New q ary quantum MDS codes with distances bigger than \frac{q}{2}$q2
The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474 1484, 2015) for known constructions. However, there have been constructed only a few q ary quantum MDS [n,n 2d+2,d]]_q$[n,n 2d+2,d]]q codes with minimum distances $d>\frac{q}{2}$d>q2 for sparse lengths $n>q+1$$n>q+1. In the case $n=\frac{q^2 1}{m}$n=q2 1m where $m|q+1$$m|q+1 or $m|q 1$$m|q 1 there are complete results. In the case $n=\frac{q^2 1}{m}$n=q2 1m while $m|q^2 1$$m|q2 1 is neither a factor of $q 1$$q 1 nor $q+1$$q+1, no q ary quantum MDS code with $d> \frac{q}{2}$d>q2 has been constructed. In this paper we propose a direct approach to construct Hermitian self orthogonal codes over \mathbf{F}_{q^2}$Fq2. Then we give some new q ary quantum codes in this case. Moreover many new q ary quantum MDS codes with lengths of the form \frac{w(q^2 1){u}$w(q2 1)u and minimum distances $d \frac{q}{2}$d>q2 are presented.
Quantum Inf. Process.