Quantum MDS codes over small fields

We consider quantum MDS (QMDS) codes for quantum systems of dimension q with lengths up to q² + 2 and minimum distances up to q + 1. We show how starting from QMDS codes of length q² + 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 n0(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 q² +2, where q = 2^m, that appears to be new.