\(4-(21,5,\lambda)\) Designs from a Group of Order \(171\)

Using basis reduction, we settle the existence problem for \(4\)-\((21,5,\lambda)\) designs with \(\lambda \in \{3,5,6,8\}\). These designs each have as an automorphism group the Frobenius group \(G\) of order \(171\) fixing two points. We also show that a \(4\)-\((21,5,1)\) design cannot have the subgroup of order \(57\) of \(G\) as an automorphism group.