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Abstract
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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 constructions for (2; 1; d)-CFF and (2; 2; d)-CFF which
improve the existing constructions. Moreover, we introduce a generalization of (2; w; d)-cover free families which is motivated by an application of CFF in the key pre-distribution schemes. Also, we investigate some properties and bounds on the parameters of this generalization.
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