The interplay of different metrics for the construction of constant dimension codes
Sascha Kurz, Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany · Advances in Mathematics of Communications · 2022
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 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.