We consider the following “efficiently decodable ” nonadaptive group testing problem. There is an unknown string x ∈ {0, 1} n with at most d ones in it. We are allowed to test any subset S ⊆ [n] of the indices. The answer to the test tells whether xi = 0 for all i ∈ S or not. The objective is to design as few tests as possible (say, t tests) such that x can be identified as fast as possible (say, poly(t)-time). Efficiently decodable non-adaptive group testing has applications in many areas, including data stream algorithms and data forensics. A non-adaptive group testing strategy can be represented by a t × n matrix, which is the stacking of all the characteristic vectors of the tests. It is well-known that if this matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input string. Furthermore, we know how to construct d-disjunct matrices with t = O(d 2 log n) efficiently. However, these matrices so far only allow for a “decoding ” time of O(nt), which can be exponentially larger than poly(t) for relatively small values of d. This paper presents a randomness efficient construction of d-disjunct matrices with t = O(d 2 log n) that can be decoded in time poly(d) · t log 2 t + O(t 2). To the best of our knowledge, this is the first result that achieves an efficient decoding time and matches the best known O(d 2 log n) bound on the number of tests. We also derandomize the construction, which results in a polynomial time deterministic construction of such matrices when d = O(log n / log log n). A crucial building block in our construction is the notion of (d, ℓ)-list disjunct matrices, which represent the more general “list group testing ” problem whose goal is to output less than d + ℓ positions in x, including all the (at most d) positions that have a one in them. List disjunct matrices turn out to be interesting objects in their own right and were also considered independently by [Cheraghchi, FCT 2009]. We present connections between list disjunct matrices, expanders, dispersers and disjunct matrices. List disjunct matrices have applications in constructing (d, ℓ)sparsity separator structures [Ganguly, ISAAC 2008] and in constructing tolerant testers for Reed-Solomon codes in the data stream model. 1

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, in-cluding 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 dis-junct matrix case (i.e. ` = 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(d2 logn+ rd) tests which are decodable in poly(t) time, where r is the maximum num-ber of test errors. This result is optimal for r = Ω(d logn), and even for r = 0 this result improves upon known results; (ii) (explicit) con-structions 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.

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