4‐*GDDs(3 n ) and generalized Steiner systems GS(2, 4, v , 3)

Generalized Steiner systems GS(2, 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 2 k  − 3, in which each codeword has length v and weight k . As to the existence of a GS(2, k , v , g ), a lot of work has been done for k  = 3, while not so much is known for k  = 4. The notion k ‐*GDD was first introduced and used to construct GS(2, 3, v , 6). In this paper, singular indirect product (SIP) construction for GDDs is modified to construct GS(2, 4, v , g ) via 4‐*GDDs. Furthermore, it is proved that the necessary conditions for the existence of a 4‐*GDD(3 n ), namely, n  ≡ 0, 1 (mod 4) and n  ≥ 8 are also sufficient. The known results on the existence of a GS(2, 4, v , 3) are then extended. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 381–393, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10047