Pairwise Balanced Designs with Block Sizes 5t + 1

Let P _1,5 denote the set of prime powers congruent to 1 modulo 5. Let Q _1,5 = P _1,5 ∪ {6}. In this paper we investigate B(Q _1,5), the values for which a pairwise balanced design exists with block sizes from Q _1,5. Such a PBD contains n = 5r + 1 points, for some positive integer r. We show that this condition is sufficient for n ≥ 4801, with at most 96 possible exceptions below this value. As a consequence, we are able to show that near resolvable designs with block size 5 exist for all n ≥ 2246 with at most 27 exceptions below this value.