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

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 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.