  In this paper we studied generalization of Hermitian function field proposed
by A.Garcia and H.Stichtenoth. We calculated a Weierstrass semigroup of the
point at infinity for the case q=2, r>=3. It turned out that unlike Hermitian
case, we have already three generators for the semigroup. We then applied this
result to codes, constructed on generalized Hermitian function fields. Further,
we applied results of C.Kirfel and R.Pellikaan to estimating a Feng-Rao
designed distance for GH-codes, which improved on Goppa designed distance.
Next, we studied the question of codes dual to GH-codes. We identified that the
duals are also GH-codes and gave an explicit formula. We concluded with some
computational results. In particular, a new record-giving [32,16,>=12]-code
over GF(8) was presented.
