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Complex orthogonal designs (CODs) have been successfully implemented in wireless systems as complex orthogonal space-time block codes (COSTBCs). Certain properties of the underlying CODs affect the performance of the codes. In addition to the main properties of a COD's rate and decoding delay, a third consideration is whether the COD can achieve transceiver signal linearization, a property that facilitates practical implementation by, for example, significantly simplifying the receiver structure for iterative decoding. It has been shown that a COD can achieve this transceiver signal linearization if the nonzero entries in any given row of the matrix are either all conjugated or all nonconjugated. This paper determines the conditions under which maximum rate CODs can achieve this desirable property. For an odd number of transmit antennas, it is shown that maximum rate CODs that achieve the lower bound on decoding delay can also achieve transceiver signal linearization. In contrast, for an even number of transmit antennas, maximum rate CODs that achieve the lower bound on delay cannot achieve this linearization. In this latter case, linearization is possible only if the COD achieves at least twice the lower bound on delay. This work highlights the tradeoffs among these three important properties.