We address the problem of designing multirate codes for a multiple-input and multiple-output (MIMO) system by restricting the receiver to be a successive decoding and interference cancellation type, when each of the input signals is encoded independently. It is assumed that the receiver knows the instantaneous fading channel states but the transmitter does not have access to them. We design a multirate coded MIMO system using punctured turbo codes and demonstrate the efficiency of the proposed scheme using simulation results. In particular, the simulation results show performance within 2 dB of MIMO channel capacity.