New results on GDDs, covering, packing and directable designs with block size 5

✍ Scribed by R. Julian R. Abel; Ahmed M. Assaf; Iliya Bluskov; Malcolm Greig; Nabil Shalaby

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 246 KB
• Volume: 18
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20253

✦ Synopsis

Abstract

This article looks at (5,λ) GDDs and (v,5,λ) pair packing and pair covering designs. For packing designs, we solve the (4__t__,5,3) class with two possible exceptions, solve 16 open cases with λ odd, and improve the maximum number of blocks in some (v, 5, λ) packings when v small (here, the Schönheim bound is not always attainable). When λ=1, we construct v=432 and improve the spectrum for v=14, 18 (mod 20). We also extend one of Hanani's conditions under which the Schönheim bound cannot be achieved (this extension affects (20__t__+9,5,1), (20__t__+17,5,1) and (20__t__+13,5,3)) packings. For covering designs we find the covering numbers C(280,5,1), C(44,5,17) and C(44,5,λ) with λ=13 (mod 20). We also know that the covering number, C(v, 5, 2), exceeds the Schönheim bound by 1 for v=9, 13 and 15. For GDDs of type g^n^, we have one new design of type 30^9^ when λ=1, and three new designs for λ=2, namely, types g^15^ with g∈{13, 17, 19}. If λ is even and a (5, λ) GDD of type g^u^ is known, then we also have a directable (5,λ) GDD of type g^u^. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:337–368, 2010