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 $\mathbb F_q$ 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 $\mathbb F_q$. 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.

[1]  Nicholas M. Katz,et al.  Sums of Betti Numbers in Arbitrary Characteristic , 2001 .

[2]  Marie Henderson,et al.  A note on the roots of trinomials over a finite field , 2004, Bulletin of the Australian Mathematical Society.

[3]  Qi Cheng,et al.  On Deciding Deep Holes of Reed-Solomon Codes , 2007, TAMC.

[4]  Guillermo Matera,et al.  Improved explicit estimates on the number of solutions of equations over a finite field , 2006, Finite Fields Their Appl..

[5]  Sudhir R. Ghorpade,et al.  Singular Varieties over Finite Fields , 2002 .

[6]  E. Kunz.  Introduction to commutative algebra and algebraic geometry , 1984 .

[7]  Thomas Ernst,et al.  GENERALIZED VANDERMONDE DETERMINANTS. , 2000 .

[8]  François Rodier.  Borne sur le degré des polynômes presque parfaitement non-linéaires , 2006, ArXiv.

[9]  Sudhir R. Ghorpade,et al.  Number of Solutions of Equations over Finite Fields and a Conjecture of Lang and Weil , 2002 .

[10]  Steven Sperber,et al.  On the degree of the $L$-function associated with an exponential sum , 1988 .

[11]  Alain Lascoux,et al.  Jacobians of Symmetric Polynomials , 2002 .

[12]  Venkatesan Guruswami,et al.  Maximum-likelihood decoding of Reed-Solomon codes is NP-hard , 2004, SODA.

[13]  Yves Aubry,et al.  Differentially 4-uniform functions , 2009, 0907.1734.

[15]  Sudhir R. Ghorpade,et al.  \'Etale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields , 2008, 0808.2169.

[16]  Daqing Wan,et al.  Generators and irreducible polynomials over finite fields , 1997, Math. Comput..

[17]  J. Brenner,et al.  Problems in Higher Algebra , 1966 .

[18]  Daqing Wan,et al.  On the subset sum problem over finite fields , 2008, Finite Fields Their Appl..

[19]  Joos Heintz,et al.  Corrigendum: Definability and Fast Quantifier Elimination in Algebraically Closed Fields , 1983, Theor. Comput. Sci..