  In this article, we show the existence of large sets
$\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The
exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the
remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq
\bar{v} < \bar{k} \leq 5$.
  The proof is constructive and consists of two parts. First, we give a
computer construction for an $\operatorname{LS}_2[3](2,4,8)$, which is a
partition of the set of all $4$-dimensional subspaces of an $8$-dimensional
vector space over the binary field into three disjoint $2$-$(8, 4, 217)_2$
subspace designs. Together with the already known
$\operatorname{LS}_2[3](2,3,8)$, the application of a recursion method based on
a decomposition of the Gra{\ss}mannian into joins yields a construction for the
claimed large sets.
