Superimposed codes are almost big distance ones

Let T(r,n) denote the maximum number of subsets of an n-set satisfying the condition that no set is covered by the union of r others, while let T*(ε,n) be the maximum size of a <ε part intersecting family, i.e. of a family where the size of the intersection of any two sets is < than the ε^th part of the smaller one. By partially answering to a question posed by Hwang and Sos (1987), we prove, that superimposed codes (r-cover-free families) are large distance ones (almost <1/r part intersecting families). More precisely, our theorem says that the rate of an r-cover-free family is ⩽ than the rate of a loglogr/r part intersecting family, i.e. for n>n₀(r)logT(r,n)/n⩽logT*(loglogr/r,n)/n