Pair covering and other designs with block size 6

✍ Scribed by R. Julian R. Abel; Iliya Bluskov; Malcolm Greig; Jan de Heer

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 218 KB
• Volume: 15
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20157

✦ Synopsis

Abstract

A t‐(v, k, λ) covering design is a set of b blocks of size k such that each t‐set of points occurs in at least λ blocks, and the covering number C~λ~(v, k, t) is the minimum value of b in any t‐(v, k, λ) covering design. In this article, we determine C~1~(v, 6, 2) for v ≡ 2 (mod 5), showing C~1~(v, 6, 2) attains the Schönheim bound. We also show C~1~(v, 6, 2) attains the Schönheim bound for v ≡ 1 (mod 5) whenever v ≥ 23986. If U~λ~(v, k, t) denotes the Schönheim bound for t‐(v, k,λ) packing designs, then we show that D~1~(v, 6, 2) = U~1~(v, 6, 2) − 1 and D~1~(v − 1, 6, 2) = U~1~(v − 1, 6, 2) if v ≡ 11 (mod 15) and v ≥ 23441, and D~1~(v, 6, 2) = U~1~(v, 6, 2) and D~1~(v − 1, 6, 2) = U~1~(v − 1, 6, 2) if v ≡ 1, 6, (mod 15) and v ≥ 811. In addition, we improve the existence results for (v, 6, 1( BIBDs and (v, K, 1) PBDs when K = H~1(5)~ = {k:k ≡ 1 (mod 5)} and when K = {6} ∪ (H~1(5)~ ∩ {prime powers}). © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 511–533, 2007