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Maximal number of rational points on a curve of genus n over GF(2).
(Formerly M2388)
1

%I M2388 #25 May 02 2024 04:31:40

%S 3,5,6,7,8,9,10,10,11,12,13,14

%N Maximal number of rational points on a curve of genus n over GF(2).

%D J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. See N_2(g) on page 51.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Robin Visser, <a href="/A005527/a005527_1.txt">Incomplete table of n, a(n) for n = 0..50</a>.

%H L. E. Dickson, <a href="https://doi.org/10.2307/1988711">Quartic Curves Modulo 2</a>, Trans. Amer. Math. Soc. 16 (1915), no. 2, 111-120.

%H Jean-Pierre Serre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k55351747/f35.item">Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini</a>, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.

%H M. Tsfasman, S. Vlǎduţ, and D. Nogin, <a href="https://doi.org/10.1090/surv/139">Algebraic Geometric Codes: Basic Notions</a>, Math. Surveys Monogr., 139 American Mathematical Society, Providence, RI, 2007. xx+338 pp.

%H G. van der Geer, E. W. Howe, K. E. Lauter, and C. Ritzenthaler, <a href="https://www.manypoints.org/">Tables of Curves with Many Points</a>.

%H J. H. van Lint and G. van der Geer, <a href="https://doi.org/10.1007/978-3-0348-9286-5">Introduction to Coding Theory and Algebraic Geometry</a>, DMV Sem., 12 Birkhäuser Verlag, Basel, 1988. 83 pp.

%H C. Xing and H. Niederreiter, <a href="https://doi.org/10.1007/s006050050036">Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points</a>, Monatsh. Math. 127 (1999), no. 3, 219-241.

%e For n = 3, Dickson (1915) proved that the genus 3 quartic curve given by x^3*y + x^2*y^2 + x*z^3 + x^2*z^2 + y^3*z + y*z^3 = 0 with 7 rational points attains the maximal number of points for a genus 3 curve over GF(2), thus a(3) = 7. - _Robin Visser_, Aug 17 2023

%K nonn,more

%O 0,1

%A _N. J. A. Sloane_.

%E Edited by _Robin Visser_, Aug 16 2023, adding terms a(10)-a(11) computed by Serre.