  For Kummer extensions defined by $y^m = f (x)$, where $f (x)$ is a separable
polynomial over the finite field $\mathbb{F}_q$, we compute the number of
Weierstrass gaps at two totally ramified places. For many totally ramified
places we give a criterion to find pure gaps at these points and present
families of pure gaps. We then apply our results to construct many points
algebraic geometric codes with good parameters.
