New Results on GDDs, Covering, Packing and Directable Designs with Block Size 5 R. Julian R. Abel, 1 Ahmed M. Assaf, 2 Iliya Bluskov, 3 Malcolm Greig, 4 Nabil Shalaby 5 1 School of Mathematics and Statistics, University of New South Wales, Sydney, N.S.W. 2052, Australia, E-mail: r.j.abel@unsw.edu.au 2 Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, E-mail: assaf1am@mail.cmich.edu 3 Department of Mathematics, University of Northern British Columbia, Prince George, BC, Canada V2N 4Z9, E-mail: bluskovi@unbc.ca 4 Greig Consulting, 103-136 Fifth Street East, North Vancouver, BC, Canada V7L 1L6, E-mail: greig@sfu.ca 5 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada A1C 5S7, E-mail: nshalaby@mun.ca, nshalaby@math.mun.ca Received July 23, 2009; revised January 12, 2010 Published online 29 July 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jcd.20253 Abstract: This article looks at (5, k) GDDs and (v, 5, k) pair packing and pair covering designs. For packing designs, we solve the (4t, 5, 3) class with two possible exceptions, solve 16 open cases with k odd, and improve the maximum number of blocks in some (v, 5, k) packings when v small (here, the Sch¨ onheim bound is not always attainable). When k = 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 Sch¨ onheim bound cannot be achieved (this extension affects (20t + 9, 5, 1), (20t + 17, 5, 1) and (20t + 13, 5, 3)) packings. For covering designs we find the covering numbers C(280, 5, 1), C(44, 5, 17) and C(44, 5, k) with k ≡ 13 (mod 20). We also know that the covering number, C(v, 5, 2), exceeds the Sch¨ onheim 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 k = 1, and three new designs for k = 2, namely, types g 15 with g ∈{13, 17, 19}. If k is even and a (5, k) GDD of type g u is known, then we also have a directable (5, k) GDD of type g u . 2010 Wiley Periodicals, Inc. J Combin Designs 18: 337–368, 2010 Contract grant sponsor: NSERC; Contract grant number: 228274-05 (to I. B.). Journal of Combinatorial Designs 337 2010 Wiley Periodicals, Inc.