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

In this paper, we study the binary splitting method for group testing and propose (1) A nonadaptive probabilistic group testing scheme that generalizes the original binary splitting method (Price and Scarlett, 2020; Cheraghchi and Nakos, 2020) 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. $\mathcal{O}(K \log N)$, where $K$ is the number of defectives and $N$ is the number of items, and (2) A non-adaptive probabilistic group testing scheme in the presence of both false positives and false negatives in test outcomes, improving and generalizing the work of Price, Scarlett and Tan (2023) in two ways: First, under $\rho$ proportion of test results flipped ($\rho$-Binary Symmetric Channel) and within the general sublinear regime $K=\Theta\left(N^{\alpha}\right)$ where $0<\alpha<1$, our scheme has $\mathcal{O}\left(\epsilon^{-1} K \log N\right)$ tests and a decoding complexity of $\mathcal{O}\left(\epsilon^{-2} K^{1+\epsilon}\right)$ where $\epsilon>0$ is a constant parameter. Second, when the false negative flipping probability $\rho^{\prime}$ satisfies $\rho^{\prime}=\mathcal{O}\left(K^{-\epsilon}\right)$ and the false positive flipping probability $\rho$ is a constant, we can simultaneously achieve $\mathcal{O}\left(\epsilon^{-1} K \log N\right)$ for both the number of tests and the decoding complexity. It remains open to achieve these under the general BSC. (The full-version of this paper is available online at https://arxiv.org/abs/2410.14566.)