The problem is:

Given a finite set $X$ with size $x$ and let $B$ be a family of $k$-element subsets of $X$, called blocks. What is the smallest possible number of blocks such that every element of $X$ is determined by the indicators of $B$ on $X$?

Does this problem has a name? What are the known upper/lower bounds? Links to papers or webpages are also welcome.

(This question was asked to me by a Chinese friend.)