# Some constructions of quantum MDS codes. (arXiv:1907.04391v1 [quant-ph])

We construct quantum MDS codes for quantum systems of dimension $q$ of length
$q^2+1$ and minimum distance $d$ for all $d \leqslant q+1$, $d \neq q$. These
codes are shown to exist by proving that there are classical generalised
Reed-Solomon codes which are contained in their Hermitian-dual. These
constructions include many constructions which were previously known but in
some cases these codes appear to be new. We go on to prove that if $d\geqslant q+2$ then there in no generalised Reed-Solomon code which is contained in its
Hermitian dual. We also construct a $[\![ 18,0,10 ]\!] _5$ quantum MDS code, a
$[\![ 18,0,10 ]\!] _7$ quantum MDS code and a $[\![ 14,0,8 ]\!] _5$ quantum
MDS code, which are the first quantum MDS codes discovered for which $d \geqslant q+3$, apart from the $[\![ 10,0,6 ]\!] _3$ quantum MDS code derived
from Glynn's code.