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 $2\leq u\leq d$ defective items, and negative otherwise. The defective items can be identified using $t=O((\ \frac{d}{u})^{u}(\frac{d}{d-u})^{d-u}(u\log\frac{d}{u}+\log\frac{1}{\epsilon})d^{2}\log n)$ tests with probability at least $1-\epsilon$ for any $\epsilon > 0$ or $t= O\left(\left(\frac{b}{u}\right)^{u}\left(\frac{d}{d-u}\right)^{d-u}\cdot d^{3}\log n \cdot \log \frac{n}{d}\right)$ tests with probability 1. The decoding time is $t\times$ poly $(d^{2}\log n)$. This result significantly improves the best known results for decoding non-adaptive threshold group testing: $O(n \log n+n\log\frac{1}{\epsilon})$ for probabilistic decoding, where $\epsilon > 0$, and $O(n^{u}\log n)$ for deterministic decoding.