Super-simple (v,5,4)(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)(v,5,2)-BIBDs with possible exceptions of v∈{75,95,115,135,195,215v∈{75,95,115,135,195,215, 231,285,365,385,515}231,285,365,385,515}. In this article, we investigate the existence of a super-simple (v,5,4)(v,5,4)-BIBD and show that such a design exists if and only if v≡0,1(mod5) and v⩾15v⩾15. In addition, we also constructed a super-simple (v,5,2)(v,5,2)-BIBD for v=75,95,v=75,95, or 385.