Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than is significantly more difficult than constructing those with minimum distance less than or equal to . Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than . In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than . For instance, some examples in these classes include q-ary -quantum MDS codes for cases: (i) and ; (ii) and ; (iii) and ; and (iv) and .