  We consider quantum MDS (QMDS) codes for quantum systems of dimension $q$
with lengths up to $q^2+2$ and minimum distances up to $q+1$. We show how
starting from QMDS codes of length $q^2+1$ based on cyclic and constacyclic
codes, new QMDS codes can be obtained by shortening. We provide numerical
evidence for our conjecture that almost all admissible lengths, from a lower
bound $n_0(q,d)$ on, are achievable by shortening. Some additional codes that
fill gaps in the list of achievable lengths are presented as well along with a
construction of a family of QMDS codes of length $q^2+2$, where $q=2^m$, that
appears to be new.
