A Large Collection of Designs From a Wreath Product on \(21\) Points

The total of \(4079\) \(2\)-designs and two \(3\)-designs on \(21\) points have been found. All these designs have the same group as an automorphism group. This group can be represented as the wreath product of \(G\) and \(H\), where \(G\) denotes the subgroup of order 3 of \(\text{PSL}(2,2)\) and \(H\) denotes the transitive subgroup of order 21 of \(\text{PSL}(3, 2)\).

In particular, \(1, 20, 101, 93, 173, 824\) and \(2867\) values of \(A\) for \(2\)-\((21,k,\lambda)\) designs have been detected, where \(k\) takes values from \(4\) through \(10\). Up to our knowledge, \(2217\) of these \(\lambda\)-values are new (\(14, 76, 65, 122, 587\), and \(1353\), for \(k\) equal to \(5, 6, …,10\), respectively). By Alltop’s extension [4], \(1353\) new \(2\)-\((21,10,A)\) designs can be extended to the same number of new \(3\)-\((22,11,\lambda)\) designs.

An extensive search with \(t > 2\) and \(k < 8\) has given only the \(3\)-\((21,6,216)\) design and the \(3\)-\((21,7,1260)\) design with the same automorphism group.