Efficient Decoding Schemes for Noisy Non-Adaptive Group Testing when Noise Depends on Number of Items in Test

The goal of non-adaptive group testing is to identify at most $d$ defective items from $N$ items, in which a test of a subset of $N$ items is positive if it contains at least one defective item, and negative otherwise. However, in many cases, especially in biological screening, the outcome is unreliable due to biochemical interaction; i.e., \textit{noise.} Consequently, a positive result can change to a negative one (false negative) and vice versa (false positive). In this work, we first consider the dilution effect in which \textit{the degree of noise depends on the number of items in the test}. Two efficient schemes are presented for identifying the defective items in time linearly to the number of tests needed. Experimental results validate our theoretical analysis. Specifically, setting the error precision of 0.001 and $d\leq16$, our proposed algorithms always identify all defective items in less than 7 seconds for $N=2^{33}\approx 9$ billion.