New LMRD bounds for constant dimension codes and improved constructions

Let V ∼= Fq be a v-dimensional vector space over the finite field Fq with q elements. By [ V k ] we denote the set of all k-dimensional subspaces in V . Its size is given by the q-binomial coefficient [ v k ]q = ∏k−1 i=0 qv−qi qk−qi for 0 ≤ k ≤ v and 0 otherwise. The set of all subspaces of V forms a metric space associated with the so-called subspace distance dS(U,W ) = dim(U + W ) − dim(U ∩ W ), cf. [20, Lemma 1]. A (v,M, d; k)q constant dimension code (CDC) C is a subset of [ V k ] of cardinality M in which for each pair of elements, called codewords, the subspace distance is lower bounded by d, i.e., we have d ≤ dS(U,W ) for all U 6= W ∈ C. The main question of subspace coding in the constant dimension case asks for the maximum cardinality M for fixed parameters q, v, d, and k of a (v,M, d; k)q code. The maximum cardinality is denoted as Aq(v, d; k). Aq(v, d; k) is known for some parameters. By definition, Aq(v, d; k) = 0 for k < 0 or v < k. If d ≤ 2, then Aq(v, d; k) = [ v k ]q. Let U ⊥ denote the orthogonal complement of U with respect to a fixed non-degenerate symmetric bilinear form on V . Since dS(U ⊥,W⊥) = dS(U,W ), we have Aq(v, d; k) = Aq(v, d; v − k), cf. [29, Remark