Generalized Steiner systems \(GS_4(2, 4, \nu, 4)\)

Generalized Steiner systems GS\(_d(t,k,v,g)\) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size \(g + 1\) with minimum Hamming distance \(d\), in which each codeword has length \(v\) and weight \(k\). It was proved that the necessary conditions for the existence of a GS\(_4(2,4,v,g)\) are also sufficient for \(g = 2, 3\) and \(6\). In this paper, a general result on the existence of a GS\(_4(2,4,v,g)\) is presented. By using this result, we prove that the necessary conditions \(v \equiv 1 \pmod{3}\) and \(v \geq 7\) are also sufficient for the existence of a GS\(_4(2, 4, v, 4)\).