Generalized Steiner systems GS 4 (2, 4, v, g) for g = 2, 3, 6

Generalized Steiner systems GS d ( t, k, v, g ) were first introduced by Etzion and used to construct optimal constant‐weight codes over an alphabet of size g  + 1 with minimum Hamming distance d , in which each codeword has length v and weight k . Much work has been done for the existence of generalized Steiner triple systems GS(2, 3, v , g ). However, for block size four there is not much known on GS d (2, 4, v , g ). In this paper, the necessary conditions for the existence of a GS d ( t, k, v, g ) are given, which answers an open problem of Etzion. Some singular indirect product constructions for GS d (2, k, v, g ) are also presented. By using both recursive and direct constructions, it is proved that the necessary conditions for the existence of a GS 4 (2, 4, v , g ) are also sufficient for g  = 2, 3, 6. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 401–423, 2001