Pairwise Balanced Designs with Block Sizes $5t+1$

Abstract

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 $n\ge 1201$, 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 $n\ge 991$, with at most $26$ possible exceptions below this value.