Bounds on the rate of separating codes

A code with words in a finite alphabet is said to be an $(s, \ell)$ separating code if for any two disjoint collections of its words of size at most $s$ and $\ell$, respectively, there exists a coordinate in which the set of symbols of the first collection do not intersect the set of symbols of the second. The main goal of the paper is obtaining new bounds on the rate of $(s, \ell)$ separating codes. Bounds on the rate of binary $(s, \ell)$ separating codes, the most important for applications, are studied in more detail. We give tables of numerical values of the best presently known bounds on the rate.