Block-transitive two-designs based on grids

TL;DR

This paper characterizes block-transitive two-designs on grid point sets, providing conditions for their existence and constructing examples, including infinite families and special cases with high symmetry.

Contribution

It offers a graph-theoretic reinterpretation of design conditions and constructs new block-transitive and flag-transitive grid-based designs with explicit parameters.

Findings

Necessary and sufficient conditions for t-designs on grids for t=2,3.

Infinite families of 2-designs with block-transitivity.

Examples of 3-designs with high symmetry.

Abstract

We study point-block incidence structures $(\mathcal{P},\mathcal{B})$ for which the point set $\mathcal{P}$ is an $m\times n$ grid. Cameron and the fourth author showed that each block $B$ may be viewed as a subgraph of a complete bipartite graph $\mathbf{K}_{m,n}$ with bipartite parts (biparts) of sizes $m, n$. In the case where $\mathcal{B}$ consists of all the subgraphs isomorphic to $B$, under automorphisms of $\mathbf{K}_{m,n}$ fixing the two biparts, they obtained necessary and sufficient conditions for $(\mathcal{P},\mathcal{B})$ to be a $2$-design, and to be a $3$-design. We first re-interpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of $\mathbf{K}_{m,m}$. We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be $t$-designs, for $t=2, 3$, and give infinite families of examples illustrating that block-transitive, point-primitive $2$-designs based on grids exist for all values of $m$, and flag-transitive, point-primitive examples occur for all even $m$. This approach also allows us to construct a small number of block-transitive $3$-designs based on grids.