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Giulietti and Korchmáros presented new curves with the maximal number of points over a field of size q6. Garcia, Güneri, and Stichtenoth extended the construction to curves that are maximal over fields of size q2n, for odd n ≥ 3. The generalized GK-curves have affine equations xq+x = yq+1 and yq2-y = zr, for r=(qn+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.
Date of Publication: Feb. 2011