The interplay of different metrics for the construction of constant dimension codes

A basic problem for constant dimension codes is to determine the maximum possible size $Aq(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}q^n$, called codewords, such that the subspace distance satisfies $dS(U,W):=2k-2\dim(U\cap W)\ge d$ 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 $Aq(n,d;k)$ 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 $Aq(10,4;5)$, $Aq(11,4;4)$, $Aq(12,6;6)$, and $Aq(15,4;4)$. We also derive general upper bounds for subcodes arising in those constructions.