  Let $k$ be a fixed finite geometric extension of the rational function field
$\mathbb{F}_q(t)$. Let $F/k$ be a finite abelian extension such that there is
an $\Fq$-rational place $\infty$ in $k$ which splits in $F/k$ and let
$\mathcal{O}_F$ denote the integral closure in $F$ of the ring of functions in
$k$ that are regular outside $\infty$. We describe algorithms for computing the
divisor class number and in certain cases for computing the structure of the
divisor class group and discrete logarithms between Galois conjugate divisors
in the divisor class group of $F$. The algorithms are efficient when $F$ is a
narrow ray class field or a small index subextension of a narrow ray class
field.\\ \\ We prove that for all prime $\ell$ not dividing $q(q-1)[F:k]$, the
structure of the $\ell$-part of the ideal class group $\p(\cO_F)$ of
$\mathcal{O}_F$ is determined by Kolyvagin derivative classes that are
constructed out of Euler systems associated with Stark units. This leads to an
algorithm to compute the structure of the $\ell$ primary part of the divisor
class group of a narrow ray class field for all primes $\ell$ not dividing
$q(q-1)[F:k]$.
