Asymptotically optimal covering designs (Q1924226)

A \((v,k,t)\) covering design is a family of \(k\)-subsets, called blocks, chosen from a \(v\)-set, such that each \(t\)-subset is contained in at least one block. In a covering design, the number of blocks, the size, is at least \({v\choose t}/{k\choose t}\). It is known that for fixed \(k\) and \(t\), there are covering designs of size \[ \begin{pmatrix} v\\ t\end{pmatrix}/\begin{pmatrix} k\\ t\end{pmatrix}(1+o(1))\quad\text{as }v\to\infty; \] see \textit{V. Rödel} [On a packing and covering problem, Eur. J. Comb. 6, 69-78 (1985; Zbl 0565.05016)]. Good coverings were constructed in the paper by the first three authors [New constructions for covering designs, J. Comb. Des. 3, No. 4, 269-284 (1995)]. In the present article, the authors prove that for two of these constructions the size of the coverings (for fixed \(k\) and \(t\)) matches the Rödl bound.