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

This article looks at (5,λ) GDDs and (ν,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 (ν,5, λ) packings when ν small (here, the Schönheim bound is not always attainable). When λ=1, we construct ν=432 and improve the spectrum for ν≡14,18(mod20). 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(ν,5,2), exceeds the Schönheim bound by 1 for ν=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 g^u is known, then we also have a directable (5, λ) GDD of type g^u.