  Giulietti and Korchm\'aros presented new curves with the maximal number of
points over a field of size q^6. Garcia, G\"uneri, 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^{q^2}-y^q = 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.
