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

✍ Scribed by G. Ge; D. Wu

Publisher: John Wiley and Sons
Year: 2003
Tongue: English
Weight: 137 KB
Volume: 11
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.10047

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✦ Synopsis

Abstract

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

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