Large sets of subspace designs

TL;DR

This paper introduces new join operations for subspaces, generalizes large set recursion methods, and constructs infinite series of nontrivial subspace designs with applications to large sets.

Contribution

It develops a framework for subspace joins, generalizes large set recursion methods, and constructs new infinite series of subspace designs and large sets.

Findings

Constructed a 2-(6,3,78)_5 design via computer.

Established two infinite two-parameter series of halvings LS_3[2](2,k,v) and LS_5[2](2,k,v).

Proved the existence of infinitely many large sets of subspace designs with t=2.

Abstract

In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Gra{\ss}mannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $\operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $\operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $\operatorname{LS}_3[2](2,k,v)$ and $\operatorname{LS}_5[2](2,k,v)$ with integers $v\geq 6$, $v\equiv 2\mod 4$ and $3\leq k\leq v-3$, $k\equiv 3\mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$.