Generalized Steiner Systems {\rm GS}(2, 4, v, 2) with v a Prime Power \equiv 7 ({\it mod}\ {12})

Abstract

Generalized Steiner systems {\rm GS}(2, 4, v, g) were first introduced by Etzion and were used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 5, in which each codeword has length v and weight 4. Etzion conjectured that the necessary conditions v\equiv 1 ({\it mod}\ {3}) and v \geq 7 are also sufficient for the existence of a {\rm GS}(2,4,v,2). Except for the example of a {\rm GS}(2,4,10,2) and some recursive constructions given by Etzion, nothing else is known about this conjecture. In this paper, Weil's theorem on character sum estimates is used to show that the conjecture is true for any prime power v\equiv 7 ({\it mod}\ {12}) except v=7, for which there does not exist a {\rm GS}(2,4,7,2).