Blocking sets in designs with block size 4 (Q1813227)

Constructions are given which produce block designs with \(\lambda=1\) or 2 and \(k=4\) admitting a blocking set. The construction includes all admissible orders \(n\) with \(\lambda=1,2\) and \(k=4\) except 8 cases. There are four separate constructions, one for \(\lambda=1\) and \(n\equiv1\) mod 12, one for \(\lambda=1\) and \(n\equiv 4\) mod 12, one for \(\lambda=2\) and \(n\equiv 7\) mod 12, and one for \(\lambda=2\) and \(n\equiv 7\) mod 12.