We enumerate the 2-\((9,4,6)\) designs and find \(270,474,142\) non-isomorphic such designs in a backtrack search. The sizes of their automorphism groups vary between \(1\) and \(360\). Out of these designs, \(19,489,464\) are simple and \(2,148,676\) are decomposable.
Citation
Patric R. J. Ostergard . There are \(270,474,142\) Nonisomorphic \(2-(9, 4,6)\) Designs[J], Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 037. 173-176. .