Recursive Constructions of Balanced Incomplete Block Designs with Block Size of \(7\), \(8\) or \(9\)

This paper gives constructions of balanced incomplete block designs and group divisible designs with \(k = 7, 8,\) or \(9\), and \(\lambda = 1\). The first objective is to give constructions for all possible cases with the exception of \(40, 78,\) and \(157\) values of \(v\). Many of these initial exceptions have now been removed by Abel. In an update section, more are removed; group divisible designs with groups of size \(k(k-1)\) are constructed for \(k = 7\) and \(8\) with \(124\) and \(87\) exceptions; it is also established that \(v \geq 294469\) and \(v \equiv 7\) mod \(42\) suffices for the existence of a resolvable balanced incomplete block design with \(k = 7\). Group divisible designs with group size \(k\) and resolvable designs are constructed.