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This paper aims at comparing the performance of numerous space-time trellis coded modulations (STTCM) over slow and fast Rayleigh fading channels in order to identify the most efficient ones. After a brief presentation of the STTCM concept, we propose to recall the criteria used to construct the codes studied herein, depending on the channel properties and the number of transmit and receive antennas. Then, we introduce our new representation of the STTCM as an extension of the one currently used to describe regular trellis codes, namely the polynomial representation. Simulation results and especially new performance relative to BER scale are then given in order to compare the behavior of Tarokh's, Baro's, Blum's, Chen's and Firmanto's codes over slow and fast Rayleigh fading channels. From these comparisons, it follows that the codes constructed with the trace criterion, namely Chen's codes, outperform all the other tested codes over both channels.