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Existence of large sets of disjoint group-divisible designs with block size three and type 2n41

✍ Scribed by L. Ji

Publisher: John Wiley and Sons
Year: 2005
Tongue: English
Weight: 142 KB
Volume: 13
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20039

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✦ Synopsis

Abstract

Large sets of disjoint group‐divisible designs with block size three and type 2^n^4^1^ (denoted by LS (2^n^4^1^)) were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡ 0 (mod 3) and do exist for any n ∉ {12, 36, 48, 144} ∪ {m > 6 : m ≡ 6,30 (mod 36)}. In this paper, we show that an LS (2^12__k__ + 6^4^1^) exists for any k ≠ 2. So, the existence of LS (2^n^4^1^) is almost solved with five possible exceptions n ∈ {12, 30, 36, 48, 144}. This solution is based on the known existence results of S (3, 4, v)s by Hanani and special S (3, {4, 6}, 6__m__)s by Mills. Partitionable H (q, 2, 3, 3) frames also play an important role together with a special known LS (2^18^4^1^) with a subdesign LS (2^6^4^1^). © 2004 Wiley Periodicals, Inc.

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