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

Author

N. J. A. Sloane, Jul 05 2010

Keywords

• nonn

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.

Crossrefs

Cf. A005525, A169869-A169883.

Programs

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

Extensions

More terms from Robin Visser, Oct 01 2023