Covering pairs by quintuples with index λ  0 (mod 4)

A (v, k. λ) covering design of order v, block size k, and index λ is a collection of k‐element subsets, called blocks, of a set V 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 \documentclass{article}\pagestyle{empty}\begin{document}$ \alpha \left({\nu,\kappa,\lambda } \right) \ge \left\lceil {\frac{\nu}{\kappa}\left\lceil {\frac{{\nu - 1}}{{\kappa - 1}}\lambda} \right\rceil} \right\rceil = \phi \left({\nu,\kappa,\lambda} \right) $\end{document}, where [χ] is the smallest integer satisfying χ ≤ χ. It is shown here that α (v, 5, λ) = ϕ(v, 5, λ) + ϵ where λ ≡ 0 (mod 4) and e= 1 if λ (v−1)≡ 0(mod 4) and λv (v−1)/4 ≡ −1 (mod 5) and e= 0 otherwise With the possible exception of (v,λ) = (28, 4). © 1993 John Wiley & Sons, Inc.