Improved non-adaptive algorithms for threshold group testing with a gap
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The basic goal of threshold group testing is to identify up to d defective items among a population of n items, where d is usually much smaller than n. The outcome of a test on a subset of items is positive if the subset has at least u defective items, negative if it has up to ℓ defective items, where 0 ≤ℓ < u, and arbitrary otherwise. This is called threshold group testing with a gap. There are a few reported studies on test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a corrected theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing. Specifically, we present a better upper bound on the number of tests for disjunct matrices compared to related work. The third and fourth contributions are a reduction in the number of tests and a reduction in the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a demonstration of the resulting improvements by simulation for previous work and the proposed schemes.
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