## Abstract

A (d,ℓ)-list disjunct matrix is a non-adaptive group testing primitive which, given a set of items with at most d "defectives," outputs a superset of the defectives containing less than ℓ non-defective items. The primitive has found many applications as stand alone objects and as building blocks in the construction of other combinatorial objects. This paper studies error-tolerant list disjunct matrices which can correct up to e _{0} false positive and e _{1} false negative tests in sub-linear time. We then use list-disjunct matrices to prove new results in three different applications. Our major contributions are as follows. (1) We prove several (almost)-matching lower and upper bounds for the optimal number of tests, including the fact that Θ(dlog(n/d)+ e _{0} + de _{1}) tests is necessary and sufficient when ℓ = Θ(d). Similar results are also derived for the disjunct matrix case (i.e. ℓ = 1). (2) We present two methods that convert error-tolerant list disjunct matrices in a black-box manner into error-tolerant list disjunct matrices that are also efficiently decodable. The methods help us derive a family of (strongly) explicit constructions of list-disjunct matrices which are either optimal or near optimal, and which are also efficiently decodable. (3) We show how to use error-correcting efficiently decodable list-disjunct matrices in three different applications: (i) explicit constructions of d-disjunct matrices with t = O(d ^{2}logn + rd) tests which are decodable in poly(t) time, where r is the maximum number of test errors. This result is optimal for r = Ω(dlogn), and even for r = 0 this result improves upon known results; (ii) (explicit) constructions of (near)-optimal, error-correcting, and efficiently decodable monotone encodings; and (iii) (explicit) constructions of (near)-optimal, error-correcting, and efficiently decodable multiple user tracing families.

Original language | English |
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Title of host publication | Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings |

Pages | 557-568 |

Number of pages | 12 |

Edition | PART 1 |

DOIs | |

State | Published - 2011 |

Event | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland Duration: 4 Jul 2011 → 8 Jul 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 6755 LNCS |

### Conference

Conference | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 |
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Country/Territory | Switzerland |

City | Zurich |

Period | 4/07/11 → 8/07/11 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science