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 9 + 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 by several authors for 2 S 9 ::; 10. The necessary conditions for the existence of a GS(2, 3, n, g) are (n l)g == o (mod 2), n(n 1)g2 == 0 (mod 6), and n 2:: 9 + 2. Recently, D. Wu et al proved that for any given 9 2:: 7, if there exists a GS(2, 3, n, g) for all n, 9 + 2 S n ::; 9g + 158, satisfying the above two congruences, then the necessary conditions are also sufficient. In this paper, the result is partially improved. It is shown that for any given g, 9 == 1,5 (mod 6) and 9 2:: 11, if there exists a GS(2, 3, n, g) for all n, n == 1,3 (mod 6) and 9 + 2 ::; n ::; 9g + 4, 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 = 11.