In this paper, we study a binary distributed detection problem under correlated noise by using wireless sensors and a fusion center (FC) with one antenna where the channel is a coherent multiple access. In order to decide between the two hypotheses, we design sensors to maximize the error exponent derived based on minimizing probability of error for Bayesian detection subject to network power constraint. We provide a closed form solution for the sensor encoders under correlated noise at the sensors. Furthermore, the effect of noise correlation at the sensors is investigated. It is shown that knowledge of noise correlation at the sensors will help to improve the system performance. Moreover, if correlation exists but the system is unaware of it, the system performance might improve depending on the correlation structure. Finally, simulations are provided to verify the analysis.