Space-time block codes (STBCs) are designed for multiple-input-multiple-output (MIMO) channels. In order to avoid errors, single-input-single-output (SISO) fading channels require long coding blocks and interleavers that result in high delays. If one wishes to increase the data rate it is necessary to take advantage of space diversity. Early STBC, that where developed by Alamouti for known channels and by Tarokh for unknown channels, have been proven to increase the performance of channels characterized by Rayleigh fading. Codes that are based on division algebras have by definition nonzero diversity and therefore are suitable for STBC in order to achieve high rates at low symbol-to-noise ratio (SNR). This work presents new high-diversity group-based STBCs with improved performance both in known and unknown channels. We describe two new sets of codes for multiple antenna communication. The first set is a set of "superquaternions" and improves considerably on the Alamouti codes. It is based on the mathematical fact that "normalized" integral quaternions are very well distributed over the unit sphere in four-dimensional (4-D) Euclidean space. The second set of codes gives arrays of 3times3 unitary matrices with full diversity. Here the idea is to use cosets of finite subgroups of division algebras that are nine-dimensional (9-D) over their center, which is a finite cyclotomic extension of the field of rational numbers. It is shown that these codes outperform Alamouti and G mr