**6. Conclusions**

In this paper, we considered a NTGT problem where the test result is positive when the number of defective samples in a pool equals or greater than a certain threshold. Recently, when performing GT for the diagnosis COVID-19 infection, if the sample's virus concentration did not sufficiently reach the threshold, false positives or false negatives can occur, so in this work we dealt with this TGT framework. In addition, a noise model was added in case pure results were flipped due to unexpected measurement noise. We took into account how many tests were needed to successfully reconstruct a small defective sample with the NTGT problem. To this end, we aimed to find the necessary and sufficient conditions for the number of tests required. For the necessary condition, we obtained the lower bound on the number of tests using Fano's inequality theorem. Next, the upper bound on performance defined by the probability of error was derived using the MAP decoding method. This result leads to the sufficient condition for identifying all defective samples in the NTGT problem. In this paper, we have shown that the necessary and sufficient conditions are consistent with the NTGT framework. In addition, we presented that the relationship between the defective rate of the input signal and the sparsity of the group matrix should be considered to design an optimal NTGT system.

**Funding:** National Research Foundation of Korea: NRF-2020R1I1A3071739.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable

**Conflicts of Interest:** The authors declare no conflict of interest.
