11.45 (1990)

Comment of 2009: In (Finite Geometry and Combinatorics (Deinze 1992), Cambridge Univ. Press, 1993, 103–119) we showed that a block-transitive group $G$ on a nontrivial 6-design is either an affine group $AGL(d, 2)$ or is between $PSL(2, q)$ and $P\Gamma L(2, q)$; in (M. Huber, J. Combin. Theory Ser. A, 117, no. 2 (2010), 196–203) it is shown that for the case $\lambda = 1$ the group $G$ may only be $P\Gamma L(2, p^e)$, where $p$ is 2 or 3 and $e$ is an odd prime power. Comment of 2013: In the case $\lambda = 1$ there are no block-transitive 7-designs (M. Huber, Discrete Math. Theor. Comput. Sci., 12, no. 1 (2010), 123–132). Comment of 2021: There are no nontrivial 6-designs with $k \leqslant 10^4$ in the case where the automorphism group is almost simple (Q. Tan, W. Liu, J. Chen, Algebra Colloq., 21, no. 2 (2014), 231–234).