Generalized Steiner Systems With Block Size Three and Group Size Four

Generalized Steiner Systems, \(\text{GS}(2, 3, n, g)\), are equivalent to maximum constant weight codes over an alphabet of size \(g+1\) with distance \(3\) and weight \(3\) in which each codeword has length \(n\). We construct Generalized Steiner Triple Systems, \(\text{GS}(2,3,n,g)\), when \(g=4\).