On packing designs with block size 5 and indexes 3 and 6

Let V be a finite set of order v. A (v, κ, λ) packing design of index λ and block size κ is a collection of κ-element subsets, called blocks, such that every 2-subset of V occurs in at most λ blocks. The packing problem is to determine the maximum number of blocks, σ(v, κ, λ), in a packing design. It is well known that σ(v, κ, λ) ≤ [v/κ[v-1/κ-1λ]] = ψ(v, κ, λ), where [x] is the largest integer satisfying x ≥ [x]. It is shown here that σ(v, 5, 6) = ψ(v, 5, 6) for all positive integers v ≥ 5 with the possible exceptions of v = 43 and that σ(v, 5, 3) = ψ(v, 5, 3) for all positive integers v ≡ 1, 5, 9, 17 (mod 20) and σ(v, 5, 3) = ψ(v, 5, 3) - 1 for all positive integers v ≡ 13 mod (20) with the possible exception of v = 17, 29, 33, 49.