A169873 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_2^n.

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

Robin Visser, Table of n, a(n) for n = 1..3000

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

Cf. A005525, A169869-A169883.

Sequence in context: A293555 A077626 A338122 * A363788 A079471 A134351

Adjacent sequences: A169870 A169871 A169872 * A169874 A169875 A169876

Keyword

nonn

Author

N. J. A. Sloane, Jul 05 2010

Extensions

More terms from Robin Visser, Oct 01 2023

Status

approved