Notes on well-distributed minimal sub-BIBDs for $\lambda=1$

In these notes we investigate BIBDs with $\lambda=1$ that present subdesigns evenly covering both blocks and vertices: we determine some of their basic properties, consequence of already existing results in the literature, with regards to their size and the number of intersections of pairs and triples of subdesigns of a specific kind. We also describe the link between these particular BIBDs and the graph isomorphism problem, based on Babai's paper, and point out the characteristics of these designs that would lead to improvements of the algorithm for the GIP.