Effective aspects of Riemann-Roch spaces in the Hilbert class field

Let $K$ be a finite field, $X$ and $Y$ two
curves over $K$, and $Y\rightarrow X$ an unramified abelian
cover with Galois group $G$. Let $D$ be a divisor on $X$
and $E$ its pullback on $Y$. Under mild conditions the linear space
associated with $E$ is a free $K[G]$-module. We study the
algorithmic aspects and applications of these modules.