New q-ary quantum MDS codes with distances bigger than $\frac{q}{2}$

Abstract

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS $[[n,n-2d+2,d]]_q$ codes with minimum distances $d>\frac{q}{2}$ 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 the case $n=\frac{q^2-1}{m}$ while $m|q^2-1$ is neither a factor of $q-1$ nor $q+1$, no q-ary quantum MDS code with $d> \frac{q}{2}$ has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over $\mathbf{F}_{q^2}$. Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form $\frac{w(q^2-1)}{u}$ and minimum distances $d > \frac{q}{2}$ are presented.