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Constructions of quantum MDS codes have been studied by many authors. Werefer to the table in page 1482 of [3] for known constructions. However thereare only few $q$-ary quantum MDS $[[n,n-2d+2,d]]_q$ codes with minimumdistances $d>\frac{q}{2}$ for sparse lengths $n>q+1$. In the case$n=\frac{q^2-1}{m}$ where $m|q+1$ or $m|q-1$ there are complete results. In thecase $n=\frac{q^2-1}{m}$ where $m|q^2-1$ is not a factor of $q-1$ or $q+1$,there is no $q$-ary quantum MDS code with $d> \frac{q}{2}$ has beenconstructed. In this paper we propose a direct approch to construct Hermitianself-orthogonal codes over ${\bf F}_{q^2}$. Thus we give some new $q$-aryquantum codes in this case. Moreover we present many new $q$-ary quantum MDScodes with lengths of the form $\frac{w(q^2-1)}{u}$ and minimum distances $d >\frac{q}{2}$.

New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$