Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes

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May 2016, 10(2): 355-365. doi: 10.3934/amc.2016010

Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes

Daniele Bartoli ^1, , Adnen Sboui ^2, and Leo Storme ^3,

1.	Dipartimento di Matematica ed Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia
2.	Institut Préparatoire aux Études d'Ingénieurs d'El-Manar, Université Tunis El Manar, Campus universitaire El Manar, B.P.244 El Manar II - 2092 Tunis, Tunisia
3.	Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received July 2014 Revised June 2015 Published April 2016

Abstract
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We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in $\mathbb{P}^n(\mathbb{F}_q)$ of small degree $d$, depending on the number of linear components contained in such curves and hypersurfaces. The obtained results have applications to the weight distribution of the projective Reed-Muller codes PRM$(q,d,n)$ over the finite field $\mathbb{F}_q$.

Keywords: small weight codewords, quadrics, Algebraic varieties, intersections, projective Reed-Muller codes..

Mathematics Subject Classification: Primary: 51E20, 94B05; Secondary: 05B2.

Citation: Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010

References:

[1]	A. Couvreur, An upper bound on the number of rational points of arbitrary projective varieties over finite fields, preprint, arXiv:1409.7544v1
[2]	S. R. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Moscow Math. J., 2 (2002), 589-631.
[3]	G. Lachaud, The parameters of projective Reed-Muller codes, Discrete Math., 81 (1990), 217-221. doi: 10.1016/0012-365X(90)90155-B.
[4]	G. Lachaud and R. Rolland, An overview of the number of points of algebraic sets over finite fields}}, preprint, arXiv:1405.3027v2 doi: 10.1016/j.jpaa.2015.05.008.
[5]	S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math., 76 (1954), 819-827.
[6]	F. Rodier and A. Sboui, Les Arrangements Minimaux et Maximaux d'Hyperplans dans $\mathbb P^n(\mathbb F_q)$, C. R. Acad. Sc. Paris Ser. I, 344 (2007), 287-290. doi: 10.1016/j.crma.2007.01.006.
[7]	F. Rodier and A. Sboui, Highest numbers of points of hypersurfaces and generalized Reed-Muller codes, Finite Fields Appl., 14 (2008), 816-822. doi: 10.1016/j.ffa.2008.02.001.
[8]	A. Sboui, Second highest number of points of hypersurfaces in $\mathbb F_q^n$, Finite Fields Appl., 13 (2007), 444-449. doi: 10.1016/j.ffa.2005.11.002.
[9]	A. Sboui, Special numbers of rational points on hypersurfaces in the $n$-dimensional projective space over a finite field, Discrete Math., 309 (2009), 5048-5059. doi: 10.1016/j.disc.2009.03.021.
[10]	J.-P. Serre, Lettre à M. Tsfasman du 24 Juillet 1989, in Journées Arithmétiques de Luminy 17-21 Juillet 1989, Astérisque, 198 (1991), 351-353.
[11]	A. B. Sørensen, Projective Reed-Muller codes, IEEE Trans. Inf. Theory, 37 (1991), 1567-1576. doi: 10.1109/18.104317.
[12]	A. B. Sørensen, On the number of rational points on codimension-1 algebraic sets in $\mathbb P^n(\mathbb F_q)$, Discrete Math., 135 (1994), 321-334. doi: 10.1016/0012-365X(93)E0009-S.
[13]	L. Storme and J. A. Thas, MDS codes and arcs in $PG(n, q)$ with $q$ even: An improvement of the bounds of Bruen, Thas and Blokhuis, J. Combin. Theory Ser. A, 62 (1993), 139-154. doi: 10.1016/0097-3165(93)90076-K.
[14]	L. Storme and H. Van Maldeghem, Cyclic arcs in PG$(2,q)$, J. Algebraic Combin., 3 (1994), 113-128. doi: 10.1023/A:1022454221497.
[15]	L. Storme and H. Van Maldeghem, Arcs fixed by a large cyclic group, Atti Sem. Mat. Fis. Univ. Modena, XLIII (1995), 273-280.

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References:

[1]	A. Couvreur, An upper bound on the number of rational points of arbitrary projective varieties over finite fields, preprint, arXiv:1409.7544v1
[2]	S. R. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Moscow Math. J., 2 (2002), 589-631.
[3]	G. Lachaud, The parameters of projective Reed-Muller codes, Discrete Math., 81 (1990), 217-221. doi: 10.1016/0012-365X(90)90155-B.
[4]	G. Lachaud and R. Rolland, An overview of the number of points of algebraic sets over finite fields}}, preprint, arXiv:1405.3027v2 doi: 10.1016/j.jpaa.2015.05.008.
[5]	S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math., 76 (1954), 819-827.
[6]	F. Rodier and A. Sboui, Les Arrangements Minimaux et Maximaux d'Hyperplans dans $\mathbb P^n(\mathbb F_q)$, C. R. Acad. Sc. Paris Ser. I, 344 (2007), 287-290. doi: 10.1016/j.crma.2007.01.006.
[7]	F. Rodier and A. Sboui, Highest numbers of points of hypersurfaces and generalized Reed-Muller codes, Finite Fields Appl., 14 (2008), 816-822. doi: 10.1016/j.ffa.2008.02.001.
[8]	A. Sboui, Second highest number of points of hypersurfaces in $\mathbb F_q^n$, Finite Fields Appl., 13 (2007), 444-449. doi: 10.1016/j.ffa.2005.11.002.
[9]	A. Sboui, Special numbers of rational points on hypersurfaces in the $n$-dimensional projective space over a finite field, Discrete Math., 309 (2009), 5048-5059. doi: 10.1016/j.disc.2009.03.021.
[10]	J.-P. Serre, Lettre à M. Tsfasman du 24 Juillet 1989, in Journées Arithmétiques de Luminy 17-21 Juillet 1989, Astérisque, 198 (1991), 351-353.
[11]	A. B. Sørensen, Projective Reed-Muller codes, IEEE Trans. Inf. Theory, 37 (1991), 1567-1576. doi: 10.1109/18.104317.
[12]	A. B. Sørensen, On the number of rational points on codimension-1 algebraic sets in $\mathbb P^n(\mathbb F_q)$, Discrete Math., 135 (1994), 321-334. doi: 10.1016/0012-365X(93)E0009-S.
[13]	L. Storme and J. A. Thas, MDS codes and arcs in $PG(n, q)$ with $q$ even: An improvement of the bounds of Bruen, Thas and Blokhuis, J. Combin. Theory Ser. A, 62 (1993), 139-154. doi: 10.1016/0097-3165(93)90076-K.
[14]	L. Storme and H. Van Maldeghem, Cyclic arcs in PG$(2,q)$, J. Algebraic Combin., 3 (1994), 113-128. doi: 10.1023/A:1022454221497.
[15]	L. Storme and H. Van Maldeghem, Arcs fixed by a large cyclic group, Atti Sem. Mat. Fis. Univ. Modena, XLIII (1995), 273-280.

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