Small & Balanced family of sets

Question

I have the following problem:

Let $\epsilon >0$, and $[n] = \{ 1,2,...,n\}$ the set of positive integers up to $n$. There exists a family of subsets $\mathcal{F} \subseteq 2^{[n]}$, such that

1. $|\mathcal{F}| = O(n)$, and
2. for every non-empty $S \subseteq [n]$, one has

$$ \epsilon |\mathcal{F}| \hspace{0.2cm} \leq \hspace{0.2cm} | \{ F \in \mathcal{F} : |F \cap S| \mbox{ is odd } \}| \hspace{0.2cm} \leq \hspace{0.2cm} (1-\epsilon)|\mathcal{F}| ?$$

Maybe it is also a "famous" problem, maybe a trivial one, I don't know.

Do you know something about this?