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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, Tunisia |
3. | Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent |
References:
[1] |
A. Couvreur, An upper bound on the number of rational points of arbitrary projective varieties over finite fields,, preprint, ().
|
[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, ().
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. |
show all references
References:
[1] |
A. Couvreur, An upper bound on the number of rational points of arbitrary projective varieties over finite fields,, preprint, ().
|
[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, ().
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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