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

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 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 − 1 is not a factor of q − 1 or q + 1, there is 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 Fq2 . Then 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 w(q−1) u and minimum distances d > q 2 .