On the Existence of Simple and Indecomposable Block Designs with Block Size \(4\)

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.