TITLE: Identifying poisoned wines
QUESTION [66 upvotes]: The standard version of this puzzle is as follows: you have $1000$ bottles of wine, one of which is poisoned. You also have a supply of rats (say). You want to determine which bottle is the poisoned one by feeding the wines to the rats. The poisoned wine takes exactly one hour to work and is undetectable before then. How many rats are necessary to find the poisoned bottle in one hour?
It is not hard to see that the answer is $10$. Way back when math.SE first started, a generalization was considered where more than one bottle of wine is poisoned. The strategy that works for the standard version fails for this version, and I could only find a solution for the case of $2$ poisoned bottles that requires $65$ rats. Asymptotically my solution requires $O(\log^2 N)$ rats to detect $2$ poisoned bottles out of $N$ bottles.
Can anyone do better asymptotically and/or prove that their answer is optimal and/or find a solution that works for more poisoned bottles? The number of poisoned bottles, I guess, should be kept constant while the total number of bottles is allowed to become large for asymptotic estimates.

REPLY [4 votes]: I have a solution that uses just 28 rats. The idea is to use a 28x1000 2-separable testing matrix $M$, where $M_{ij}$ is 1 if rat $i$ drinks bottle $j$ and 0 if he doesn't drink it. A matrix is 2-separable if the boolean OR of any of its two columns is unique. You can read more about such matrices here: https://en.wikipedia.org/wiki/Disjunct_matrix
You can find the actual matrix $M$ here: https://pastebin.com/4yDd1XvG
I plan to write a short report describing the method I used to find $M$.
UPDATE:
A 27x1065 2-separable matrix has been found so it is possible to solve this problem with 27 rats. See https://oeis.org/A054961