Combining subspace codes

In the context of constant--dimension subspace codes, an important problem is to determine the largest possible size $Aq(n, d; k)$ of codes whose codewords are $k$-subspaces of $\mathbb{F}q^n$ with minimum subspace distance $d$. 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 $Aq(10, 4; 5)$, $Aq(12, 4; 4)$, $Aq(12, 6, 6)$ and $Aq(16, 4; 4)$.