  It has been a great challenge to construct new quantum MDS codes. In
particular, it is very hard to construct quantum MDS codes with relatively
large minimum distance. So far, except for some sparse lengths, all known
$q$-ary quantum MDS codes have minimum distance less than or equal to $q/2+1$.
In the present paper, we provide a construction of quantum MDS codes with
minimum distance bigger than $q/2+1$. In particular, we show existence of
$q$-ary quantum MDS codes with length $n=q^2+1$ and minimum distance $d$ for
any $d\le q-1$ and $d= q+1$(this result extends those given in
\cite{Gu11,Jin1,KZ12}); and with length $(q^2+2)/3$ and minimum distance $d$
for any $d\le (2q+2)/3$ if $3|(q+1)$. Our method is through Hermitian
self-orthogonal codes. The main idea of constructing Hermitian self-orthogonal
codes is based on the solvability in $\F_q$ of a system of homogenous equations
over $\F_{q^2}$.
