Quantum MDS codes over small fields
Markus Grassl, Martin Rötteler · 2015
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 = 2m, that appears to be new.