On additive MDS codes over small fields
Let $ C $ be a $ (n,q^{2k},n-k+1)_{q^2} $ additive MDS code which is linear over $ {\mathbb F}_q $. We prove that if $ n \geq q+k $ and $ k+1 $ of the projections of $ C $ are linear over $ {\mathbb F}_{q^2} $ then $ C $ is linear over $ {\mathbb F}_{q^2} $. We use this geometrical theorem, other ge...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/383024 |
| Acceso en línea: | https://hdl.handle.net/2117/383024 https://dx.doi.org/10.3934/amc.2021024 |
| Access Level: | acceso abierto |
| Palabra clave: | Geometry Error-correcting codes (Information theory) MDS codes MDS conjecture quantum codes additive codes stabiliser codes arcs Geometria finita Codis de correcció d'errors (Teoria de la informació) Classificació AMS::51 Geometry::51E Finite geometry and special incidence structures Classificació AMS::94 Information And Communication, Circuits::94B Theory of error-correcting codes and error-detecting codes Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria |
| Sumario: | Let $ C $ be a $ (n,q^{2k},n-k+1)_{q^2} $ additive MDS code which is linear over $ {\mathbb F}_q $. We prove that if $ n \geq q+k $ and $ k+1 $ of the projections of $ C $ are linear over $ {\mathbb F}_{q^2} $ then $ C $ is linear over $ {\mathbb F}_{q^2} $. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over $ {\mathbb F}_q $ for $ q \in \{4,8,9\} $. We also classify the longest additive MDS codes over $ {\mathbb F}_{16} $ which are linear over $ {\mathbb F}_4 $. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $ q \in \{ 2,3\} $. |
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