Large sets of subspace designs

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 $vgeq 6$, $vequiv 2mod 4$ and $3leq kleq v-3$, $kequiv 3mod 4$. 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 $t = 2$.

Derived and Residual Subspace Designs

A generalization of forming derived and residual designs from $t$-designs to subspace designs is proposed. A $q$-analog of a theorem by Van Trung, van Leijenhorst and Driessen is proven, stating that if for some (not necessarily realizable) parameter set the derived and residual parameter set are realizable, the same is true for the reduced parameter set. As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no $q$-analog of the large Witt design.