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Multiple-input multiple-output (MIMO) system with multicode transmission can provide high speed data services by transmitting independent parallel substreams from multiple antennas and through multicode channelization. In this paper, we first introduce an iterative two-stage successive interference cancellation (SIC) detection scheme for a multicode MIMO system. The proposed technique cancels the interference signals successively in the space domain followed by the code domain. Next, we develop various transmit power allocation schemes over different data substreams for the proposed detection process to improve error rate performance. The joint transmit power allocation is derived to make the post-detection signal-to-interference-plus-noise ratio (SINR) become the same for all substreams in both the space and code domains. As a computationally efficient scheme, we propose a two-stage power allocation scheme, which allocates the total transmit power to the substreams in the code domain at the first stage, and allocates this code domain power to the substreams in the space domain at the second stage. Furthermore, variable and constant power ratio (PR) schemes are derived to reduce the feedback overhead. In particular, the constant PR scheme utilizes the transmit power ratio determined by the long-term statistical properties of the fading channel amplitudes, and achieves significantly reduced feedback rate. Numerical results show that the proposed transmit power allocation schemes for the two-stage SIC significantly outperform the equal power allocation scheme.