  We investigate complete arcs of degree greater than two, in projective planes
over finite fields, arising from the set of rational points of a generalization
of the Hermitian curve. The degree of the arcs is closely related to the number
of rational points of a class of Artin-Schreier curves which is calculated by
using exponential sums via Coulter's approach. We also single out some examples
of maximal curves.
