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

TL;DR

This paper constructs an infinite series of nontrivial $q$-analogs of $t$-wise balanced designs with block dimension 2, using Borel subgroups, applicable for all $t \,\geq 1$ and prime powers $q$, extending to classical set designs when $q=1$.

Contribution

The paper introduces a novel construction method for $q$-analogs of $t$-wise balanced designs with specific parameters, expanding the known classes of such combinatorial designs.

Findings

Constructed infinite series of $t$-$(n,K,\,\lambda;q)$ designs with $|K|=2$

Designs exist for all $t\geq 1$ and prime powers $q$

Includes classical $t$-wise balanced designs as special cases when $q=1$

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.