We consider the problem of nonadaptive noiseless group testing of N items of which K are defective. We describe four detection algorithms, the COMP algorithm of Chan et al., two new algorithms, DD and SCOMP, which require stronger evidence to declare an item defective, and an essentially optimal but computationally difficult algorithm called SSS. We consider an important class of designs for the group testing problem, namely those in which the test structure is given via a Bernoulli random process. In this class of Bernoulli designs, by considering the asymptotic rate of these algorithms, we show that DD outperforms COMP, that DD is essentially optimal in regimes where K ≥ √N, and that no algorithm can perform as well as the best nonrandom adaptive algorithms when K > N^0.35. In simulations, we see that DD and SCOMP far outperform COMP, with SCOMP very close to the optimal SSS, especially in cases with larger K.