Block-transitive designs based on grids

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