Generalized Hermitian Codes over GF(2^r)

Abstract

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.