Hφholdt, van Lint and Pellikaan proposed a generalization of one-point AG codes, called the evaluation codes. We show that an evaluation code from a weight function can be constructed as Miura's generalization of one-point AG codes. Hence we can construct a one-point AG code as good as a given evaluation code from a weight function.

Recently, Garcia and Stichtenoth proposed sequences of algebraic function fields with finite constant fields such that their sequences attain the Drinfeld-Vl bound. In the sequences, the third algebraic function fields are Artin-Schreier extensions of Hermitian function fields. On the other hand, Miura presented powerful tools to construct one-point algebraic geometric (AG) codes from algebraic function fields. In this paper, we clarify rational functions of the third algebraic function fields which correspond to generators of semigroup of nongaps at a specific place of degree one. Consequently, we show generator matrices of the one-point AG codes with respect to the third algebraic function fields for any dimension by using rational functions of monomial type and rational points.