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Tables of curves with many points


Authors: Gerard van der Geer and Marcel van der Vlugt
Journal: Math. Comp. 69 (2000), 797-810
MSC (1991): Primary 11G20, 14G15; Secondary 14H05
DOI: https://doi.org/10.1090/S0025-5718-99-01143-6
Published electronically: August 18, 1999
MathSciNet review: 1654002
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Abstract | References | Similar Articles | Additional Information

Abstract: These tables record results on curves with many points over finite fields. For relatively small genus ($0\leq g \leq 50$) and $q$ a small power of $2$ or $3$ we give in two tables the best presently known bounds for $N_{q}(g)$, the maximum number of rational points on a smooth absolutely irreducible projective curve of genus $g$ over a field $\mathbb{F}_{q}$ of cardinality $q$. In additional tables we list for a given pair $(g,q)$ the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.


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  • [A] R. Auer: Ray class fields of global function fields with many rational places. Report University of Oldenburg, 1998.
  • [D] L.E. Dickson: Geometrical and invariantive theory of quartic curves modulo $2$. Am. J. Math. 37 (1915), 337-354
  • [Do] J. Doumen: Master's thesis. Leiden University, 1998.
  • [E] N. Elkies: Private communication, 1997.
  • [F-T] Rainer Fuhrmann and Fernando Torres, The genus of curves over finite fields with many rational points, Manuscripta Math. 89 (1996), no. 1, 103–106. MR 1368539, https://doi.org/10.1007/BF02567508
  • [G-S] A. Garcia, H. Stichtenoth: A class of polynomials over finite fields. Preprint, 1998.
  • [G-S-X] A. Garcia, H. Stichtenoth, C.P. Xing: On subfields of the Hermitian function field. Preprint 1998.
  • [G-V1] Gerard van der Geer and Marcel van der Vlugt, Curves over finite fields of characteristic 2 with many rational points, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 6, 593–597 (English, with English and French summaries). MR 1240806
  • [G-V2] G. van der Geer and M. van der Vlugt, Generalized Hamming weights of codes and curves over finite fields with many points, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 417–432. MR 1360517
  • [G-V3] Gerard van der Geer and Marcel van der Vlugt, Quadratic forms, generalized Hamming weights of codes and curves with many points, J. Number Theory 59 (1996), no. 1, 20–36. MR 1399697, https://doi.org/10.1006/jnth.1996.0086
  • [G-V4] Gerard van der Geer and Marcel van der Vlugt, How to construct curves over finite fields with many points, Arithmetic geometry (Cortona, 1994) Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 169–189. MR 1472497
  • [G-V5] G. van der Geer, M. van der Vlugt: Tables for the function $N_{q}(g)$. Regularly updated tables at:
  • [G-V6] G. van der Geer, M. van der Vlugt: Generalized Reed-Muller codes and curves with many points. J. of Number Theory 72 (1998) 257-268. CMP 99:04
  • [G-V7] G. van der Geer, M. van der Vlugt: Constructing curves over finite fields with many rational points by solving linear equations. Report W 97-29, Leiden University 1997.
  • [G] V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259 (1981), no. 6, 1289–1290 (Russian). MR 628795
  • [H] J.P. Hansen: Group codes and algebraic curves. Mathematica Gottingensis, Schriftenreihe SFB Geometrie und Analysis, Heft 9, 1987.
  • [H-Le B] Gaétan Haché and Dominique Le Brigand, Effective construction of algebraic geometry codes, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1615–1628. Special issue on algebraic geometry codes. MR 1391019, https://doi.org/10.1109/18.476233
  • [H-S] Johan P. Hansen and Henning Stichtenoth, Group codes on certain algebraic curves with many rational points, Appl. Algebra Engrg. Comm. Comput. 1 (1990), no. 1, 67–77. MR 1325513, https://doi.org/10.1007/BF01810849
  • [I] Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721–724 (1982). MR 656048
  • [L1] Kristin Lauter, Ray class field constructions of curves over finite fields with many rational points, Algorithmic number theory (Talence, 1996) Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 187–195. MR 1446511, https://doi.org/10.1007/3-540-61581-4_54
  • [L2] K. Lauter: Non-existence of a curve over $\mathbb{F}_{3}$ of genus $5$ with $14$ rational points. Preprint 1998.
  • [L3] K. Lauter: Improved upper bounds for the number of rational points on algebraic curves over finite fields. Preprint, University of Michigan, 1999.
  • [M-Z-Z] Oscar Moreno, Dmitrii Zinoviev, and Victor Zinoviev, On several new projective curves over 𝐹₂ of genus 3, 4, and 5, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1643–1648. Special issue on algebraic geometry codes. MR 1391022, https://doi.org/10.1109/18.476236
  • [N-X1] Harald Niederreiter and Chaoping Xing, Quasirandom points and global function fields, Finite fields and applications (Glasgow, 1995) London Math. Soc. Lecture Note Ser., vol. 233, Cambridge Univ. Press, Cambridge, 1996, pp. 269–296. MR 1433154, https://doi.org/10.1017/CBO9780511525988.022
  • [N-X2] Harald Niederreiter and Chaoping Xing, Explicit global function fields over the binary field with many rational places, Acta Arith. 75 (1996), no. 4, 383–396. MR 1387872
  • [N-X3] Harald Niederreiter and Chaoping Xing, Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places, Acta Arith. 79 (1997), no. 1, 59–76. MR 1438117
  • [N-X4] H. Niederreiter, C. P. Xing: Drinfeld modules of rank 1 and algebraic curves with many rational points II. Acta Arithm. 81 (1997), 81-100. CMP 97:14
  • [N-X5] Harald Niederreiter and Chaoping Xing, Global function fields with many rational places over the ternary field, Acta Arith. 83 (1998), no. 1, 65–86. MR 1489567
  • [N-X6] H. Niederreiter, C. P. Xing: Algebraic curves with many rational points over finite fields of characteristic $2$. To appear in: Proc.Number Theory Conference (Zakopane 1997), de Gruyter, Berlin.
  • [N-X7] H. Niederreiter, C. P. Xing: A general method of constructing global function fields with many rational places. To appear in: Algorithmic Number Theory (Portland 1998), Lecture Notes in Comp. Science, Springer, Berlin.
  • [N-X8] H. Niederreiter, C. P. Xing: Nets, $(t,s)$-sequences and algebraic geometry. To appear in Pseudo- and quasi-random point sets, P.Hellekalek, G. Larcher, Eds. Lecture Notes in Statistics, Springer, New York, 1998.
  • [O-S] F. Özbudak, H. Stichtenoth: Curves with many points and configurations of hyperplanes over finite fields. Preprint 1998.
  • [Sch] R. Schoof: Algebraic curves and coding theory. UTM 336, Univ.of Trento, 1990.
  • [Seg] Beniamino Segre, Introduction to Galois geometries, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8 (1967), 133–236 (English, with Italian summary). MR 0238846
  • [Sem] S. Sémirat: Genus theory for quadratic fields and applications. Preprint Université Paris VI, 1998.
  • [S1] Jean-Pierre Serre, Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 9, 397–402 (French, with English summary). MR 703906
    Jean-Pierre Serre, Œuvres. Vol. I, Springer-Verlag, Berlin, 1986 (French). 1949–1959. MR 926689
    Jean-Pierre Serre, Œuvres. Vol. II, Springer-Verlag, Berlin, 1986 (French). 1960–1971. MR 926690
    Jean-Pierre Serre, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986 (French). 1972–1984. MR 926691
  • [S2] Jean-Pierre Serre, Nombres de points des courbes algébriques sur 𝐹_{𝑞}, Seminar on number theory, 1982–1983 (Talence, 1982/1983) Univ. Bordeaux I, Talence, 1983, pp. Exp. No. 22, 8 (French). MR 750323
    Jean-Pierre Serre, Œuvres. Vol. I, Springer-Verlag, Berlin, 1986 (French). 1949–1959. MR 926689
    Jean-Pierre Serre, Œuvres. Vol. II, Springer-Verlag, Berlin, 1986 (French). 1960–1971. MR 926690
    Jean-Pierre Serre, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986 (French). 1972–1984. MR 926691
  • [S3] Jean-Pierre Serre, Œuvres. Vol. I, Springer-Verlag, Berlin, 1986 (French). 1949–1959. MR 926689
    Jean-Pierre Serre, Œuvres. Vol. II, Springer-Verlag, Berlin, 1986 (French). 1960–1971. MR 926690
    Jean-Pierre Serre, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986 (French). 1972–1984. MR 926691
  • [S4] J-P. Serre: Rational points on curves over finite fields. Notes of lectures at Harvard University 1985.
  • [S5] J-P. Serre: Letter to G. van der Geer, September 1, 1997.
  • [Sh] V. Shabat: Unpublished manuscript, University of Amsterdam, 1997/98.
  • [St1] Henning Stichtenoth, Self-dual Goppa codes, J. Pure Appl. Algebra 55 (1988), no. 1-2, 199–211. MR 968575, https://doi.org/10.1016/0022-4049(88)90046-1
  • [St2] H. Stichtenoth: Algebraic-geometric codes associated to Artin-Schreier extensions of $\mathbb{F}_{q}(z)$. In: Proc. 2nd Int. Workshop on Alg. and Comb. Coding Theory, Leningrad (1990), 203-206.
  • [S-V] Karl-Otto Stöhr and José Felipe Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986), no. 1, 1–19. MR 812443, https://doi.org/10.1112/plms/s3-52.1.1
  • [Su] M. Suzuki: Private communication, 1998.
  • [Wi] M. Wirtz : Konstruktion und Tabellen linearer Codes. Westfälische Wilhelms-Universität Münster, 1991.
  • [Wo] Jacques Wolfmann, Nombre de points rationnels de courbes algébriques sur des corps finis associées à des codes cycliques, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 8, 345–348 (French, with English summary). MR 909562
  • [X-N] C. P. Xing, H. Niederreiter: Drinfeld modules of rank 1 and algebraic curves with many rational points. Report Austrian Academy of Sciences, Vienna, 1996.

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Additional Information

Gerard van der Geer
Affiliation: Faculteit WINS, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Email: geer@wins.uva.nl

Marcel van der Vlugt
Affiliation: Mathematisch Instituut, Rijksuniversiteit te Leiden, Niels Bohrweg 1, 2300 RA Leiden, The Netherlands
Email: vlugt@wi.leidenuniv.nl

DOI: https://doi.org/10.1090/S0025-5718-99-01143-6
Received by editor(s): October 2, 1997
Received by editor(s) in revised form: April 28, 1998
Published electronically: August 18, 1999
Article copyright: © Copyright 2000 American Mathematical Society