Super-simple (v, 5, 4) designs

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. Recently, Gronau et al determined the existence of super-simple (v, 5, 2)-BIBDs with possible exceptions of v ∈ {75, 95, 115, 135, 195, 215, 231, 285, 365, 385, 515}. In this article, we investigate the existence of a super-simple (v, 5, 4)-BIBD and show that such a design exists if and only if v ≡ 0, 1 (mod 5) and v ≥ 15. In addition, we also constructed a super-simple (v, 5, 2)-BIBD for v = 75, 95, or 385.