Super-simple balanced incomplete block designs with block size 5 and index 3

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple (v,5,3)(v,5,3) balanced incomplete block design and show that such a design exists if and only if v≡1,5(mod20) and v≥21v≥21 except possibly when v=45,65v=45,65.