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

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