Optimal monotone encodings

Moran, Naor, and Segev have asked what is the minimal r = r(n,k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1,2,...,n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multiuser tracing ((k, α)-FUT (fraction user-tracing) families). We show that r(n, k) = Θ(k log(n/k) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant.