Coin-weighting problem

We see that a collection of $m$ subsets $S_1, S_2, \dots, S_m$ is determining if there exists a function $f$ such that $$f(|S_1 \cap T|, |S_2 \cap T|, \dots, |S_m \cap T|) = T$$ for every $T \subseteq \{1, 2, \dots, n\}$.

The number of 'inputs' of $f$ above is $(n+1)^m$, because $|S_i \cap T|$ has at most $n+1$ possible values (ranging from $0$ to $n$). On the other hand, notice that there are $2^n$ possible subsets of $\{1, 2, \dots, n\}$. Hence, $f$ must have at least $2^n$ possible 'inputs'. Therefore, $(n+1)^m \geq 2^n$, which is equivalent to $m \geq \frac{n}{\log_2(n+1)}$.