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Orthogonal frequency division multiplexing (OFDM) is an effective technique to deal with frequency-selective channels since it facilitates low complexity equalization and decoding. Many existing OFDM designs successfully exploit the multipath diversity offered by frequency-selective channels. However, most of them require maximum likelihood (ML) or near-ML detection at the receiver, which is of high complexity. On the other hand, empirical results have shown that linear detectors have low complexity but offer inferior performance. In this paper, we analytically quantify the diversity orders of linear equalizers for linear precoded OFDM systems, and prove that they are unable to collect full diversity. To improve the performance of linear equalizers, we further propose to use a lattice reduction (LR) technique to help collect diversity. The LR-aided linear equalizers are shown to achieve maximum diversity order (i.e., the one collected by the ML detector), but with low complexity that is comparable to that of conventional linear equalizers. The theoretical findings are corroborated by simulation results.