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

Sascha Kurz
2022 Advances in Mathematics of Communications
<p style='text-indent:20px;'>A basic problem for constant dimension codes is to determine the maximum possible size <inline-formula><tex-math id="M1">\begin{document}$A_q(n,d;k)$\end{document}</tex-math></inline-formula> of a set of <inline-formula><tex-math id="M2">\begin{document}$k$\end{document}</tex-math></inline-formula>-dimensional subspaces in <inline-formula><tex-math id="M3">\begin{document}$\mathbb{F}_q^n$\end{document}</tex-math></inline-formula>, called codewords, such that
more » ... e subspace distance satisfies <inline-formula><tex-math id="M4">\begin{document}$d_S(U,W): = 2k-2\dim(U\cap W)\ge d$\end{document}</tex-math></inline-formula> for all pairs of different codewords <inline-formula><tex-math id="M5">\begin{document}$U$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$W$\end{document}</tex-math></inline-formula>. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for <inline-formula><tex-math id="M7">\begin{document}$A_q(n,d;k)$\end{document}</tex-math></inline-formula> 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 <inline-formula><tex-math id="M8">\begin{document}$A_q(10,4;5)$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$A_q(11,4;4)$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$A_q(12,6;6)$\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$A_q(15,4;4)$\end{document}</tex-math></inline-formula>. We also derive general upper bounds for subcodes arising in those constructions.</p>