Singularities of symmetric hypersurfaces and Reed-Solomon codes

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

Singularities of symmetric hypersurfaces and Reed-Solomon codes

Antonio Cafure ^1, , Guillermo Matera ^2, and Melina Privitelli ^2,

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

Abstract
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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.

Keywords: Reed-Solomon codes, deep holes, Finite fields, singular hypersurfaces, rational points., symmetric polynomials.

Mathematics Subject Classification: Primary: 11G25, 14G15; Secondary: 14G50,05E0.

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. Google Scholar
[2]	Y. Aubry and F. Rodier, Differentially 4-uniform functions,, in, (2010), 1. Google Scholar
[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. doi: 10.1016/j.ffa.2005.03.003. Google Scholar
[4]	Q. Cheng and E. Murray, On deciding deep holes of Reed-Solomon codes,, in, (2007), 296. doi: 10.1007/978-3-540-72504-6_27. Google Scholar
[5]	R. Coulter and M. Henderson, A note on the roots of trinomials over a finite field,, Bull. Austral. Math. Soc., 69 (2004), 429. doi: 10.1017/S0004972700036200. Google Scholar
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[7]	D. K. Faddeev and I. S. Sominskii, "Problems in Higher Algebra,", Freeman, (1965). Google Scholar
[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. Google Scholar
[9]	S. Ghorpade and G. Lachaud, Number of solutions of equations over finite fields and a conjecture of Lang and Weil,, in, (2002), 269. Google Scholar
[10]	V. Guruswami and A. Vardy, Maximum-likelihood decoding of Reed-Solomon codes is NP-hard,, IEEE Trans. Inform. Theory, 51 (2005), 2249. doi: 10.1109/TIT.2005.850102. Google Scholar
[11]	J. Heintz, Definability and fast quantifier elimination in algebraically closed fields,, Theoret. Comput. Sci., 24 (1983), 239. doi: 10.1016/0304-3975(83)90002-6. Google Scholar
[12]	N. Katz, Sums of Betti numbers in arbitrary characteristic,, Finite Fields Appl., 7 (2001), 29. doi: 10.1006/ffta.2000.0303. Google Scholar
[13]	E. Kunz, "Introduction to Commutative Algebra and Algebraic Geometry,'', Birkhäuser, (1985). Google Scholar
[14]	A. Lascoux and P. Pragracz, Jacobian of symmetric polynomials,, Ann. Comb., 6 (2002), 169. doi: 10.1007/PL00012583. Google Scholar
[15]	J. Li and D. Wan, On the subset sum problem over finite fields,, Finite Fields Appl., 14 (2008), 911. doi: 10.1016/j.ffa.2008.05.003. Google Scholar
[16]	Y.-J. Li and D. Wan, On error distance of Reed-Solomon codes,, Sci. China Ser. A, 51 (2008), 1982. doi: 10.1007/s11425-008-0066-3. Google Scholar
[17]	R. Lidl and H. Niederreiter, "Finite Fields,'' 2^nd edition,, Addison-Wesley, (1997). Google Scholar
[18]	F. Rodier, Borne sur le degré des polynômes presque parfaitement non-linéaires,, in, (2009), 169. Google Scholar
[19]	I. R. Shafarevich, "Basic Algebraic Geometry: Varieties in projective space,'', Springer, (1994). Google Scholar
[20]	D. Wan, Generators and irreducible polynomials over finite fields,, Math. Comp., 66 (1997), 1195. doi: 10.1090/S0025-5718-97-00835-1. Google Scholar

show all references

References:

[1]	A. Adolphson and S. Sperber, On the degree of the L-function associated with an exponential sum,, Compos. Math., 68 (1988), 125. Google Scholar
[2]	Y. Aubry and F. Rodier, Differentially 4-uniform functions,, in, (2010), 1. Google Scholar
[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. doi: 10.1016/j.ffa.2005.03.003. Google Scholar
[4]	Q. Cheng and E. Murray, On deciding deep holes of Reed-Solomon codes,, in, (2007), 296. doi: 10.1007/978-3-540-72504-6_27. Google Scholar
[5]	R. Coulter and M. Henderson, A note on the roots of trinomials over a finite field,, Bull. Austral. Math. Soc., 69 (2004), 429. doi: 10.1017/S0004972700036200. Google Scholar
[6]	T. Ernst, Generalized Vandermonde determinants,, report 2000: 6 Matematiska Institutionen, (2000). Google Scholar
[7]	D. K. Faddeev and I. S. Sominskii, "Problems in Higher Algebra,", Freeman, (1965). Google Scholar
[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. Google Scholar
[9]	S. Ghorpade and G. Lachaud, Number of solutions of equations over finite fields and a conjecture of Lang and Weil,, in, (2002), 269. Google Scholar
[10]	V. Guruswami and A. Vardy, Maximum-likelihood decoding of Reed-Solomon codes is NP-hard,, IEEE Trans. Inform. Theory, 51 (2005), 2249. doi: 10.1109/TIT.2005.850102. Google Scholar
[11]	J. Heintz, Definability and fast quantifier elimination in algebraically closed fields,, Theoret. Comput. Sci., 24 (1983), 239. doi: 10.1016/0304-3975(83)90002-6. Google Scholar
[12]	N. Katz, Sums of Betti numbers in arbitrary characteristic,, Finite Fields Appl., 7 (2001), 29. doi: 10.1006/ffta.2000.0303. Google Scholar
[13]	E. Kunz, "Introduction to Commutative Algebra and Algebraic Geometry,'', Birkhäuser, (1985). Google Scholar
[14]	A. Lascoux and P. Pragracz, Jacobian of symmetric polynomials,, Ann. Comb., 6 (2002), 169. doi: 10.1007/PL00012583. Google Scholar
[15]	J. Li and D. Wan, On the subset sum problem over finite fields,, Finite Fields Appl., 14 (2008), 911. doi: 10.1016/j.ffa.2008.05.003. Google Scholar
[16]	Y.-J. Li and D. Wan, On error distance of Reed-Solomon codes,, Sci. China Ser. A, 51 (2008), 1982. doi: 10.1007/s11425-008-0066-3. Google Scholar
[17]	R. Lidl and H. Niederreiter, "Finite Fields,'' 2^nd edition,, Addison-Wesley, (1997). Google Scholar
[18]	F. Rodier, Borne sur le degré des polynômes presque parfaitement non-linéaires,, in, (2009), 169. Google Scholar
[19]	I. R. Shafarevich, "Basic Algebraic Geometry: Varieties in projective space,'', Springer, (1994). Google Scholar
[20]	D. Wan, Generators and irreducible polynomials over finite fields,, Math. Comp., 66 (1997), 1195. doi: 10.1090/S0025-5718-97-00835-1. Google Scholar

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