login
A169873
Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_2^n.
1
6, 10, 18, 33, 53, 97, 172, 321, 603, 1153, 2227, 4353, 8553, 16897, 33491, 66561, 132519, 264193, 527183, 1052673, 2102943, 4202497, 8400192, 16793601, 33577603, 67141633, 134264067, 268500993, 536963592, 1073872897, 2147669011, 4295229441, 8590305319, 17180393473, 34360479823
OFFSET
1,1
REFERENCES
J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
LINKS
Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
Gerard van der Geer et al., Tables of curves with many points
Gerard van der Geer and Marcel van der Vlugt, Tables of curves with many points, Math. Comp. 69 (2000) 797-810.
PROG
(Sage)
def a(n):
if n==2: return 10
elif (n%2 == 0): return 2^n + 1 + 2^(n/2+2)
elif ((floor(2^(n/2+1))%2 == 0) or (2^n-1).is_square()
or (4*2^n-3).is_square() or (4*2^n-7).is_square()):
if (frac(2^(n/2+1)) > ((sqrt(5)-1)/2)): return 2^n + 2*floor(2^(n/2+1))
else: return 2^n + 2*floor(2^(n/2+1)) - 1
else: return 2^n + 1 + 2*floor(2^(n/2+1)) # Robin Visser, Oct 01 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 05 2010
EXTENSIONS
More terms from Robin Visser, Oct 01 2023
STATUS
approved