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This paper studies the mutual information improvement attained by rotated multidimensional (multi-D) constellations via a unitary precoder G in Rayleigh fading. At first, based on the symmetric cut-off rate of the N-D signal space, we develop a design criterion with regard to the precoder G. It is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high-rate regimes. Based on parameterizations of unitary matrices, we then construct good unitary precoder G in 4-D signal space using a simple optimization problem, which involves only four real variables and it is applicable to any modulation scheme. To further illustrate the potential of multi-D constellation and to show the practical use of mutual information improvement, we propose a simple yet powerful bit-interleaved coded modulation (BICM) scheme in which a (multi-D) mapping technique employed in a multi-D rotated constellation is concatenated with a short-memory high-rate convolutional code. By using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique provides an exceptionally good error performance. In particular, both EXIT chart analysis and simulation results indicate that a turbo pinch-off and a bit error rate around 10-6 happen at a signal-to-noise ratio that is well below the coded modulation and BICM capacities using traditional signal sets. For example, with code rates ranging from 2/3 to 7/8, the proposed system can operate 0.82 dB-2.93 dB lower than the BICM capacity with QPSK and Gray labeling. The mutual information gain offered by rotated constellations can be therefore utilized to design simple yet near Shannon limit systems in the high-rate regions.