrices of primitive polynomials to construct points and subspaces systematically. Then, we constructed point-hyperplane incidence matrices and processed them computationally in MATLAB to determine the basic parameters of our new codes. Through this process, we obtained two specific codes: a ternary code from  with parameters.  that can correct up to  errors, and a binary code from  with parameters  that can correct up to  errors. By applying the Sphere Packing Bound, it turned out that although these codes are non-perfect, they nevertheless possess an extremely high error-correcting capacity. Overall, we were able to assemble a workable algebraic scheme that transforms the complex structure of higher-dimensional spaces into reliable error-correcting codes, offering useful theoretical insights beyond the familiar low-dimensional models.

References

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Authors

Abdulmuttalib N. M. S Ibrahim

Department of Mathematics, College of Education for Pure Science, University of Mosul, Iraq

Nada Y. K. Yahya

Department of Mathematics, College of Education for Pure science, University of Mosul

Identifiers

DOI: 10.33899/jes.v35i3.61424

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[1]
“Incidence Matrix Based Construction of Linear Error-Correcting Codes from PG(9,q) and PG(10,q)”, JES, vol. 35, no. 3, pp. 40–49, Jul. 2026, doi: 10.33899/jes.v35i3.61424.
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This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

[1]
“Incidence Matrix Based Construction of Linear Error-Correcting Codes from PG(9,q) and PG(10,q)”, JES, vol. 35, no. 3, pp. 40–49, Jul. 2026, doi: 10.33899/jes.v35i3.61424.