Pairwise Balanced Designs with Block Sizes \(5t + 1\)

In this paper we construct pairwise balanced designs (PBDs) having block sizes which are prime powers congruent to \(1\) modulo \(5\) together with \(6\). Such a PBD contains \(n = 5r + 1\) points, for some positive integer \(r\). We show that this condition is sufficient for \(n \geq 1201\), with at most \(74\) possible exceptions below this value. As an application, we prove that there exists an almost resolvable BIB design with \(n\) points and block size five whenever \(n \geq 991\), with at most \(26\) possible exceptions below this value.