  Given $d$ defective items in a population of $n$ items with $d \ll n$, in
threshold group testing without gap, the outcome of a test on a subset of items
is positive if the subset has at least $u$ defective items and negative
otherwise, where $1 \leq u \leq d$. The basic goal of threshold group testing
is to quickly identify the defective items via a small number of tests. In
non-adaptive design, all tests are designed independently and can be performed
in parallel. The decoding time in the non-adaptive state-of-the-art work is a
polynomial of $(d/u)^u (d/(d-u))^{d - u}, d$, and $\log{n}$. In this work, we
present a novel design that significantly reduces the number of tests and the
decoding time to polynomials of $\min\{u^u, (d - u)^{d - u}\}, d$, and
$\log{n}$. In particular, when $u$ is a constant, the number of tests and the
decoding time are $O(d^3 (\log^2{n}) \log{(n/d)} )$ and $O\big(d^3 (\log^2{n})
\log{(n/d)} + d^2 (\log{n}) \log^3{(n/d)} \big)$, respectively. For a special
case when $u = 2$, with non-adaptive design, the number of tests and the
decoding time are $O(d^3 (\log{n}) \log{(n/d)} )$ and $O(d^2 (\log{n} +
\log^2{(n/d)}) )$, respectively. Moreover, with 2-stage design, the number of
tests and the decoding time are $O(d^2 \log^2{(n/d)} )$.
