Optimal and Lattice Codes in a MIMO System With Complete Feedback

Space time coding for a multiple-input, multiple-output (MIMO) system with complete feedback and maximum-likelihood (ML)-decoding is considered. First, a criterion for designing an optimal codebook with complete feedback in the case of high signal to noise ratio (SNR) is given; it turns out that designing an optimal codebook with complete feedback in the case of high SNR is equivalent to efficiently packing a finite number of points into a dynamic ellipsoid in a finite dimensional real Euclidean space. Second, based on this design criterion, an optimal tradeoff between diversity and multiplexing is then derived. Finally, practical design using lattice codes is addressed, and surprisingly, it is shown that the lattice codes can achieve the optimal tradeoff. It is also demonstrated that it is not necessary for an optimal lattice code to use all the subchannels generated by the system. This is in striking contrast with a general optimal code, which generally utilizes all the subchannels.