  A basic problem for constant dimension codes is to determine the maximum
possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in
$\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies
$d_S(U,W):=2k-2\dim(U\cap W)\ge d$ for all pairs of different codewords $U$,
$W$. Constant dimension codes have applications in e.g.\ random linear network
coding, cryptography, and distributed storage. Bounds for $A_q(n,d;k)$ are the
topic of many recent research papers. Providing a general framework we survey
many of the latest constructions and show up the potential for further
improvements. As examples we give improved constructions for the cases
$A_q(10,4;5)$, $A_q(11,4;4)$, $A_q(12,6;6)$, and $A_q(15,4;4)$. We also derive
general upper bounds for subcodes arising in those constructions.
