  In a previous paper we showed that, for any $n \ge m+2$, most sets of $n$
points in $\RR^m$ are determined (up to rotations, reflections, translations
and relabeling of the points) by the distribution of their pairwise distances.
But there are some exceptional point configurations which are not
reconstructible from the distribution of distances in the above sense. In this
paper, we present a reconstructibility test with running time $O(n^{11})$. The
cases of orientation preserving rigid motions (rotations and translations) and
scalings are also discussed.
