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Mathematics
Volume 14
Issue 2
10.3390/math14020228
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Codes over the Dickson Near-Field of Order Nine

by
Altaf Alshuhail
1,*,
Maryam Fahd A. Alshammari
1 and
Patrick Solé
2
1
Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia
2
Lab I2M (CNRS, Aix Marseille University), 13009 Marseilles, France
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(2), 228; https://doi.org/10.3390/math14020228
Submission received: 14 December 2025 / Revised: 5 January 2026 / Accepted: 6 January 2026 / Published: 7 January 2026
(This article belongs to the Special Issue Discrete Mathematics in Coding Theory)
Versions Notes

Abstract

A near-field is an algebraic structure akin to a division ring where distributivity is relaxed on one side. The smallest such object is the Dickson near-field of order nine. We lay the foundations of linear codes over that near-field. This includes a systematic form for their generator matrices based on a one-sided Gauss pivot algorithm. Dual codes are defined and a formula is given for the parity check matrix. LCD codes are characterized by a direct sum property. Self-orthogonal codes are classified in short lengths.
Keywords: Dickson near-field; near-vector space; Gauss pivot; generator matrix; dual code; LCD codes Dickson near-field; near-vector space; Gauss pivot; generator matrix; dual code; LCD codes

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MDPI and ACS Style

Alshuhail, A.; Alshammari, M.F.A.; Solé, P. Codes over the Dickson Near-Field of Order Nine. Mathematics 2026, 14, 228. https://doi.org/10.3390/math14020228

AMA Style

Alshuhail A, Alshammari MFA, Solé P. Codes over the Dickson Near-Field of Order Nine. Mathematics. 2026; 14(2):228. https://doi.org/10.3390/math14020228

Chicago/Turabian Style

Alshuhail, Altaf, Maryam Fahd A. Alshammari, and Patrick Solé. 2026. "Codes over the Dickson Near-Field of Order Nine" Mathematics 14, no. 2: 228. https://doi.org/10.3390/math14020228

APA Style

Alshuhail, A., Alshammari, M. F. A., & Solé, P. (2026). Codes over the Dickson Near-Field of Order Nine. Mathematics, 14(2), 228. https://doi.org/10.3390/math14020228

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