Generalized Steiner Triple Systems with Group Size \(g=7,8\)

Generalized Steiner triple systems, \(GS(2,3,n,g)\) are used to construct maximum constant weight codes over an alphabet of size \(g+1\) with distance \(3\) and weight \(3\) in which each codeword has length \(n\). The existence of \(GS(2,3,n,g)\) has been solved for \(g = 2,3,4,5,6,9\). The necessary conditions for the existence of a \(GS(2,3,n,g)\) are \((n-1)g \equiv 0 \pmod{2}\), \(n(n-1)g \equiv 0 \pmod{6}\), and \(n \geq g+2\). In this paper, the existence of a \(GS(2,3,7,g)\) for any given \(g \geq 7\) is investigated. It is proved that if there exists a \(GS(2,3,n,g)\) for all \(n\), \(g+2 \leq n \leq 9g+158\), satisfying the two congruences, then the necessary conditions are also sufficient. As an application it is proved that the necessary conditions for the existence of a \(GS(2,3,n,g)\) are also sufficient for \(g = 7,8\).