Further results on the existence of super-simple pairwise balanced designs with block sizes 3 and 4

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple pairwise balanced designs are useful in constructing other types of super-simple designs which can be applied to codes and designs. In this paper, the super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple \begin{document}$(v, \{3,4\}, λ)$\end{document} -PBD for \begin{document}$λ = 7,9$\end{document} and \begin{document}$λ = 2k$\end{document} , \begin{document}$k≥1$\end{document} , are sufficient with seven possible exceptions. In the end, several optical orthogonal codes and superimposed codes are given.