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 (4t,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 (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,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 gn, we have one new design of type 309 when λ=1, and three new designs for λ=2, namely, types g15 with g∈{13, 17, 19}. If λ is even and a (5, λ) GDD of type gu is known, then we also have a directable (5,λ) GDD of type gu. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:337–368, 2010