Existence of Generalized Steiner Systems GS(2, 4, v, 2)

Generalized Steiner systems GS(2, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance five, in which each codeword has length v and weight four. Etzion conjectured its existence for any integer v ≥ 7 and v ≡ 1 (mod 3). The conjecture has been verified for prime powers v > 7 and v ≡ 7 (mod 12) by the latter two of the present authors. It has also been pointed out that there does not exist a GS(2, 4, 7, 2). In this paper, constructions using frame generalized Steiner systems, two holey perfect bases and orthogonal Latin squares are discussed. With these constructions the conjecture is confirmed with the exception for v = 7 and three possible exceptions for v ∈ {13, 52, 58}.