Design theory
An existence theorem for cyclic triplewhist tournaments
Selected papers of the 14th British conference on Combinatorial conference
Optimal constant weight codes over Zk and generalized designs
Discrete Mathematics
Some combinational constructions for optical orthogonal codes
Discrete Mathematics
Existence of (q,6,1) Difference Families withq a Prime Power
Designs, Codes and Cryptography
On cyclically resolvable cyclic Steiner 2-designs
Journal of Combinatorial Theory Series A
Constant Weight Codes and Group Divisible Designs
Designs, Codes and Cryptography
A survey on maximum distance Holey packings
Discrete Applied Mathematics
Existence of Generalized Steiner Systems GS(2,4,",2)
Designs, Codes and Cryptography
Constructions for generalized Steiner systems GS(3, 4, v, 2)
Designs, Codes and Cryptography
Combinatorial designs and the theorem of Weil on multiplicative character sums
Finite Fields and Their Applications
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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).