American Institute of Mathematical Sciences

February  2012, 6(1): 69-94. doi: 10.3934/amc.2012.6.69

Singularities of symmetric hypersurfaces and Reed-Solomon codes

 1 Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, Los Polvorines (B1613GSX) Buenos Aires, Argentina, and, Ciclo Básico Común, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón III (1428) Buenos Aires, Argentina, and, National Council of Research and Technology (CONICET), Buenos Aires, Argentina 2 Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, Los Polvorines (B1613GSX) Buenos Aires, Argentina, and, National Council of Research and Technology (CONICET), Buenos Aires, Argentina, Argentina

Received  October 2010 Revised  December 2011 Published  January 2012

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.
Citation: Antonio Cafure, Guillermo Matera, Melina Privitelli. Singularities of symmetric hypersurfaces and Reed-Solomon codes. Advances in Mathematics of Communications, 2012, 6 (1) : 69-94. doi: 10.3934/amc.2012.6.69
References:
 [1] A. Adolphson and S. Sperber, On the degree of the L-function associated with an exponential sum, Compos. Math., 68 (1988), 125-159. [2] Y. Aubry and F. Rodier, Differentially 4-uniform functions, in "Arithmetic, Geometry, Cryptography and Coding Theory 2009" (eds. D. Kohel and R. Rolland), Amer. Math. Soc., (2010), 1-8. [3] A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), 155-185. doi: 10.1016/j.ffa.2005.03.003. [4] Q. Cheng and E. Murray, On deciding deep holes of Reed-Solomon codes, in "Theory and Applications of Models of Computation,'' Springer, Berlin, (2007), 296-305. doi: 10.1007/978-3-540-72504-6_27. [5] R. Coulter and M. Henderson, A note on the roots of trinomials over a finite field, Bull. Austral. Math. Soc., 69 (2004), 429-432. doi: 10.1017/S0004972700036200. [6] T. Ernst, Generalized Vandermonde determinants, report 2000: 6 Matematiska Institutionen, Uppsala Universitet, 2000; available online at http://www2.math.uu.se/research/pub/Ernst1.pdf [7] D. K. Faddeev and I. S. Sominskii, "Problems in Higher Algebra," Freeman, San Francisco, 1965. [8] S. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J., 2 (2002), 589-631. [9] S. Ghorpade and G. Lachaud, Number of solutions of equations over finite fields and a conjecture of Lang and Weil, in "Number Theory and Discrete Mathematics" (eds. A.K. Agarwal et al.), Hindustan Book Agency, (2002), 269-291. [10] V. Guruswami and A. Vardy, Maximum-likelihood decoding of Reed-Solomon codes is NP-hard, IEEE Trans. Inform. Theory, 51 (2005), 2249-2256. doi: 10.1109/TIT.2005.850102. [11] J. Heintz, Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci., 24 (1983), 239-277. doi: 10.1016/0304-3975(83)90002-6. [12] N. Katz, Sums of Betti numbers in arbitrary characteristic, Finite Fields Appl., 7 (2001), 29-44. doi: 10.1006/ffta.2000.0303. [13] E. Kunz, "Introduction to Commutative Algebra and Algebraic Geometry,'' Birkhäuser, Boston, 1985. [14] A. Lascoux and P. Pragracz, Jacobian of symmetric polynomials, Ann. Comb., 6 (2002), 169-172. doi: 10.1007/PL00012583. [15] J. Li and D. Wan, On the subset sum problem over finite fields, Finite Fields Appl., 14 (2008), 911-929. doi: 10.1016/j.ffa.2008.05.003. [16] Y.-J. Li and D. Wan, On error distance of Reed-Solomon codes, Sci. China Ser. A, 51 (2008), 1982-1988. doi: 10.1007/s11425-008-0066-3. [17] R. Lidl and H. Niederreiter, "Finite Fields,'' 2nd edition, Addison-Wesley, Massachusetts, 1997. [18] F. Rodier, Borne sur le degré des polynômes presque parfaitement non-linéaires, in "Arithmetic, Geometry, Cryptography and Coding Theory,'' Amer. Math. Soc., (2009), 169-181. [19] I. R. Shafarevich, "Basic Algebraic Geometry: Varieties in projective space,'' Springer, Berlin, 1994. [20] D. Wan, Generators and irreducible polynomials over finite fields, Math. Comp., 66 (1997), 1195-1212. doi: 10.1090/S0025-5718-97-00835-1.

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References:
 [1] A. Adolphson and S. Sperber, On the degree of the L-function associated with an exponential sum, Compos. Math., 68 (1988), 125-159. [2] Y. Aubry and F. Rodier, Differentially 4-uniform functions, in "Arithmetic, Geometry, Cryptography and Coding Theory 2009" (eds. D. Kohel and R. Rolland), Amer. Math. Soc., (2010), 1-8. [3] A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), 155-185. doi: 10.1016/j.ffa.2005.03.003. [4] Q. Cheng and E. Murray, On deciding deep holes of Reed-Solomon codes, in "Theory and Applications of Models of Computation,'' Springer, Berlin, (2007), 296-305. doi: 10.1007/978-3-540-72504-6_27. [5] R. Coulter and M. Henderson, A note on the roots of trinomials over a finite field, Bull. Austral. Math. Soc., 69 (2004), 429-432. doi: 10.1017/S0004972700036200. [6] T. Ernst, Generalized Vandermonde determinants, report 2000: 6 Matematiska Institutionen, Uppsala Universitet, 2000; available online at http://www2.math.uu.se/research/pub/Ernst1.pdf [7] D. K. Faddeev and I. S. Sominskii, "Problems in Higher Algebra," Freeman, San Francisco, 1965. [8] S. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J., 2 (2002), 589-631. [9] S. Ghorpade and G. Lachaud, Number of solutions of equations over finite fields and a conjecture of Lang and Weil, in "Number Theory and Discrete Mathematics" (eds. A.K. Agarwal et al.), Hindustan Book Agency, (2002), 269-291. [10] V. Guruswami and A. Vardy, Maximum-likelihood decoding of Reed-Solomon codes is NP-hard, IEEE Trans. Inform. Theory, 51 (2005), 2249-2256. doi: 10.1109/TIT.2005.850102. [11] J. Heintz, Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci., 24 (1983), 239-277. doi: 10.1016/0304-3975(83)90002-6. [12] N. Katz, Sums of Betti numbers in arbitrary characteristic, Finite Fields Appl., 7 (2001), 29-44. doi: 10.1006/ffta.2000.0303. [13] E. Kunz, "Introduction to Commutative Algebra and Algebraic Geometry,'' Birkhäuser, Boston, 1985. [14] A. Lascoux and P. Pragracz, Jacobian of symmetric polynomials, Ann. Comb., 6 (2002), 169-172. doi: 10.1007/PL00012583. [15] J. Li and D. Wan, On the subset sum problem over finite fields, Finite Fields Appl., 14 (2008), 911-929. doi: 10.1016/j.ffa.2008.05.003. [16] Y.-J. Li and D. Wan, On error distance of Reed-Solomon codes, Sci. China Ser. A, 51 (2008), 1982-1988. doi: 10.1007/s11425-008-0066-3. [17] R. Lidl and H. Niederreiter, "Finite Fields,'' 2nd edition, Addison-Wesley, Massachusetts, 1997. [18] F. Rodier, Borne sur le degré des polynômes presque parfaitement non-linéaires, in "Arithmetic, Geometry, Cryptography and Coding Theory,'' Amer. Math. Soc., (2009), 169-181. [19] I. R. Shafarevich, "Basic Algebraic Geometry: Varieties in projective space,'' Springer, Berlin, 1994. [20] D. Wan, Generators and irreducible polynomials over finite fields, Math. Comp., 66 (1997), 1195-1212. doi: 10.1090/S0025-5718-97-00835-1.
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