$q$-Analogs of $t$-Wise Balanced Designs from Borel Subgroups

Abstract

A $t\text{-}(n,K,\lambda;q)$ design, also called the $q$-analog of a $t$-wise balanced design, is a set ${\mathcal B}$ of subspaces with dimensions contained in $K$ of the $n$-dimensional vector space ${\mathbb F}_q^n$ over the finite field with $q$ elements such that each $t$-subspace of ${\mathbb F}_q^n$ is contained in exactly $\lambda$ elements of ${\mathcal B}$. In this paper we give a construction of an infinite series of nontrivial $t\text{-}(n,K,\lambda;q)$ designs with $|K|=2$ for all dimensions $t\ge 1$ and all prime powers $q$ admitting the standard Borel subgroup as group of automorphisms. Furthermore, replacing $q=1$ gives an ordinary $t$-wise balanced design defined on sets.