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Polar codes, ever since their introduction, have been shown to be very effective for various wireless communication channels. This, together with their relatively low implementation complexity, has made polar codes an attractive coding scheme for wireless communications.On the other hand, within the realm of non-coherent wireless MIMO communication, Grassmannian signalling has been shown to approach the ergodic capacity of frequency-flat block fading channels at high SNR. In this paper, we combine these two components together using multilevel polar coding to create a system that produces the best known performance for a coded non-coherent channel. This combination requires that the signal constellation be set-partition labelled, so, a novel set partitioning algorithm, which works for regular and irregular multidimensional constellations, such as the Grassmannian constellation, is proposed. Finally, we develop a methodology for designing polar codes for a noisy channel and this is used to further improve the performance of our system. In simulation, we compare the error rate performance of our design with that of existing schemes and show that significant gains are possible over the previously known best technique, which is based on turbo codes.We then provide a complexity analysis of our receiver in comparison with other existing methods and show that it is able to provide these gains at a considerably lower complexity.