Abstract
The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over \(\mathbb {F}_{q}\,\). Research on bounds for A(q) is closely connected with the so-called asymptotic main problem in Coding Theory. In this paper, we study some generalizations of this number for non-irreducible curves, their connection with A(q) and their application in Coding Theory. We also discuss the possibility of constructing codes from non-irreducible curves, both from theoretical and practical point of view.
Partially supported by the project MTM2015-65764-C3-1-P (MINECO/FEDER).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abhyankar, S.S.: Irreducibility criterion for germs of analytic functions of two complex variables. Adv. Math. 74, 190–257 (1989)
Atiyah, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Reading (1969)
Beelen, P., Pellikaan, R.: The Newton-polygon of plane curves with many rational points. Des. Codes Crypt. 21, 41–67 (2000)
Campillo, A.: Algebroid Curves in Positive Characteristic. Lecture Notes in Mathematics, vol. 813. Springer, Berlin/New York (1980)
Campillo, A., Farrán, J.I.: Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes. Math. Comput. 71, 1759–1780 (2001)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: “Singular 4-1-0”, A computer algebra system for polynomial computations, Centre for Computer Algebra, TU Kaiserslautern (2016). Available via http://www.singular.uni-kl.de/
Drinfeld, V.G., Vlăduţ, S.G.: Number of points of an algebraic curve. Funktsional’-nyi Analiz i Ego Prilozhenia 17, 53–54 (1983)
García, A., Stichtenoth, H., Thomas, M.: On towers and composita of towers of function fields over finite fields. Finite Fields Appl. 3, 257–274 (1997)
Greuel, G.-M., Lossen, Ch., Shustin, E.: Plane curves of minimal degree with prescribed singularities. Invent. Math. 133(3), 539–580 (1998)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Lachaud, G.: Les Codes Géométriques de Goppa. Sém. Bourbaki, 37ème année 641, 1984–1985 (1986). Astérisque
Serre, J.P.: Sur le nombre des points rationnels d’un courbe algébrique sur un corps fini. C. R. Acad. Sc. Paris 296, 397–402 (1983)
Tsfasman, M.A.: Goppa codes that are better than Varshamov-Gilbert bound. Prob. Peredachi Inform. 18, 3–6 (1982)
Tsfasman, M.A., Vlăduţ, S.G.: Algebraic-Geometric Codes. Mathematics and Its Applications, vol. 58. Kluwer Academic, Amsterdam (1991)
Tsfasman, M.A., Vlăduţ, S.G., Zink, Th.: Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound. Math. Nachr. 109, 21–28 (1982)
Weil, A.: Basic Number Theory. Grundlehren der Mathematischen Wissenschaften, Bd. 144. Springer, New York (1974)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Farrán, J.I. (2018). Asymptotics of Reduced Algebraic Curves Over Finite Fields. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-96827-8_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96826-1
Online ISBN: 978-3-319-96827-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)