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This paper investigates optimal transmission of space-time block codes (STBCs) in distributed multiple-input multiple-output (D-MIMO) Rayleigh fading channels. The optimal diversity performance is achieved through transmit power allocation implemented at the receiver based on transmit and receive correlations to minimize the average symbol error rate (SER). Evaluation of SER performance of uncoded STBCs over a generalized distributed antenna (DA) topology is first presented, with exact analytical SER expressions derived for MQAM and MPSK symbols. SER upper bounds are also derived, based on which two criteria for complexity reduced antenna subset selection with sub-optimal power allocation are further proposed, whose performance approaches optimal over correlated D-MIMO channels. Moreover, a novel simplified but close SER approximation scheme is devised to significantly facilitate optimal SER calculation. We continue to thoroughly analyze how the optimal diversity is affected by large scale fading, targeted data rate, antenna correlations and transmit power. Finally, we develop a surprisingly close and useful analogy between open loop STBCs in co-located MIMO and optimal STBCs in D-MIMO with minimum feedback (i.e., n bits for n DAs in Criterion 2 with power allocation scheme 2 which equally allocates power to the selected DAs). Extensive simulation results have been presented to demonstrate the effectiveness of our analysis.