Singularities of symmetric hypersurfaces and reed-solomon codes
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 hypersurfac...
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2012
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_19305346_v6_n1_p69_Cafure http://hdl.handle.net/20.500.12110/paper_19305346_v6_n1_p69_Cafure |
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paper:paper_19305346_v6_n1_p69_Cafure2023-06-08T16:30:22Z Singularities of symmetric hypersurfaces and reed-solomon codes Deep holes Finite fields Rational points Reed-solomon codes Singular hypersurfaces Symmetric polynomials 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. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_19305346_v6_n1_p69_Cafure http://hdl.handle.net/20.500.12110/paper_19305346_v6_n1_p69_Cafure |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Deep holes Finite fields Rational points Reed-solomon codes Singular hypersurfaces Symmetric polynomials |
| spellingShingle |
Deep holes Finite fields Rational points Reed-solomon codes Singular hypersurfaces Symmetric polynomials Singularities of symmetric hypersurfaces and reed-solomon codes |
| topic_facet |
Deep holes Finite fields Rational points Reed-solomon codes Singular hypersurfaces Symmetric polynomials |
| description |
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. |
| title |
Singularities of symmetric hypersurfaces and reed-solomon codes |
| title_short |
Singularities of symmetric hypersurfaces and reed-solomon codes |
| title_full |
Singularities of symmetric hypersurfaces and reed-solomon codes |
| title_fullStr |
Singularities of symmetric hypersurfaces and reed-solomon codes |
| title_full_unstemmed |
Singularities of symmetric hypersurfaces and reed-solomon codes |
| title_sort |
singularities of symmetric hypersurfaces and reed-solomon codes |
| publishDate |
2012 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_19305346_v6_n1_p69_Cafure http://hdl.handle.net/20.500.12110/paper_19305346_v6_n1_p69_Cafure |
| _version_ |
1768543057407377408 |