Q: How were the designs constructed ? A: Using DISCRETA, a software for the construction of t-designs with prescribed automorphism group. You can get to the DISCRETA homepage by following this LINK TO THE DISCRETA HOMEPAGE.
see also for information on 6-designs and designs on up to 8 points:
Bernd Schmalz: t-Designs zu vorgegebener Automorphismengruppe, Januar 1992 (Dissertation) (Prof. Laue)
B. Schmalz, The $t$-designs with prescribed automorphism group, new simple 6-designs. J. Combinatorial Designs 1 (1993), 125-170.