Two-Point Coordinate Rings for GK-Curves

Giulietti and Korchmáros presented new curves with the maximal number of points over a field of size
$q^{6}$
. Garcia, Güneri, and Stichtenoth extended the construction to curves that are maximal over fields of size
$q^{2n}$
, for odd
$n \geq 3$
. The generalized GK-curves have affine equations
$x^{q}+x = y^{q+1}$
and
$y^{q^{2}}-y = z^{r}$
, 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.