An explicit construction for large sets of infinite dimensional $q$-Steiner systems

Let $V$ be a vector space over the finite field ${\mathbb F}_q$. A $q$-Steiner system, or an $S(t,k,V)_q$, is a collection ${\mathcal B}$ of $k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of $V$ is contained in a unique element of ${\mathcal B}$. A large set of $q$-Steiner systems, or an $LS(t,k,V)_q$, is a partition of the $k$-dimensional subspaces of $V$ into $S(t,k,V)_q$ systems. In the case that $V$ has infinite dimension, the existence of an $LS(t,k,V)_q$ for all finite $t,k$ with $1