A construction of designs on points from multiply homogeneous permutation groups of degree n

A simple criterion to construct a t-design on n + 1 points from a t-ply homogeneous permutation group of degree n using some orbits of the group is obtained. The design is not simple in general. Applying the criterion to 2-dimensional projective special linear groups PSL ( 2 , q ) for q ⩾ 19 acting on the projective line of q + 1 points, simple 3- ( q + 2 , 1 2 ( q − 1 ) , 1 24 ( q − 1 ) ( q − 3 ) ( q − 5 ) ) designs are obtained if q ≡ 3 or 7 (mod 12), and simple 3- ( q + 2 , 1 2 ( q + 1 ) , 1 8 ( q − 1 ) 2 ( q − 3 ) ) designs are obtained if q ≡ 3 (mod 4).