Two-point coordinate rings for GK-curves

TL;DR

This paper provides a new proof of the maximality of generalized GK-curves and introduces methods for efficiently computing their two-point coordinate rings, extending the understanding of these algebraic curves over finite fields.

Contribution

It offers a novel proof of the maximality of generalized GK-curves and develops methods to efficiently determine their two-point coordinate rings.

Findings

New proof of maximality for generalized GK-curves

Efficient methods for computing two-point coordinate rings

Extension of GK-curve properties to larger finite fields

Abstract

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.