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Recently, a hardware-friendly fixed-complexity Lenstra-Lenstra-Lovász (fcLLL) algorithm has been developed for use in the linear receivers of multiple-input-multiple-output (MIMO) systems. In this paper, we propose a novel fcLLL-based detection method that significantly outperforms conventional fcLLL-based linear receivers at a lower fixed complexity. The main idea is that the complexity of fcLLL can be significantly reduced by disregarding a single column of the channel matrix. We also propose column selection methods that consider both the error performance and complexity simultaneously. Simulations are performed to demonstrate the efficacy of the proposed method.