Super-simple group divisible designs with block size 4 and index 9

A design is said to be super-simple if the intersection of any two blocks has at most two elements. In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple GDDs are useful in constructing super-simple BIBDs. The existence of super-simple ( 4 , λ ) ‐ GDDs has been determined for λ = 2 – 6 . In this paper, we investigate the existence of a super-simple (4,9)-GDD of group type gu and show that such a design exists if and only if u ≥ 4 , g ( u − 2 ) ≥ 18 and u ( u − 1 ) g 2 ≡ 0 (mod 4).