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 2 − 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 F q 2 . 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 2 −1) u and minimum distances d > q 2 .

A basic problem for the constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of a set of k-dimensional subspaces in F n q such that the subspace distance satisfies d(U, V ) = 2k − 2 dim(U ∩ V ) ≥ d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximal rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem of the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A q (n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

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