Quantum MDS Codes over Small Fields

Understand

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.