Construction for indecomposable simple \((v, k, \lambda)\)-BIBDs (Q1923533)

A \((v,k,\lambda)\)-BIBD is simple if it contains no repeated blocks and is decomposable if the blocks of the design can be partitioned into two designs with parameters \((v,k,\lambda_1)\) and \((v,k,\lambda_2)\) where \(\lambda_1+\lambda_2= \lambda\). It is indecomposable if it is not decomposable. This paper gives a new construction for indecomposable simple \((v,k,\lambda)\)-BIBD. As an application, the author proves that there exists an indecomposable simple \((v,3,\lambda)\)-BIBD for all \(v\geq 24\lambda-5\) satisfying the necessary conditions. This greatly improves the previous best bound which was of order \(\lambda^4\).