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

Abstract

In this article, we show the existence of large sets $${\text {LS}}_2[3](2,k,v)$$ 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 $${\text {LS}}_2[3](2,4,8)$$ , 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 $${\text {LS}}_2[3](2,3,8)$$ , the application of a recursion method based on a decomposition of the Grasmannian into joins yields a construction for the claimed large sets.