Optimal monotone encodings

Noga Alon, Rani Hod

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1343-1353
Number of pages11
JournalIEEE Transactions on Information Theory
Issue number3
StatePublished - 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Monotone encoding
  • Multiuser tracing
  • Superimposed codes


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