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We investigate multirate multicarrier code-division multiple-access (MC-CDMA) downlink systems with multicode transmission. Specifically, we design a new linear equalizer that outperforms the minimum mean-squared error (MMSE) per-carrier equalizer, without requiring the spreading codes of the interfering users. The complexity of this new linear equalizer is similar to that of the MMSE per-user equalizer, which however needs the spreading codes of the interfering users. Simulation results validate the effectiveness of the proposed equalizer, especially when there are few interfering users.