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 vertical bar q + 1 or m vertical bar q - 1 there are complete results. In the case n = q(2)-1/m while m vertical bar 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 newq-ary quantum codes in this case. Moreover many newq-ary quantum MDS codes with lengths of the form w(q(2)-1)/u and minimum distances d > q/2 are presented.