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In this paper, a novel concept called a uniquely factorable constellation (UFC) is proposed for the systematic design of noncoherent full-diversity constellations for a wireless communication system that is equipped with a single transmitter antenna and multiple receiver antennas [single input-multiple output (SIMO)], where neither the transmitter nor the receiver knows channel-state information. It is proved that such a UFC design guarantees the unique blind identification of channel coefficients and transmitted signals in a noise-free case by processing only two received signals, as well as full diversity for the noncoherent maximum-likelihood (ML) receiver in a noise case. By using the Lagrange four-square theorem, an algorithm is developed to efficiently and effectively design various sizes of energy-efficient unitary UFCs to optimize the coding gain. In addition, a closed-form optimal energy scale is found to maximize the coding gain for the unitary training scheme based on the commonly used quadratic-amplitude modulation (QAM) constellations. Comprehensive computer simulations show that, using the same generalized likelihood ratio test (GLRT) receiver, the error performance of the unitary UFC designed in this paper outperforms the differential scheme, the optimal unitary training scheme presented in this paper, and the signal-to-noise ratio (SNR) efficient training scheme using the QAM constellation, which, so far, yields the best error performance in the current literature. However, with the same training ML receiver, the error performance of the UFCs is worse than the training scheme based on the QAM constellations.