Blocking sets in (υ, {2, 4}, 1)-designs

The existence of blocking sets in (υ, {2, 4}, 1)-designs is examined. We show that for υ = 0, 3, 5, 6, 7, 8, 9, 11 (mod 12), blocking sets cannot exist. We prove that for each υ = 1, 2, 4 (mod 12) there is a (υ, {2, 4}, 1)-design with a blocking set with three possible exceptions. The case υ = 10 (mod 12) is still open; we consider the first four values of υ in this situation.