Efficiently Decodable Non-Adaptive Threshold Group Testing

We consider non-adaptive threshold group testing for identification of up to d defective items in a set of n items, where a test is positive if it contains at least 2 ≤ u ≤ d defective items, and negative otherwise. The defective items can be identified using t = O ((d/u)
^u
(d/d-u)
^d-u
(u log d/u + log 1/∈)·d
²
log n) tests with probability at least 1 - ∈ for any ∈ > 0 or t = O((d/u)
^u
(d/d-u)
^d-u
d
³
log n · log d/n) tests with probability 1. The decoding time is t × poly(d
²
log n). This result significantly improves the best known results for decoding non-adaptive threshold group testing: O(n log n + n log 1/∈) for probabilistic decoding, where ∈ > 0, and O(n
^u
log n) for deterministic decoding.