## Abstract

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 language | English |
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Pages (from-to) | 1343-1353 |

Number of pages | 11 |

Journal | IEEE Transactions on Information Theory |

Volume | 55 |

Issue number | 3 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

### Bibliographical note

Funding Information:Manuscript received April 29, 2008; revised October 23, 2008. Current version published February 25, 2009. This work was supported in part by the Israel Science Foundation and by a USA-Israeli BSF Grant. The material in this paper was presented in part at the 35th International Colloquium on Automata, Languages and Programming, Reykjavik, Iceland, July 2008.

### Funding

Manuscript received April 29, 2008; revised October 23, 2008. Current version published February 25, 2009. This work was supported in part by the Israel Science Foundation and by a USA-Israeli BSF Grant. The material in this paper was presented in part at the 35th International Colloquium on Automata, Languages and Programming, Reykjavik, Iceland, July 2008.

Funders | Funder number |
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Israel Science Foundation | |

USA-Israeli BSF |

## Keywords

- Monotone encoding
- Multiuser tracing
- Superimposed codes