Super-simple cyclic designs with small values

Super-simple cyclic designs are useful on constructing codes and designs such as superimposed codes, perfect hash families and optical orthogonal codes with index two. In this paper, we show that there exists a super-simple cyclic (v,4,λ) for 7⩽v⩽41 and all admissible λ with two definite exceptions of (v,λ)=(9,3),(13,5) and one possible exception of (v,λ)=(39,18). Some useful algorithms are explained for computer search and new designs are displayed.