## 1 Introduction  
Multi-user (MU) massive multiple input multiple output (MIMO) is one of core techniques for high error performance and spectral efficiency in wireless communication systems without additional bandwidths and transmit powers. Compared to traditional MU-MIMO systems which use only two, four, or eight antennas, the massive MIMO systems make the fading channels asymptotically orthogonal, since the base station (BS) has tens to hundreds of antennas [1; 2; 3; 4]. The various performances were studied in the massive MIMO systems in [5; 6; 7; 8; 9]. Achievable user rate and throughput performances were analyzed in a multi-hop massive MIMO system [5], in a dual-polarized massive MIMO system [6] and in a transceiver hardware impairment massive MIMO system [7]. Additionally, the method for pilot allocation for enhanced channel estimation was studied in [8; 9]. The orthogonal frequency division multiplexing (OFDM) technique has the advantage of robustness in the frequency selective fading channels. In the OFDM, each narrowband channel goes through a frequency flat channel since the OFDM uses a cyclic prefix (CP). The CP produces the effect of a fading channel with multiplications at the receiver. Like the 4G long term evolution (LTE), 5G new radio (NR) uses the transmission waveform as CP-OFDM with various numerologies. Therefore, the combination of the massive MIMO and the OFDM gives a tremendous data rate. At the BS in the massive MIMO system, the transmit symbols are preprocessed for eliminating inter user interference in the downlink system, and the received symbols are decoded for eliminating inter antenna interference in the uplink system. The linear zero-forcing (ZF) is a suboptimal pre-coding or detection scheme, since the massive MIMO system has a high loading factor \((\alpha)\) which is defined as the ratio of the number of BS antennas to the number of total users. In [1; 2], the ergodic spectral efficiency for the ZF is analyzed in the massive MIMO system and suboptimal performance is proven in both downlink and uplink systems. However, the ZF calculates the inversion of the gram matrix, and it requires \(O\left(K^{3}\right)\) multiplications when the number of total users is \(K\). The huge complexity with respect to the number of users is prohibited in the massive MIMO system since dozens of active users should be supported. Thus, several pre-coding and detection schemes for reducing the complexity of the ZF were surveyed in [10; 11; 12; 13]. Among various schemes, ref [12] proved that the Gauss-Seidel (GS) requires very low transmit power for obtaining target error performance compared to other schemes such as the Neumann series (NS), the Jacobi method, the Richardson method, and optimized coordinate descent. In [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25], the pre-coding and detection schemes based on the GS were studied in the massive MIMO systems. The GS has high error performance using the iterative method and requires \(O\left(K^{2}\right)\) multiplications since an algorithm requires only matrix-vector multiplications. For efficient signal transmission, the authors in [14] applied the GS pre-coding in the downlink system for the first time and proved that the GS has a \(\sqrt{2}\) times higher convergence rate than the NS. For reducing the complexity and error propagation of the GS, the authors in [15] proposed a reduced signal dimension using an interference cancellation and effective decoding scheme using the grey region in the uplink system. For accelerating the convergence rate of the GS, the authors in [16; 17; 18; 19] proposed the band matrix and the stair matrix as an initial matrix, respectively, in the uplink system. For better error performance of the GS, the authors in [20; 21; 22; 23] applied the soft-output detector in the uplink system.