In the paper "On two problems of information theory",  Erdõs and Rényi give lower bounds on the minimum number of weighings one must do to determine the number of false coins in a set of $n$ coins. The trivial lower bound they initially provide is:

$n / \log_2 (n + 1)$. 

This isn't hard to see why through various information-theoretic or combinatorial arguments. The problem is how to construct such sets to do these weighings? Are there algorithms that utilize a constructive proof to achieve these lower bounds without relying on randomness? Are there randomized algorithms that achieve these bounds?