On the Existence of Simple and Indecomposable Block Designs with Block Size $4$

Abstract

It is proved in this paper that for $\lambda =4$ and $5$, the necessary conditions for the existence of a simple $B(4,\lambda ;v)$ are also sufficient. It is also proved that for $\lambda =4$ and $5$, the necessary conditions for the existence of an indecomposable simple $B(4,\lambda ;v)$ are also sufficient, with the unique exception $(v,\lambda )=(7,4)$ and $10$ possible exceptions.