In this correspondence, a generalization of Hermitian function field proposed by Garcia and Stichtenoth is studied. A Weierstrass semigroup of the point at infinity for the case q = 2, r ges 3 is calculated. It turns out that unlike for the Hermitian case, there are already three generators for the semigroup. This result then is applied to codes, constructed on generalized Hermitian (GH) function fields. Further, results of Kirfel and Pellikaan are applied to estimating a Feng-Rao designed distance for GH codes, which improve on the Goppa designed minimum distance. Next, the question of codes dual to GH codes is studied. It is shown that the duals are also GH codes and an explicit formula is given. In particular, this formula enables one to calculate the parameters of a dual code. A new record-giving [32,16, ges 12]-code over GF(8) is presented as one of the examples