We investigate the stochastic resonance phenomenon in a discrete Hopfield neural network for transmitting binary amplitude modulated signals, wherein the binary information is represented by two stored patterns. Based on the potential energy function and the input binary signal amplitude, the observed stochastic resonance phenomena involve two general noise-improvement mechanisms. A suitable amount of added noise assists or accelerates the switch of the network state vectors to follow input binary signals more correctly, yielding a lower probability of error. Moreover, at a given added noise level, the probability of error can be further reduced by the increase of the number of neurons. When the binary signals are corrupted by external heavy-tailed noise, it is found that the Hopfield neural network with a large number of neurons can outperform the matched filter in the region of low input signal-to-noise ratios per bit.