Abstract
In this paper we solve three open problems on maximal curves with Frobenius dimension 3. In particular, we prove the existence of a maximal curve with order sequence (0, 1, 3, q).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdón M., Torres F.: On \({F_{q^2}}\) -maximal curves of genus (q−3)q/6. Beiträge Algebra Geom. 46(1), 241–260 (2005)
Cossidente A., Korchmáros G., Torres F.: On curves covered by the Hermitian curve. J. Algebra 216, 56–76 (1999)
Fuhrmann R., Torres F.: The genus of curves over finite fields with many rational points. Manuscr. Math. 89, 103–106 (1996)
Fuhrmann R., Garcia A., Torres F.: On maximal curves. J. Number Theory 67, 29–51 (1997)
Garcia A.: Curves over finite fields attaining the Hasse–Weil upper bound. In: European Congress of Mathematics, vol. II (Barcelona, 2000), Progr. Math., vol. 202, pp. 199–205. Birkhäuser, Basel (2001).
Garcia A.: On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin (2002).
Garcia A., Stichtenoth H.: Algebraic function fields over finite fields with many rational places. IEEE Trans. Inform. Theory 41, 1548–1563 (1995)
Garcia A., Stichtenoth H. (eds): Topics in geometry, coding theory and cryptography. In: Algebra and Applications, vol. 6. Springer, Dordrecht (2007).
Garcia A., Stichtenoth H., Xing C.P.: On subfields of the Hermitian function field. Compos. Math. 120, 137–170 (2000)
Giulietti M., Korchmáros G.: A new family of maximal curves over a finite field. Math. Ann. 343, 229–245 (2009)
Hefez A., Kakuta N.: On the geometry of non-classical curves. Bol. Soc. Bras. Mat. 23(1)(2), 79–91 (1992).
Hirschfeld J.W.P., Korchmáros G., Torres F.: Algebraic Curves Over a Finite Field. Princeton Univ. Press, Princeton and Oxford (2008).
Homma M.: Duality of spaces and their tangent surfaces in characteristic p > 0. Ark. Math. 28(2), 221–235 (1991)
Korchmáros G., Torres F.: Embedding of a maximal curve in a Hermitian variety. Compos. Math. 128, 95–113 (2001)
Rück H.G., Stichtenoth H.: A characterization of Hermitian function fields over finite fields. J. Reine Angew. Math. 457, 185–188 (1994)
Stichtenoth H.: Algebraic Function Fields and Codes. Springer-Verlag, New York, Berlin, Heidelberg (1993)
Stichtenoth H., Xing C.P.: The genus of maximal function fields. Manuscr. Math. 86, 217–224 (1995)
Torres F.: Algebraic curves with many points over finite fields. In: Martínez-Moro, E., Munuera, C., Ruano, D. (eds) Advances in Algebraic Geometry Codes, pp. 221–256. World Scientific Publishing Company, Singapore (2008)
van der Geer G.: Curves over finite fields and codes: In: European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math., vol. 202, pp. 225–238. Birkhäuser, Basel (2001).
van der Geer G.: Coding theory and algebraic curves over finite fields: a survey and questions. In: Applications of Algebraic Geometry to Coding Theory, Physics and Computation, NATO Sci. Ser. II Math. Phys. Chem., vol. 36, pp. 139–159. Kluwer, Dordrecht (2001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Leo Storme.
Dedicated to the memory of András Gács (1969–2009).
Rights and permissions
About this article
Cite this article
Fanali, S., Giulietti, M. On some open problems on maximal curves. Des. Codes Cryptogr. 56, 131–139 (2010). https://doi.org/10.1007/s10623-010-9389-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-010-9389-5