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Abstract:

We determine conditions on q for the nonexistence of deep holes of the standard Reed-Solomon code of dimension k over F q generated by polynomials of degree k + d. Our conditions rely on the existence of q-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over F q. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of q-rational points is established. © 2012 AIMS-SDU.

Registro:

Documento: Artículo
Título:Singularities of symmetric hypersurfaces and reed-solomon codes
Autor:Cafure, A.; Matera, G.; Privitelli, M.
Filiación:Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento J.M, Gutiérrez 1150, Los Polvorines (B1613GSX), Buenos Aires, Argentina
Ciclo Básico Común, Universidad de Buenos Aires Ciudad Universitaria, Pabellón III (1428), Buenos Aires, Argentina
National Council of Research and Technology (CONICET), Buenos Aires, Argentina
Palabras clave:Deep holes; Finite fields; Rational points; Reed-solomon codes; Singular hypersurfaces; Symmetric polynomials
Año:2012
Volumen:6
Número:1
Página de inicio:69
Página de fin:94
DOI: http://dx.doi.org/10.3934/amc.2012.6.69
Título revista:Advances in Mathematics of Communications
Título revista abreviado:Adv. Math. Commun.
ISSN:19305346
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_19305346_v6_n1_p69_Cafure

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Citas:

---------- APA ----------
Cafure, A., Matera, G. & Privitelli, M. (2012) . Singularities of symmetric hypersurfaces and reed-solomon codes. Advances in Mathematics of Communications, 6(1), 69-94.
http://dx.doi.org/10.3934/amc.2012.6.69
---------- CHICAGO ----------
Cafure, A., Matera, G., Privitelli, M. "Singularities of symmetric hypersurfaces and reed-solomon codes" . Advances in Mathematics of Communications 6, no. 1 (2012) : 69-94.
http://dx.doi.org/10.3934/amc.2012.6.69
---------- MLA ----------
Cafure, A., Matera, G., Privitelli, M. "Singularities of symmetric hypersurfaces and reed-solomon codes" . Advances in Mathematics of Communications, vol. 6, no. 1, 2012, pp. 69-94.
http://dx.doi.org/10.3934/amc.2012.6.69
---------- VANCOUVER ----------
Cafure, A., Matera, G., Privitelli, M. Singularities of symmetric hypersurfaces and reed-solomon codes. Adv. Math. Commun. 2012;6(1):69-94.
http://dx.doi.org/10.3934/amc.2012.6.69