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Multiuser diversity techniques are used in multicarrier data systems to enhance downlink cell throughput. This requires downlink channel information from the users that is opportunistically used by a base station to send data to the users with good channel condition. Channel feedback from the user to the base station incurs high overhead especially when many users are in the cell and each user needs to report channel information over multiple channels, as in OFDMA systems. To reduce the quantity of feedback information without significant throughput degradation, a practical strategy is to deliver feedback on a partial set of channels with the best channel quality. We call it best feedback, and this reporting scheme carried out for best four or five among 24 channels has been already adopted in the IEEE 802.16e standard. Considering real feedback conditions, we investigate the performance of a best feedback scheme and derive the optimal number of channels for which information needs to be fed back to keep the throughput gap (compared to a full feedback scheme) within a target margin. From the optimal condition, we propose an adaptive best feedback algorithm, where the number of reported channels is adjusted to adapt to the number of users in the cell. We also propose an adjusted periodic feedback algorithm, where users are divided into groups and scheduling is carried out group by group, so a user can report feedback information when his group is scheduled, thereby reducing the frequency of feedback transmission. To support differentiated performance, we further propose heterogeneous feedback algorithms where users are divided into heterogeneous groups, each with a different group size or a different feedback period. Numerical results validate our analysis and provide meaningful insights into the design of various channel feedback schemes.