  In this paper we introduce digital nets over finite abelian groups which
contain digital nets over finite fields and certain rings as a special case. We
prove a MacWilliams type identity for such digital nets. This identity can be
used to compute the strict $t$-value of a digital net over finite abelian
groups. If the digital net has $N$ points in the $s$ dimensional unit cube
$[0,1]^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$
operations and the weight enumerator polynomial can be computed in
$\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of
integers. By precomputing some values the number of operations of computing the
weight enumerator polynomial can be reduced further.
