We study point-block incidence structures (,) for which the point set is an 𝑚 × 𝑛 grid. Cameron and the fourth author showed that each block 𝐵 may be viewed as a subgraph of a complete bipartite graph 𝐊 𝑚,𝑛 with bipartite parts (biparts) of sizes 𝑚, 𝑛. In the case where consists of all the subgraphs isomorphic to 𝐵, under automorphisms of 𝐊𝑚,𝑛 fixing the two biparts, they obtained necessary and sufficient conditions for ( , ) to be a 2-design, and to be a 3-design. We first rein- terpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of 𝐊𝑚,𝑚. We find necessary and sufficient conditions, again in terms of graph the- oretic parameters, for these incidence structures to be 𝑡-designs, for 𝑡 = 2, 3, and give infinite families of exam- ples 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.
