  A binary code is called a superimposed cover-free $(s,\ell)$-code if the code
is identified by the incidence matrix of a family of finite sets in which no
intersection of $\ell$ sets is covered by the union of $s$ others. A binary
code is called a superimposed list-decoding $s_L$-code if the code is
identified by the incidence matrix of a family of finite sets in which the
union of any $s$ sets can cover not more than $L-1$ other sets of the family.
For $L=\ell=1$, both of the definitions coincide and the corresponding binary
code is called a superimposed $s$-code. Our aim is to obtain new lower and
upper bounds on the rate of the given codes. In particular, we derive lower
bounds on the rates of a superimposed cover-free $(s,\ell)$-code and
list-decoding $s_L$-code based on the ensemble of constant weight binary codes.
Also, we establish an upper bound on the rate of superimposed list-decoding
$s_L$-code.
