2022/11/09 by Seyed Hassan Alavi, Ashraf Daneshkhah, Alice Devillers +1 · 1 citation
Engineering · Computer Science · #graph theory and CDMA systems #VLSI and FPGA Design Techniques #Digital Image Processing Techniques
paper · doi:10.1112/blms.12744
Abstract We study point‐block incidence structures for which the point set is an grid. Cameron and the fourth author showed that each block may be viewed as a subgraph of a complete bipartite graph with bipartite parts (biparts) of sizes . In the case where consists of all the subgraphs isomorphic to , under automorphisms of fixing the two biparts, they obtained necessary and sufficient conditions for 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 . We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be ‐designs, for , and give infinite families of examples illustrating that block‐transitive, point‐primitive 2‐designs based on grids exist for all values of , and flag‐transitive, point‐primitive examples occur for all even . This approach also allows us to construct a small number of block‐transitive 3‐designs based on grids.