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The focus of the paper is on the design of space-time codes for a general multiple-input, multiple-output detection problem, when multiple observations are available at the receiver. The figure of merit used for optimization purposes is the convex combination of the Kullback-Leibler divergences between the densities of the observations under the two hypotheses, and different system constraints are considered. This approach permits to control the average sample number (i.e., the time for taking a decision) in a sequential probability ratio test and to asymptotically minimize the probability of miss in a likelihood ratio test: the solutions offer an interesting insight in the optimal transmit policies, encapsulated in the rank of the code matrix, which rules the amount of diversity to be generated, as well as in the power allocation policy along the active eigenmodes. A study of the region of achievable divergence pairs, whose availability permits optimization of a wide range of merit figures, is also undertaken. A set of numerical results is finally given, in order to analyze and discuss the performance and validate the theoretical results.