-
Previous Article
The geometric structure of relative one-weight codes
- AMC Home
- This Issue
-
Next Article
Arbitrarily varying multiple access channels with conferencing encoders: List decoding and finite coordination resources
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, 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. |
show all references
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. |
[1] |
Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333 |
[2] |
Martino Borello, Olivier Mila. Symmetries of weight enumerators and applications to Reed-Muller codes. Advances in Mathematics of Communications, 2019, 13 (2) : 313-328. doi: 10.3934/amc.2019021 |
[3] |
Olav Geil, Stefano Martin. Relative generalized Hamming weights of q-ary Reed-Muller codes. Advances in Mathematics of Communications, 2017, 11 (3) : 503-531. doi: 10.3934/amc.2017041 |
[4] |
Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271 |
[5] |
Jesús Carrillo-Pacheco, Felipe Zaldivar. On codes over FFN$(1,q)$-projective varieties. Advances in Mathematics of Communications, 2016, 10 (2) : 209-220. doi: 10.3934/amc.2016001 |
[6] |
Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233 |
[7] |
Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543 |
[8] |
Daniele Bartoli, Lins Denaux. Minimal codewords arising from the incidence of points and hyperplanes in projective spaces. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021061 |
[9] |
Claude Carlet. Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022047 |
[10] |
Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim and Julius L. Shaneson. Hodge genera and characteristic classes of complex algebraic varieties. Electronic Research Announcements, 2008, 15: 1-7. doi: 10.3934/era.2008.15.1 |
[11] |
Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015 |
[12] |
Peter Beelen, David Glynn, Tom Høholdt, Krishna Kaipa. Counting generalized Reed-Solomon codes. Advances in Mathematics of Communications, 2017, 11 (4) : 777-790. doi: 10.3934/amc.2017057 |
[13] |
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 |
[14] |
Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021 |
[15] |
Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039 |
[16] |
Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006 |
[17] |
László Mérai, Igor E. Shparlinski. Unlikely intersections over finite fields: Polynomial orbits in small subgroups. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1065-1073. doi: 10.3934/dcds.2020070 |
[18] |
Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2021, 15 (2) : 207-218. doi: 10.3934/amc.2020053 |
[19] |
Johan Rosenkilde. Power decoding Reed-Solomon codes up to the Johnson radius. Advances in Mathematics of Communications, 2018, 12 (1) : 81-106. doi: 10.3934/amc.2018005 |
[20] |
José Moreira, Marcel Fernández, Miguel Soriano. On the relationship between the traceability properties of Reed-Solomon codes. Advances in Mathematics of Communications, 2012, 6 (4) : 467-478. doi: 10.3934/amc.2012.6.467 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]