Codes over the Dickson Near-Field of Order Nine
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.
Authors
- Patrick Solé (ORCID: https://orcid.org/0000-0002-4078-8301)
- Altaf Alshuhail
- Maryam Fahd A. Alshammari
Institutions
- Centre National de la Recherche Scientifique (FR)
- University of Ha'il (SA)
- Centrale Méditerranée (FR)
- Institut Polytechnique de Bordeaux (FR)
Publication Details
- Journal
- Mathematics
- Published
- 2026-01-07
- DOI
- https://doi.org/10.3390/math14020228
- Primary Topic
- Coding theory and cryptography
- Type
- article
- Field-Weighted Citation Impact
- 0.00