Sublinear-Time Non-Adaptive Group Testing with $O(k \log n)$ Tests via Bit-Mixing Coding

The group testing problem consists of determining a small set of defective items from a larger set of items based on tests on groups of items, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. While rigorous group testing algorithms have long been known with runtime at least linear in the number of items, a recent line of works has sought to reduce the runtime to ${\rm poly}(k \log n)$, where $n$ is the number of items and $k$ is the number of defectives. In this paper, we present such an algorithm for non-adaptive probabilistic group testing termed {\em bit mixing coding} (BMC), which builds on techniques that encode item indices in the test matrix, while incorporating novel ideas based on erasure-correction coding. We show that BMC achieves asymptotically vanishing error probability with $O(k \log n)$ tests and $O(k^2 \cdot \log k \cdot \log n)$ runtime, in the limit as $n \to \infty$ (with $k$ having an arbitrary dependence on $n$). This closes a recently-proposed open problem of simultaneously achieving ${\rm poly}(k \log n)$ decoding time using $O(k \log n)$ tests without any assumptions on $k$. In addition, we show that the same scaling laws can be attained in a commonly-considered noisy setting, in which each test outcome is flipped with constant probability.