On covering designs with block size 5 and index 5

Let V be a finite set of order υ. A (υ, κ, λ) covering 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 least λ blocks. The covering problem is to determine the minimum number of blocks, α(υ, κ, λ), in a covering design. It is well known that {Mathematical expression}, where [x] is the smallest integer satisfying x≤[X]. It is shown here that α(υ, 5, 5)=φ{symbol}(υ, 5, 5) for all positive integers υ≥5 with the possible exception of υ=24, 28, 56, 104, 124, 144, 164, 184.