Cover-free families have been widely studied over recent decades due to their applications in numerous subjects. In this paper, we introduce the concept of (s,t;d)-union–intersection-bounded families, which is a generalization of t-cover-free families. We provide a general upper bound on the maximum size of an (s,t;d)-union–intersection-bounded family, and show a probabilistic lower bound for the case that the ground set is sufficiently large. They have the same order of magnitude for certain cases. We also discuss the applications of (s,t;d)-union–intersection-bounded families in broadcast encryption, and derive a better upper bound for (1,t;d)-union–intersection-bounded families (also known as superimposed distance codes).
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