  An (r;w; d) -cover-free family (CFF) is a family of subsets of a finite set
such that the intersection of any r members of the family contains at least d
elements that are not in the union of any other w members. The minimum number
of elements for which there exists an (r;w; d)-CFF with t blocks is denoted by
N((r;w; d); t). In this paper, we determine the exact value of N((r;w; d); t)
for some special parameters. Also, we present two constructions for (2; 1; d)-
CFF and (2; 2; d)-CFF which improve the existing constructions. Moreover, we
introduce a generalization of cover-free families which is motivated by an
application of CFF in the key predistribution schemes. Also, we investigate
some properties and bounds on the parameters of this generalization.
