Designs, Geometries, and Hamada's Conjecture

Block designs are highly structured combinatorial objects with links to finite geometries and error-correcting codes. We will discuss Hamada's conjecture, which concerns the characterization of designs with geometric parameters via the ranks of their incidence matrices. This conjecture has been proved -- and disproved! -- for specific parameters, which highlights the problem of classifying geometric designs. We will demonstrate the constructions of several infinite families of counterexamples, and examine what they teach us about pseudo-geometric designs.