A new family of Castle and Frobenius nonclassical curves

Motivated by the study of minimal value set polynomials, we construct ${\mathbb{F}}_q$-Frobenius nonclassical curves with a large number of ${\mathbb{F}}_q$-rational points. For some of these curves, we determine the Weierstrass semigroup at the unique point at infinity. In particular, we prove that they yield new examples of Castle curves.