Generalized hashing and parent-identifying codes

Let C be a code of length n over an alphabet of q letters. For a pair of integers 2⩽t<u, C is (t,u)-hashing if for any two subsets T,U⊂C, satisfying T⊂U, |T|=t, |U|=u, there is a coordinate 1⩽i⩽n such that for any x∈T, y∈U−x, x and y differ in the ith coordinate. This definition, generalizing the standard notion of a t-hashing family, is motivated by an application in designing the so-called parent identifying codes, used in digital fingerprinting. In this paper, we provide lower and upper bounds on the best possible rate of (t,u)-hashing families for fixed t,u and growing n. We also describe an explicit construction of (t,u)-hashing families. The obtained lower bound on the rate of (t,u)-hashing families is applied to get a new lower bound on the rate of t-parent identifying codes.