Sparse Graph Codes for Non-adaptive Quantitative Group Testing
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This paper considers the problem of quantitative group testing with a non-adaptive testing strategy. In a quantitative group testing scheme, a set of tests are designed to identify defective items among a large population of items, where the outcome of a test shows the number of defective items in the tested group. There are two models for the defective items: (i) deterministic, and (ii) randomized. In the deterministic model, the exact number of the defective items, K, is known, whereas in the randomized model each item is defective with probability K/N, independent of the other items, where N is the total number of items. In this work, we propose a quantitative non-adaptive group testing algorithm using sparse graph codes over bi-regular bipartite graphs with left-degree ℓ and right degree r and binary t-error-correcting BCH codes. We show that for both the deterministic and randomized models, our algorithm requires at most m=c(t)K(t(ℓ N/c(t)K+1)+1)+1 tests to recover all the defective items with probability approaching one (as N and K grow unbounded), where c(t) is a constant that depends only on t. The results of our theoretical analysis reveal that using a t-error-correcting binary BCH code for t∈{1,2,3}, when compared to t≥ 4, leads to a fewer number of tests. Simulation results show that the proposed strategy significantly reduces the required number of tests for identifying all the defective items with probability approaching one compared to a recently proposed scheme.
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