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We consider the use of linear constellation precoding and linear detection in a multicarrier OFDM system with multiple receive antennas to obtain improved performance over multipath fading channels at a low complexity. We split the full set of subcarriers into smaller groups and spread the data symbols assigned to each group via precoding matrices. We adopt the diversity-multiplexing tradeoff (DMT) framework and derive the DMT-optimal split of the subcarriers (DMT-optimal grouping) and the DMT-optimal number of symbols assigned to each group (DMT-optimal symbol loading). We determine necessary and sufficient precoder design conditions to achieve DMT optimality and give specific constructions of such precoders. Next, we consider a multiuser OFDMA system and derive an algorithm which divides the available subcarriers among the active users in order to maximize the diversity order of the system (or joint) error probability. We also extend our analysis to OFDM systems equipped with multiple transmit antennas. Finally, we obtain important insights on the role of outer codes in an OFDM system employing linear precoding and linear equalization.