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

Abstract

In this article, we show the existence of large sets ${\mathrm{LS}}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v\ge 8$ and $0\le k\le v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2\le \bar{v}<\bar{k}\le 5$. The proof is constructive and consists of two parts. First, we give a computer construction for an ${\mathrm{LS}}_2[3](2,4,8)$, 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 ${\mathrm{LS}}_2[3](2,3,8)$, the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.