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The non-existence of block-transitive subspace designs

  • Author(s): Hawtin, Daniel R.;
  • Lansdown, Jesse
  • et al.

Published Web Location Commons 'BY' version 4.0 license

Let $q$ be a prime power and $V\cong\mathbb{F}_q^d$. A $t$-$(d,k,\lambda)_q$ 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 $\mathrm{Aut}(\mathcal{D})$ acts transitively on $\mathcal{B}$. It is shown here that if $t\geq 2$ and $\mathcal{D}$ is a block-transitive $t$-$(d,k,\lambda)_q$ design then $\mathcal{D}$ is trivial, that is, $\mathcal{B}$ is the set of all $k$-dimensional subspaces of $V$.

Mathematics Subject Classifications: 05E18, 05B99

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