Generalized Steiner systems GS(2, 4, v, 2) with v a prime power = (mod 12)

Generalized Steiner systems GS(2, 4, v, g) were first introduced by Etzion and were used to construct optimal constant weight codes over an alphabet o f size g+1 with minimum Hamming distance 5, in which each codeword has leng th v and weight 4. Etzion conjectured that the necessary conditions v = 1 ( mod 3) and v greater than or equal to 7 are also sufficient for the existen ce of a GS(2,4,v,2). Except for the example of a GS(2,4,10,2) and some recu rsive constructions given by Etzion, nothing else is known about this conje cture. In this paper, Weil's theorem on character sum estimates is used to show that the conjecture is true for any prime power v = 7 (mod 12) except v=7, for which there does not exist a GS(2,4,7,2).