Nonadaptive Noise-Resilient Group Testing with Order-Optimal Tests and Fast-and-Reliable Decoding

TL;DR

The paper tackles nonadaptive GT under noise by introducing Gacha GT, a modular framework that unites list-decodable and list-recoverable coding ideas with expander-based test designs. The core innovation is a folded Reed--Solomon construction that yields probabilistic list decoding with a hash-table–style test encoding, enabling $m=\mathcal{O}_Z(\sigma k \log(n) 2^{\mathcal{O}(\tau)})$ tests and decoding time $\mathcal{O}_Z(\sigma k\mathrm{poly}(\log(\sigma n)) 2^{\mathcal{O}(\tau)})$, while ensuring $k\exp(-\sigma \log_2(n)^{1-1/\tau})$ average misclassifications. By composing a hierarchy of gadgets (parallel, serial, and pyramid formations) and applying denoising steps (including Barg--Zémor capacity-achieving codes and channel downgrading), Gacha achieves robust performance across a wide range of parameter regimes and binary-input channels, improving partial-recovery, exact-recovery, and worst-case GT scenarios. The framework provides a modular path to scalable, noise-resilient GT with near-optimal test complexity and fast decoding, with potential impact on applications from heavy hitters to IoT device identification. Overall, Gacha demonstrates that carefully integrated coding-theoretic gadgets can drive order-optimal nonadaptive GT under realistic noisy conditions.

Abstract

Group testing (GT) is the Boolean version of spare signal recovery and, due to its simplicity, a marketplace for ideas that can be brought to bear upon related problems, such as heavy hitters, compressed sensing, and multiple access channels. The definition of a "good" GT varies from one buyer to another, but it generally includes (i) usage of nonadaptive tests, (ii) limiting to $O(k \log n)$ tests, (iii) resiliency to test noise, (iv) $O(k \mathrm{poly}(\log n))$ decoding time, and (v) lack of mistakes. In this paper, we propose $Gacha~GT$. Gacha is an elementary and self-contained, versatile and unified scheme that, for the first time, satisfies all criteria for a fairly large region of parameters, namely when $\log k < \log(n)^{1-1/O(1)}$. Outside this parameter region, Gacha can be specialized to outperform the state-of-the-art partial-recovery GTs, exact-recovery GTs, and worst-case GTs. The new idea Gacha brings to the market is a redesigned Reed--Solomon code for probabilistic list-decoding at diminishing code rates over reasonably-large alphabets. Normally, list-decoding a vanilla Reed--Solomon code is equivalent to the nontrivial task of identifying the subsets of points that fit low-degree polynomials. In this paper, we explicitly tell the decoder which points belong to the same polynomial, thus reducing the complexity and enabling the improvement on GT.

Figures (16)

• Figure 1: Gacha GT (this work): reshape the $\nu$-bit phone number into a $\sqrt\nu \times \sqrt\nu$ square array; copy-and-past each row to a random batch of tests. This combines ideas from Figures \ref{['fig:horizontal']} and \ref{['fig:vertical']}.
• Figure 2: How Section \ref{['sec:circle']} synthesizes corrupted words. Each $6$-digit number represents a symbol of $6\sqrt\nu$ bits. Each column is a codeword. Each row in Figure \ref{['fig:QR']} becomes a symbol here.
• Figure 3: Left: folded Reed--solomon code in literature. Right: our customized Reed--Solomon code.
• Figure 4: Left: a small GT scheme as a building block. Right: a large GT scheme built by disconnected, independent copies of $A$.
• Figure 5: Left: Proposition \ref{['pro:serial']} can be seen as a construction based on a complete bipartite graph. Right: Proposition \ref{['pro:expander']} can be seen as a construction based on a "good" graph.

Theorems & Definitions (31)

• Theorem 1: Main theorem
• Lemma 3: Birthdays are collision-free
• proof
• Lemma 4: Evaluation pairs are ample
• proof
• Proposition 5: Decoder is reliable
• proof
• Theorem 6: Toy Gacha
• Proposition 7: Repeat to increase throughput
• proof