On the Existence of Simple \(3\)-\((30, 7, 15)\) and \(3\)-\((26, 12, 55)\) Designs

Abstract

For each of the parameter sets \((30, 7, 15)\) and \((26, 12, 55)\), a simple \(3\)-design is given. They have \(\text{PSL}(2, 29)\) and \(\text{PSL}(2, 25)\) as their automorphism group, respectively. Each of the two simple \(3\)-designs is the first one ever known with the parameter set given and \(4\) in each of the two parameter sets is minimal for the given \(v\) and \(k\).