Universal Sets and Cover-Free Families

We propose a polynomial time construction of an $(n,d)$-universal set over alphabet $Σ=\{0,1\}$, of size $d\cdot 2^{d+o(d)}\cdot\log n$. This is an improvement over the size, $d^{5}2^{2.66d}\log n$, of an $(n,d)$-universal set constructed by Bshouty, \cite{BshoutyTesters}, over alphabet $Σ=\{0,1\}$.

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