  Roughly speaking, an $(n,(r,s))$-Cover Free Family (CFF) is a small set of
$n$-bit strings such that: "in any $d:=r+s$ indices we see all patterns of
weight $r$". CFFs have been of interest for a long time both in discrete
mathematics as part of block design theory, and in theoretical computer science
where they have found a variety of applications, for example, in parametrized
algorithms where they were introduced in the recent breakthrough work of Fomin,
Lokshtanov and Saurabh under the name `lopsided universal sets'.
  In this paper we give the first explicit construction of cover-free families
of optimal size up to lower order multiplicative terms, {for any $r$ and $s$}.
In fact, our construction time is almost linear in the size of the family.
Before our work, such a result existed only for $r=d^{o(1)}$. and $r=
\omega(d/(\log\log d\log\log\log d))$. As a sample application, we improve the
running times of parameterized algorithms from the recent work of Gabizon,
Lokshtanov and Pilipczuk.
