Entry details for q = 23 = 8, g = 6
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Lower bound Nmin = 33

Submitted by Gerrit Oomens
Date 01-01-1900
Reference H. Stichtenoth
Algebraic-geometric codes associated to Artin-Schreier extensions of F_q(z)
In: Proc. 2nd Int. Workshop on Alg. and Comb. Coding Theory, Leningrad (1990), p. 203-206.
Tags Fibre products of Artin-Schreier curves

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Explicit example     
Everett Howe
12-08-2019 01:28
Here is an example of a genus-6 curve over F_8 with 33 points:

y^2 + y = x^5 + x^3
z^2 + z = r*x^5 + r^2*x^3

(Here r is an element of F_8 satisfying r^3 + r + 1 = 0.)
Upper bound Nmax = 34

Submitted by Everett Howe
Date 05-03-2010
Reference Everett W. Howe, Kristin E. Lauter
New methods for bounding the number of points on curves over finite fields
Geometry and Arithmetic (C. Faber, G. Farkas, and R. de Jong, eds.), European Mathematical Society, 2012, pp. 173–212
The program IsogenyClasses.magma (available at the URL linked to above) shows that a genus-6 curve over F_8 cannot have more than 35 points, and if such a curve has 35 points it must have real Weil polynomial (x + 5)^3 * (x^3 + 11*x^2 + 37*x + 37) and be a double cover of a genus-3 curve with real Weil polynomial (x^3 + 11*x^2 + 37*x + 37). The program 8-6.magma, also available at the URL linked to above, enumerates the ordinary genus-6 double covers of all such plane quartics, and shows that none of them has 35 points. Therefore N_8(6) is at most 34.
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