Combining subspace codes

In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size \begin{document}$ A_q(n, d; k) $\end{document} of codes whose codewords are \begin{document}$ k $\end{document} -subspaces of \begin{document}$ {\mathbb F}_q^n $\end{document} with minimum subspace distance \begin{document}$ d $\end{document} . Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant-dimension subspace codes for many parameters, including \begin{document}$ A_q(10, 4; 5) $\end{document} , \begin{document}$ A_q(12, 4; 4) $\end{document} , \begin{document}$ A_q(12, 6, 6) $\end{document} and \begin{document}$ A_q(16, 4; 4) $\end{document} .