American Institute of Mathematical Sciences

Advanced
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

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, ().   Google Scholar

[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.   Google Scholar

[3]

G. Lachaud, The parameters of projective Reed-Muller codes,, Discrete Math., 81 (1990), 217.  doi: 10.1016/0012-365X(90)90155-B.  Google Scholar

[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.  Google Scholar

[5]

S. Lang and A. Weil, Number of points of varieties in finite fields,, Amer. J. Math., 76 (1954), 819.   Google Scholar

[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.  doi: 10.1016/j.crma.2007.01.006.  Google Scholar

[7]

F. Rodier and A. Sboui, Highest numbers of points of hypersurfaces and generalized Reed-Muller codes,, Finite Fields Appl., 14 (2008), 816.  doi: 10.1016/j.ffa.2008.02.001.  Google Scholar

[8]

A. Sboui, Second highest number of points of hypersurfaces in $\mathbb F_q^n$,, Finite Fields Appl., 13 (2007), 444.  doi: 10.1016/j.ffa.2005.11.002.  Google Scholar

[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.  doi: 10.1016/j.disc.2009.03.021.  Google Scholar

[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.   Google Scholar

[11]

A. B. Sørensen, Projective Reed-Muller codes,, IEEE Trans. Inf. Theory, 37 (1991), 1567.  doi: 10.1109/18.104317.  Google Scholar

[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.  doi: 10.1016/0012-365X(93)E0009-S.  Google Scholar

[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.  doi: 10.1016/0097-3165(93)90076-K.  Google Scholar

[14]

L. Storme and H. Van Maldeghem, Cyclic arcs in PG$(2,q)$,, J. Algebraic Combin., 3 (1994), 113.  doi: 10.1023/A:1022454221497.  Google Scholar

[15]

L. Storme and H. Van Maldeghem, Arcs fixed by a large cyclic group,, Atti Sem. Mat. Fis. Univ. Modena, XLIII (1995), 273.   Google Scholar

show all references

References:
[1]

A. Couvreur, An upper bound on the number of rational points of arbitrary projective varieties over finite fields,, preprint, ().   Google Scholar

[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.   Google Scholar

[3]

G. Lachaud, The parameters of projective Reed-Muller codes,, Discrete Math., 81 (1990), 217.  doi: 10.1016/0012-365X(90)90155-B.  Google Scholar

[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.  Google Scholar

[5]

S. Lang and A. Weil, Number of points of varieties in finite fields,, Amer. J. Math., 76 (1954), 819.   Google Scholar

[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.  doi: 10.1016/j.crma.2007.01.006.  Google Scholar

[7]

F. Rodier and A. Sboui, Highest numbers of points of hypersurfaces and generalized Reed-Muller codes,, Finite Fields Appl., 14 (2008), 816.  doi: 10.1016/j.ffa.2008.02.001.  Google Scholar

[8]

A. Sboui, Second highest number of points of hypersurfaces in $\mathbb F_q^n$,, Finite Fields Appl., 13 (2007), 444.  doi: 10.1016/j.ffa.2005.11.002.  Google Scholar

[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.  doi: 10.1016/j.disc.2009.03.021.  Google Scholar

[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.   Google Scholar

[11]

A. B. Sørensen, Projective Reed-Muller codes,, IEEE Trans. Inf. Theory, 37 (1991), 1567.  doi: 10.1109/18.104317.  Google Scholar

[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.  doi: 10.1016/0012-365X(93)E0009-S.  Google Scholar

[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.  doi: 10.1016/0097-3165(93)90076-K.  Google Scholar

[14]

L. Storme and H. Van Maldeghem, Cyclic arcs in PG$(2,q)$,, J. Algebraic Combin., 3 (1994), 113.  doi: 10.1023/A:1022454221497.  Google Scholar

[15]

L. Storme and H. Van Maldeghem, Arcs fixed by a large cyclic group,, Atti Sem. Mat. Fis. Univ. Modena, XLIII (1995), 273.   Google Scholar

[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]

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
[9]

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
[10]

Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006
[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]

László Mérai, Igor E. Shparlinski. Unlikely intersections over finite fields: Polynomial orbits in small subgroups. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1065-1073. doi: 10.3934/dcds.2020070
[16]

Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195
[17]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[18]

Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443
[19]

Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020053
[20]

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

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences