One- and Two-Point Codes Over Kummer Extensions

We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by y
^m
= f (x)
^λ
, where f (x) is a separable polynomial over F
_q
. In addition, we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one- and two-point algebraic geometric codes with good parameters.