On subsets of the normal rational curve
A normal rational curve of the (k-1) -dimensional projective space over Fq is an arc of size q+1 , since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k-6 of a normal rational curve cannot be extended to an arc of size q+2 . In...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2017 |
| 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:2117/111658 |
| Acceso en línea: | https://hdl.handle.net/2117/111658 https://dx.doi.org/10.1109/TIT.2017.2671344 |
| Access Level: | acceso abierto |
| Palabra clave: | Finite geometries MDS code normal rational curve Reed-Solomon code Geometries finites Classificació AMS::51 Geometry::51E Finite geometry and special incidence structures Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica |
| Sumario: | A normal rational curve of the (k-1) -dimensional projective space over Fq is an arc of size q+1 , since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k-6 of a normal rational curve cannot be extended to an arc of size q+2 . In fact, we prove something slightly stronger. Suppose that q is odd and E is a (2k-3) -subset of an arc G of size 3k-6 . If G projects to a subset of a conic from every (k-3) -subset of E , then G cannot be extended to an arc of size q+2 . Stated in terms of error-correcting codes we prove that a k -dimensional linear maximum distance separable code of length 3k-6 over a field Fq of odd characteristic, which can be extended to a Reed–Solomon code of length q+1 , cannot be extended to a linear maximum distance separable code of length q+2 . |
|---|