A Family of \(4\)-Designs on \(38\) Points

Using a modification of the Kramer-Mesner method, \(4-(38,5,\lambda)\) designs are constructed with \(\text{PSL}(2,37)\) as an automorphism group and with \(\lambda\) in the set \(\{6,10,12,16\}\). It turns out also that there exists a \(4-(38,5,16)\) design with \(\text{PGL}(2,37)\) as an automorphism group.