  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 $t = O \left( \left( \frac{d}{u}
\right)^u \left( \frac{d}{d - u} \right)^{d-u} \left(u \log{\frac{d}{u}} +
\log{\frac{1}{\epsilon}} \right) \cdot d^2 \log{n} \right)$ tests with
probability at least $1 - \epsilon$ for any $\epsilon > 0$ or $t = O \left(
\left( \frac{d}{u} \right)^u \left( \frac{d}{d -u} \right)^{d - u} d^3 \log{n}
\cdot \log{\frac{n}{d}} \right)$ tests with probability 1. The decoding time is
$t \times \mathrm{poly}(d^2 \log{n})$. 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.
