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In this paper, we address the problem of space-time codebook design for noncoherent communications in multiple-antenna wireless systems. In contrast with other approaches, the channel matrix is modeled as an unknown deterministic parameter at both the receiver and the transmitter, and the Gaussian observation noise is allowed to have an arbitrary correlation structure, known by the transmitter and the receiver. In order to handle the unknown deterministic space-time channel, a generalized likelihood ratio test (GLRT) receiver is considered. A new methodology for space-time codebook design under this noncoherent setup is proposed. It optimizes the probability of error of the GLRT receiver's detector in the high signal-to-noise ratio (SNR) regime by solving a high-dimensional nonlinear nonsmooth optimization problem in a two-step approach. First, a convex semidefinite programming (SDP) relaxation of the codebook design problem yields a rough estimate of the optimal codebook. This is then refined through a geodesic descent optimization algorithm that exploits the Riemannian geometry imposed by the power constraints on the space-time codewords. The results obtained through computer simulations illustrate the advantages of our method. For the specific case of spatio-temporal white observation noise, our codebook constructions replicate the performance of state-of-the-art known solutions. The main point here is that our methodology permits extending the codebook construction to any given correlated noise environment. The simulation results show good performance of these new designed codes in colored noise setups.