On covering designs with block size 5 and index 11 ≤ λ ≤ 21: The case ν ≡ 0 (mod 4)

Let V be a finite set of order v. A (v,k,λ) covering design of index λ and block size k is a collection of k-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(v,k,λ), in a covering design. It is well known that α(v,k,λ) ≥ = φ(v,k,λ), where ⌈ x ⌉ is the smallest integer satisfying x ≤ ⌈ x ⌉. It is shown here that with the possible exception of (v,λ) = (44, 13), (28, 17), (44, 17), α(v,5,λ) = φ(v,5,λ) + e provided v ≡ 0 (mod 4) and 11 ≤ λ ≤ 21 where e = 1 if λ(v - 1) ≡ 0 (mod 4) and λv(v-1)/4 ≡ - 1 (mod 5) and e=0 otherwise.