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.