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

Abstract: Constructions of quantum MDS codes have been studied by many authors. We refer to the table in page 1482 of [3] for known constructions. However there are only few $q$-ary quantum MDS $[[n,n-2d+2,d]]<em>q$ codes with minimum distances $d&gt;\frac{q}{2}$ for sparse lengths $n&gt;q+1$. In the case $n=\frac{q<sup>2-1}{m}$ where $m|q+1$ or $m|q-1$ there are complete results. In the case $n=\frac{q<sup>2-1}{m}$ where $m|q<sup>2-1$ is not a factor of $q-1$ or $q+1$, there is no $q$-ary quantum MDS code with $d&gt; \frac{q}{2}$ has been constructed. In this paper we propose a direct approch to construct Hermitian self-orthogonal codes over ${\bf F}</em>{q<sup>2}$. Thus we give some new $q$-ary quantum codes in this case. Moreover we present many new $q$-ary quantum MDS codes with lengths of the form $\frac{w(q<sup>2-1)}{u}$ and minimum distances $d &gt; \frac{q}{2}$.