Asymptotic Upper Bound for the Rate of $(w,r)$ Cover-Free Codes

A binary code is called a $(w,r)$ cover-free code if it is the incidence matrix of a family of sets where the intersection of any $w$ of the sets is not covered by the union of any other $r$ sets. Such a family is called a $(w,r)$ cover-free family. We obtain a new recurrent inequality for the rate of $(w,r)$ cover-free codes, which improves previously known upper bounds on the rate.