2011/09/10 by Antonio Cafure, Cafure, Antonio, Guillermo Matera +3 · 6 citations
Computer Science · Mathematics · #Coding theory and cryptography #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1109.2265
We determine conditions on q for the nonexistence of deep holes of the\nstandard Reed-Solomon code of dimension k over Fq generated by polynomials of\ndegree k+d. Our conditions rely on the existence of q-rational points with\nnonzero, pairwise-distinct coordinates of a certain family of hypersurfaces\ndefined over Fq. We show that the hypersurfaces under consideration are\ninvariant under the action of the symmetric group of permutations of the\ncoordinates. This allows us to obtain critical information concerning the\nsingular locus of these hypersurfaces, from which the existence of q-rational\npoints is established.\n