Multi-carrier direct sequence code division multiple access (MC-DS-CDMA) systems can support multi-rate services with a great deal of flexibility by dynamically changing timeand frequency-domain spreading gains. However, in a frequency-selective fading channel, the multi-rate users using the time spreading codes with an ancestor-to-child relation, even with orthogonal frequency spreading codes, may still be affected by different quantities of multiple access interference (MAI). How to design an effective radio resource management strategy for the MC-DS-CDMA systems subject to different levels of MAIs becomes a new and important issue. In this paper, based on the error rate analysis in the physical layer, we observe and define a new performance metric, called the MAI coefficient, which can quantitatively predict the MAI effect imposed on a particular code in a multi-rate MC-DS-CDMA system. Furthermore, from the radio resource management perspective, we develop an interference avoidance code assignment scheme based on the MAI coefficient. By simulations and analysis, we demonstrate that the proposed interference avoidance code assignment method can effectively reduce the MAI in a multi-rate MC-DS-CDMA system, while maintaining very good call blocking rate performance.