Explicit Riemann-Roch spaces in the Hilbert class field

Let $\mathbf{K}$ be a finite field, $X$ and $Y$ two curves over $\mathbf{K}$, and $Y\to X$ 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 $\mathbf{K}[G]$-module. We study the algorithmic aspects and applications of these modules.