The Non-Existence of Block-Transitive Subspace Designs

TL;DR

This paper proves that for certain parameters, non-trivial block-transitive subspace designs do not exist, implying all such designs are trivial and encompass all possible subspaces.

Contribution

It establishes the non-existence of non-trivial block-transitive subspace designs for t≥2, advancing understanding of symmetry properties in q-analogues of block designs.

Findings

Block-transitive designs are trivial for t≥2.

Non-trivial block-transitive subspace designs do not exist.

All k-subspaces form the design when block-transitive.

Abstract

Let $q$ be a prime power and $V\cong{\mathbb F}_q^n$. A $t$-$(n,k,\lambda)_q$ design, or simply a subspace design, is a pair ${\mathcal D}=(V,{\mathcal B})$, where ${\mathcal B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $\lambda$ elements of ${\mathcal B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group ${\rm Aut}({\mathcal D})$ acts transitively on ${\mathcal B}$. It is shown here that if $t\geq 2$ and ${\mathcal D}$ is a block-transitive $t$-$(n,k,\lambda)_q$ design then ${\mathcal D}$ is trivial, that is, ${\mathcal B}$ is the set of all $k$-dimensional subspaces of $V$.