Singularities of symmetric hypersurfaces and an application to Reed-Solomon codes
Singularities of symmetric hypersurfaces and an application to Reed-Solomon codes
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发表时间：
2011-09
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通讯作者：
A. Cafure;G. Matera;Melina Privitelli
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作者：
A. Cafure;G. Matera;Melina Privitelli
我们确定了由k+d次多项式生成的k / F_q维的标准Reed-Solomon码不存在深孔的条件。我们的条件依赖于在F_q上定义的某个超曲面族的非零、两两不同坐标的q有理点的存在。我们证明了所考虑的超曲面在坐标的对称置换群的作用下是不变的。这允许我们获得关于这些超曲面奇异轨迹的关键信息，由此建立了q有理点的存在性。
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.