4-* GDDs ( 3 n ) and Generalized Steiner Systems GS

Generalized Steiner systems GSð2; k; v; gÞ were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size gþ 1 with minimum Hamming distance 2k 3, in which each codeword has length v and weight k. As to the existence of a GSð2; k; v; gÞ, a lot of work has been done for k 1⁄4 3, while not so much is known for k 1⁄4 4. The notion k-*GDD was first introduced and used to construct GSð2; 3; v; 6Þ. In this paper, singular indirect product (SIP) construction for GDDs is modified to construct GSð2; 4; v; gÞ via 4-*GDDs. Furthermore, it is proved that the necessary conditions for the existence of a 4-*GDDð3Þ, namely, n 0; 1 ðmod 4Þ and n 8 are also sufficient. The known results on the existence of a GSð2; 4; v; 3Þ 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