This thesis consists of two main areas: space-time coding for multiple-input multiple-output (MIMO) wireless systems and group coding generated from affine reflection groups.;We present systematic designs of full diversity unitary space-time codes with high coding gain, using group codes (permutation codes variant II and finite reflection group codes), the Bruhat decomposition and character tables of groups. We also analyze the pairwise error probability of space-time codes in a keyhole channel, and develop a fast optimization to the decision metric of multiple-symbol detection for differential unitary space-time modulation using a stack algorithm.;Group codes generated from affine reflection group codes are related to high dimensional constellations, that is, lattice codes. High dimensional constellations have been studied widely for problems in communication systems, with priority on finding those with good minimum nearest-neighbour distance and simple decoding algorithm.;We derive formulas for the initial vector and best minimum distance of affine reflection group codes of any rank. We also propose efficient decoding algorithms (from the geometrical analysis of groups and from their trellis diagram representations). This includes an original method to decode the codewords which lie outside the bounded region. Finally, we investigate their applications to orthogonal frequency division multiplexing (OFDM) systems to reduce a peak-to-average power ratio.;MIMO systems, using multiple antennas at the transmitter and receiver, employ multiplexing to increase the rate and diversity to improve the performance and spectral efficiency of data transmission in wireless communications. Space-time coding has been developed to achieve high data rate and reliability in MIMO wireless systems. The research on space-time coding (in code designs, performance analysis, decoding algorithms and its applications) has been growing rapidly in past decade.
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