Sparse Graph Codes for Non-adaptive Quantitative Group Testing

This paper considers the problem of Quantitative Group Testing (QGT). Consider a set of N items among which K items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. In this work, we propose a non-adaptive QGT scheme using sparse graph codes over bi-regular bipartite graphs and binary t-error-correcting BCH codes. The proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, we show that for the sub-linear regime where $\frac{K}{N}$ vanishes as K, $N\rightarrow\infty$, the proposed scheme requires at most $m\approx 1.19K\log_{2}\left(4.74\frac{N}{K}\right)$ tests to recover all the defective items with probability approaching one as K, $N\rightarrow\infty$. This bound can be achieved by $t=2$. The testing and recovery algorithms of the proposed scheme for any $t\leq 4$ have the computational complexity of $O\left(K\log^{2}\frac{N}{K}\right)$ and $O\left(K\log\frac{N}{K}\right)$, respectively. Our simulation results also show that the proposed scheme significantly outperforms a non-adaptive semi-quantitative group testing scheme recently proposed by Abdalla et at. in terms of the required number of tests for identifying all the defective items with high probability.