A new series of large sets of subspace designs over the binary field

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.