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

Daniele Bartoli; Adnen  Sboui; Leo Storme

doi:10.3934/amc.2016010

Article Contents

2016, Volume 10, Issue 2: 355-365. Doi: 10.3934/amc.2016010

This issue Previous Article Arbitrarily varying multiple access channels with conferencing encoders: List decoding and finite coordination resources Next Article The geometric structure of relative one-weight codes

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

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
3.
Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received: July 2014

Revised: June 2015

Published: May 2016

Abstract / Introduction Related Papers Cited by

Abstract

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 $\mathbb{F}_q$.

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Mathematics Subject Classification: Primary: 51E20, 94B05; Secondary: 05B25.

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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.