Two-Point Coordinate Rings for GK-Curves
Giulietti and Korchmaros presented new curves with the maximal number of points over a field of size q(6). Garcia, Guneri, and Stichtenoth extended the construction to curves that are maximal over fields of size q(2n), for odd n >= 3. The generalized GK-curves have affine equations x(q) + x = y(q+1) and y(q2) - y = z(r), for r = (q(n) + 1)/(q + 1). We give a new proof for the maximality of the generalized GK-curves and we outline methods to efficiently obtain their two-point coordinate ring.
algebraic geometric codes maximal curves weierstrass points