Efficiently decodable defective items detected by new model of noisy group testing

Detection of defective items from a colossal number of items, commonly referred to as "Group Testing", has been widely studied because of its promising applications in various fields. To detect at most d defective items from N items, a subset of N items is formed into a test and tested. If the total number of tests needed is t, a set of t tests can be seen as a t × N measurement matrix, M = (m
_ij
), in which each row represents a test; m
_ij
= 1 iff item j belongs to test i. Ideally, the outcome for a test is positive iff there is at least a defective item in that test. However, in many cases, especially in chemical screening, the outcome is not reliable; i.e., a positive outcome can change to a negative one (false negative) and vice versa (false positive) due to noise. Normally, noise is a random variable. However, in chemical screening, the degree of noise depends on the number of items in a test. We call this phenomenon new model of group testing. Despite the frequent occurrence of this effect in chemical screening, group testing using the new model is not well studied. Moreover, there is no decoding scheme for this model. In this paper, we present probabilistic existence of successful decoding of a defective item detected by a new model of noisy group testing when there is at most 0 ≤ e
₀
≤ log N false positives and 0 ≤ e
₁
≤ log N false negatives. If e
₀
= o(t) and ei = o(t) as t goes to infinity, the probability of successful decoding goes to 1. The decoding time is O(eo + e
₁
) χ poly(t). Moreover, a matrix M associated with this decoding scheme can be explicitly constructed in time O(tN) where t = O (log N + e
₀
+ e
₁
). Based on this result, we generalized our result for decoding d defective items by using a divide and conquer (D&C) strategy.