## 1 Introduction  
The development of visible-light communication technology (VLC) [1; 2; 3; 4] has always focused on improving the transmission rate and quality under harsh channel environments and limited bandwidth. It has been verified that the most effective way of improving the data transmission rate and quality on the visible-light channel is by adopting the multiple-input multiple-output (MIMO) [5; 6; 7; 8] technology. In other words, multiple antennas or antenna arrays are placed at the sending and receiving ends of the visible-light communication system to transmit information. MIMO technology is a key technology widely used in the new generation of visible-light communication systems to double the capacity and spectral efficiency of communication systems without increasing their bandwidth. The MIMO technology has been adopted in indoor optical communication systems. [PERSON] et al. proposed a novel generalized light-emitting diode (LED) refractive index modulation method for VLC systems based on multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM). By adopting spatial multiplexing and LED index modulation, this method circumvents the typical spectral efficiency loss triggered by time-domain and frequency-domain shaping in OFDM signals; however, the complexity is extremely high [9]. Therefore, [PERSON] et al. proposed OSM technology based on MIMO technology, which reduced the complexity but did not significantly improve the transmission rate [10]. In addition to utilizing conventional modulation symbols to transmit information, optical generalized spatial modulation (OSM) [11; 12; 13; 14] also hides part of the information in the index of the transmitting antenna, thereby improving the MIMO technology. [PERSON] et al. proposed three low-complexity detection schemes for SM MIMO systems. The MRC-based scheme can achieve near-optimal performance while reducing the complexity of the ML-based SM receiver; however, it does not significantly improve the transmission rate [15]. Accordingly, researchers have proposed an optical generalized spatial modulation (OGSM) algorithm [16; 17; 18; 19; 20]. However, the definition of GSM in the current literature is not accurate. [PERSON] re-define GSM as follows: for the one where the activated transmitters transmit the same signal, it is defined as \"GSM\"; for the one where the activated transmitters transmit different signals, it is defined as \"GSSMP\" [21]. [PERSON] et al. optimized the amplitude phase modulation (APM) symbol constellation in a VLC system based on the GSM, and adopted the statistical convergence gradient descent (SCGD) algorithm. Although the numerical simulation verified the superior performance of the optimal constellation, the algorithm was too complex [22]. [PERSON] et al. proposed a generalized precoder design formula and iterative algorithm for the spatial modulation of MIMO systems with CSIT, to address the high complexity of the maximum-likelihood detection algorithm in large-scale GSM scenarios. Although this algorithm significantly reduces the complexity, it does not optimize the transmission performance [23]. [PERSON] et al. proposed a generalized spatial modulation (GSM) multiple-input, multiple-output coding scheme based on an active space and cooperative constellation. Compared with the conventional GSM algorithm, this algorithm improves the power efficiency of indoor visible-light communication; however, its complexity is high [24]. [PERSON] et al. proposed a novel convolutional coding-optical generalized spatial modulation-space diversity (CC-OGSM-SD) serial relay system under the M distribution [25]. Based on the OGSM technology, it was proposed to improve the bit-error performance using an antenna selection algorithm. [PERSON] proposes an antenna selection algorithm based on norm selection, which requires known channel parameters, but the actual channel is time-varying [26]; [PERSON] et al.