Efficiently Decodable Error-Correcting List Disjunct Matrices and Applications - (Extended Abstract)

A (d, )-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 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 e0 false positive and e1 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(d log(n/d) + e0 + de1) tests is necessary and sufficient when = Θ(d). Similar results are also derived for the disjunct matrix case (i.e. = 1). (2) We present two methods that convert errortolerant 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 log n+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(d log n), 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.