## 1 Introduction  
The demand for high data rate wireless services combined with the requirement for an increased Quality of Service (QoS) is driving the research and innovation in the wireless communications networks. Multiple-Input Multiple-Output (MIMO) techniques are characterised by an increased capacity as well as improved performance and hence the research in the MIMO design has seen a great interest for the last few years [1; 2; 3]. Recently, large-scale MIMOs have been proposed in order to scale up the MIMO gains [4; 5], where, for example a large number of antennas can be implemented at the base station for supporting a large number of users. MIMOs can be used to attain a multiplexing gain, where different transmit antennas transmit different symbols at the same time and using the same frequency. This increases the system throughput at the expense of having a more complex receiver for the sake of removing the interference imposed by the simultaneously transmitted symbols. As the number of transmit antennas increases, the detector complexity increases with the increase in the interference. Therefore, in large-scale MIMO systems, one of the main challenges is to reduce the detection complexity [6; 7; 8]. The Maximum Likelihood (ML) detector offers the best possible Bit Error Ratio (BER) performance at the expense of an increased complexity, which becomes impractical in terms of hardware implementation as the number of antennas and constellation size increases. On the other hand, linear detectors, such as the Zero Forcing (ZF) detector, are simple to implement, which is at the expense of a significant performance degradation compared to the ML detector. Hence, Lattice Reduction (LR) aided detectors have been proposed for MIMO systems in order to achieve a near-ML performance with significantly lower complexity [9; 10]. Several algorithms have been proposed in the literature for performing Lattice Reduction such as the Lenstra, Lanstra, and Lovasz (LLL) algorithm [11], the Korkine-Zolotareff (KZ) algorithm [12] and the Element-based Lattice Reduction (ELR) algorithm [13]. The difference between these algorithms lies in their complexity, which also affects their performance when used for MIMO detection. The ELR algorithm has been proposed as a reduced complexity LR algorithm, while performing better than the LLL-aided MIMO detectors, when the number of antennas in significantly high [13]. In large scale MIMOs, the ELR algorithm requires less number of arithmetic operations for the LR basis update than the LLL algorithm [13]. LLL-aided linear detectors were employed in [10; 14] to improve the performance of the conventional linear detectors dispensing with LR. On the other hand, Sphere Decoder (SD) [15] was proposed in order to reduce the complexity of ML, while obtaining a near-ML performance. The SD searches only through those points that fall inside a sphere of radius \(r\) rather than do a full search as in the ML detector. There are many approaches to perform the tree search in the SD, where in this paper, we use the K-best algorithm [16; 17]. LLL-aided K-best detector has been proposed in [18] in order to reduce the performance gap between the LR-aided detectors and their ML counterpart [18]. Recently, the ELR algorithm has been proposed for attaining an improved BER performance in large-scale MIMOs, while maintaining lower complexity compared to the LLL-aided detector [13]. Therefore, motivated by the results of [18], where the performance of the K-Best detector was improved by performing LR of the channel matrix before the detection, in [19] we proposed to improve the large-scale MIMO detection performance by applying the ELR algorithm with the K-Best detector, which we refer to as ELR-aided K-Best detector. We have shown in [19] that the ELR-aided K-Best detector is capable of achieving an improved BER performance compared to the LLL-aided K-Best detector, while requiring a significantly lower complexity.