  We construct quantum MDS codes with parameters $ [\![ q^2+1,q^2+3-2d,d ]\!]
_q$ 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 contain
their Hermitian dual. These constructions include many constructions which were
previously known but in some cases these codes are new. We go on to prove that
if $d\geqslant q+2$ then there is no generalised Reed-Solomon
$[n,n-d+1,d]_{q^2}$ code which contains its Hermitian dual. We also construct
an $ [\![ 18,0,10 ]\!] _5$ quantum MDS code, an $ [\![ 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.
