New q -ary quantum MDS codes with distances bigger than q/2

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>q/2 for sparse lengths n>q+1 . In the case n=q^2-1/m where m|q+1 or m|q-1 there are complete results. In the case n=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> q/2 has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over 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 w(q^2-1)/u and minimum distances d > q/2 are presented.