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Orthogonal designs have received considerable attention in the development of efficient modulation and coding methods for future multi-antenna wireless communication systems due to their special properties. In this paper, we propose a class of space-time block codes constructed by combining orthogonal designs with sphere packing for an arbitrary number of transmit antennas. The structure of the orthogonal designs is exploited to guarantee full diversity, and sphere packing is used to improve the coding advantage. Space-time block code construction from block-orthogonal designs is also considered: the full-diversity property is ensured by rotating the sphere packing underlying the code, and the optimal rotation angle is determined for a class of sphere packing. Code design examples are provided for two and four transmit antennas and various transmission rates. The simulation results show that by jointly designing the symbols in the orthogonal designs, the performance of the block codes can be significantly increased.