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...

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Detalles Bibliográficos
Autores: Ball, Simeon Michael|||0000-0003-4845-2084, De Beule, Jan
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
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spelling On subsets of the normal rational curveBall, Simeon Michael|||0000-0003-4845-2084De Beule, JanFinite geometriesMDS codenormal rational curveReed-Solomon codeGeometries finitesClassificació AMS::51 Geometry::51E Finite geometry and special incidence structuresÀrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraicaA 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 .20172017-06-0120172017-12-11journal articlehttp://purl.org/coar/resource_type/c_6501CVoRhttp://purl.org/coar/version/c_e19f295774971610info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/111658https://dx.doi.org/10.1109/TIT.2017.2671344reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengMinisterio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2014-54745-P ESTRUCTURAS DISCRETAS, GEOMETRICAS Y ALEATORIASopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1116582026-05-27T15:37:01Z
dc.title.none.fl_str_mv On subsets of the normal rational curve
title On subsets of the normal rational curve
spellingShingle On subsets of the normal rational curve
Ball, Simeon Michael|||0000-0003-4845-2084
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
title_short On subsets of the normal rational curve
title_full On subsets of the normal rational curve
title_fullStr On subsets of the normal rational curve
title_full_unstemmed On subsets of the normal rational curve
title_sort On subsets of the normal rational curve
dc.creator.none.fl_str_mv Ball, Simeon Michael|||0000-0003-4845-2084
De Beule, Jan
author Ball, Simeon Michael|||0000-0003-4845-2084
author_facet Ball, Simeon Michael|||0000-0003-4845-2084
De Beule, Jan
author_role author
author2 De Beule, Jan
author2_role author
dc.subject.none.fl_str_mv 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
topic 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
description 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 .
publishDate 2017
dc.date.none.fl_str_mv 2017
2017-06-01
2017
2017-12-11
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
CVoR
http://purl.org/coar/version/c_e19f295774971610
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/111658
https://dx.doi.org/10.1109/TIT.2017.2671344
url https://hdl.handle.net/2117/111658
https://dx.doi.org/10.1109/TIT.2017.2671344
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2014-54745-P ESTRUCTURAS DISCRETAS, GEOMETRICAS Y ALEATORIAS
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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