Note on noisy group testing: Asymptotic bounds and belief propagation reconstruction

An information theoretic perspective on group testing problems has recently been proposed by Atia and Saligrama, in order to characterise the optimal number of tests. Their results hold in the noiseless case, where only false positives occur, and where only false negatives occur. We extend their results to a model containing both false positives and false negatives, developing simple information theoretic bounds on the number of tests required. Based on these bounds, we obtain an improved order of convergence in the case of false negatives only. Since these results are based on (computationally infeasible) joint typicality decoding, we propose a belief propagation algorithm for the detection of defective items and compare its actual performance to the theoretical bounds.