Super-simple balanced incomplete block designs with block size 4 and index 9

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple designs are also useful in constructing codes and designs such as superimposed codes and perfect hash families. The existence of super-simple (v,4,λ)(v,4,λ)-BIBDs have been determined for λ=2–6λ=2–6. In this paper, we investigate the existence of a super-simple (v,4,9)(v,4,9)-BIBD and show that such a design exists if and only if v≡0,1(mod4) and v⩾20v⩾20. Applications of the results to optical orthogonal codes are also mentioned.