Further results on the existence of generalized Steiner triple systems with group size g≡1, 5(mod 6)

Generalized Steiner triple systems, 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. The necessary conditions for the existence of a GS(2, 3, n, g) are (n − 1)g ≡ 0 (mod 2), n(n − 1)g ≡ 0 (mod 6), and n ≥ g + 2. Recently, we proved that for any given g, g ≡ 1, 5 (mod 6) and g ≥ 11, if there exists a GS(2, 3, n, g) for all n, n ≡ 1, 3 (mod 6) and g + 2 ≤ n ≤ 9g + 4, then the necessary conditions are also sufficient. In this paper, the above result is improved and two new results are obtained. First, we show that for any given g, g ≡ 1, 5 (mod 6) and g ≥ 17, if there exists a GS(2, 3, n, g) for all n, n ≡ 1, 3 (mod 6) and g + 2 ≤ n ≤ 7g + 6, then the necessary conditions are also sufficient. Second, we prove that the necessary conditions for the existence of a GS(2, 3, n, g) are also sufficient for g = 13.