Noisy Nonadaptive Group Testing with Binary Splitting: New Test Design and Improvement on Price-Scarlett-Tan's Scheme

In Group Testing, the objective is to identify $K$ defective items out of $N$, $K\ll N$, by testing pools of items together and using the least amount of tests possible. Recently, a fast decoding method based on binary splitting (Price and Scarlett, 2020) has been proposed that simultaneously achieve optimal number of tests and decoding complexity for Non-Adaptive Probabilistic Group Testing (NAPGT). However, the method works only when the test results are noiseless. In this paper, we further study the binary splitting method and propose (1) A NAPGT scheme that generalizes the original binary splitting method from the noiseless case into tests with $\rho$ proportion of false positives (the $\rho$-False Positive Channel), where $\rho$ is a constant, with asymptotically-optimal number of tests and decoding complexity, i.e. $\mathcal{O}(K\log N)$, and (2) A NAPGT scheme in the presence of both false positives and false negatives in test outcomes, improving and generalizing the work of Price, Scarlett and Tan~\cite{price2023fast} in two ways: First, under $\rho$-proportion of test results flipped ($\rho$-Binary Symmetric Channel) and within the general sublinear regime $K=\Theta(N^\alpha)$ where $0<\alpha<1$, our algorithm has a decoding complexity of $\mathcal{O}(\epsilon^{-2}K^{1+\epsilon})$ where $\epsilon>0$ is a constant parameter. Second, when the false negative flipping probability $\rho'$ satisfies $\rho'=\mathcal{O}(K^{-\epsilon})$ and the false positive flipping probability $\rho$ is a constant, we can simultaneously achieve $\mathcal{O}(\epsilon^{-1}K\log N)$ for both the number of tests and the decoding complexity. It remains open to achieve these optimals under the general BSC.