Minimal value set polynomials and a generalization of the Hermitian curve

Abstract: We use a recent characterization of minimal value set polynomials and $q$- Frobenius nonclassical curves to construct curves that generalize the Hermitian curve. The genus $g$ and the number $N$ of $\mathbb{F}_q$-rational points of the curves are computed and, for a special family of these curves, we determine the Weierstrass semigroup at the unique point at infinity. These special curves yield new examples of Castle curves and improve on a previous example of Garcia-Stichtenoth of curves with large ratio $N/g$.