t-Designs from the large Mathieu groups

A t-design (X, @?) is a @u-set X together with a family @? of k-subsets from X, called blocks, such that each subset of X of size t is contained in exactly @l members of @?. A t-design with the above parameters is also called a t-(@u, k, @l) design. Here, we allow repeated k-sets in @?, i.e. @? is a multiset. We describe the action of the Mathieu groups M"n, n = 24, 23, 22, on the power sets of the respective X (Chang Choi and John H. Conway have done this for M"2"4), and then determine all of the quadruples of parameters t, n, k, @l with 2 =< t < k =< 1/2n for which there is a t-(n, k, @l) design with M"n as automorphism group. Among the many new t-designs found there is, for example, an 11-(24, 12, 6) design which is the union of three orbits of 12-sets under M"2"4, two of which are repeated six times.