In this paper, we consider lattice reduction (LR) aided linear detection in a DSTTD-OFDM (double space-time transmit diversity- orthogonal frequency division multiplexing) system with antenna shuffling. We first derive an antenna shuffling criterion for the LR-aided DSTTD-OFDM system. Next, we propose a combined reduced-feedback and adaptive LR algorithm by exploiting the correlation between OFDM subcarriers in the frequency domain. The LR-aided DSTTD OFDM system with this algorithm requires low computational effort for the LR operation and small feedback information. Simulation results show that a significant improvement could be achieved in the proposed system compared to previous (non-LR-aided) systems under spatially correlated channels. Also, the proposed complexity-reduced approach could greatly lower the system complexity while exhibiting a slight performance loss.