## 1 Introduction
### Related Works  
Most research on multi-user VLC systems currently concentrates on mitigating MUI [7]. Various precoding strategies have been devised to achieve specific goals, such as minimizing the maximum mean square error (MSE) [7], maximizing the minimum signal-to-interference-plus-noise ratio (SINR) [8], and optimizing the maximum achievable sum rate of the systems [9]. Currently, the design of precoding schemes in VLC systems predominantly focuses on the linear design part due to its ease of operation [10; 11; 12; 13; 14; 15; 16; 17]. [PERSON] et al. [10] introduced a precoding design based on maximizing the rates of multi-user multiple-input single-output (MISO) VLC systems. Subsequently, they expanded this non-alternating method to optimize robust minimum mean square error (MMSE) precoding [11]. According to the characteristics of multi-user scenarios, [PERSON] et al. [12] proposed the selectivity of the receiving angle for each user. Then, they combined this with a linear zero-forcing (ZF) precoding scheme to further enhance the system's performance. By capitalizing on LED characteristics and block diagonalization precoding (BDP) technology, [PERSON] et al. [13] designed a weighted-adjustment block diagonalization precoding (WA-BDP) scheme for multi-user MIMO VLC systems. They further introduced spatial dimming (SD) built on LED selection and proposed an SD-WA-BDP scheme, maximizing LED utilization for simultaneous illumination and communication purposes. [PERSON] et al. [14] investigated the development of linear coordinated precoding strategies to enable coordinated multi-point (CoMP) communication in VLC networks. [PERSON] et al. [15] studied the robust joint design of transmission precoders and reception filters for multi-user MIMO VLC systems, aiming to maximize the minimum SINR. Later, [PERSON] et al. [16] optimized these robust transmission precoders to maximize the sum capacity under worst-case scenarios. Based on the properties of Farey sequences, [PERSON] et al. [17] proposed an optimal precoding scheme designed for multi-user MISO VLC systems, accounting for dual constraints related to illumination and communication. There has also been extensive research on nonlinear precoding designs [18; 19; 20]. Compared to linear precoding schemes, nonlinear precoding incurs higher energy consumption and complexity but achieves higher signal-to-noise ratios (SNR). [PERSON] et al. [18] proposed ZF dirty-paper coding to attain the maximum channel capacity in MIMO systems, whereas [PERSON] et al. [19] introduced Tomlinson-Harashima precoding (THP) to reduce complexity. [PERSON] et al. [20] developed a successive interference cancellation (SIC)-based precodingscheme with sub-connected architecture (SIC-SA) to address severe MUI issues in MIMO VLC systems. In addition to the research on precoding design in IM/DD MISO or MIMO VLC systems, there has been extensive research on multi-carrier VLC systems to achieve improved spectral efficiency [21; 22]. [PERSON] et al. [21] proposed a spatial modulation scheme based on ZF and MMSE precoding for indoor MU-MIMO orthogonal frequency-division multiplexing (OFDM) VLC systems using direct current bias. After incorporating their designed transmit antenna selection algorithm, the results demonstrated significant improvements over traditional modulation schemes. [PERSON] et al. [22] focused on each subcarrier of OFDM and used a complex channel matrix in the frequency domain to calculate the corresponding precoding matrix, aiming to eliminate inter-user interference. The results demonstrated the superiority of frequency domain-based precoding algorithms in scenarios with low light power and close user distances. However, current analyses of MIMO VLC systems mainly rely on AWGN channel models, which are not appropriate for photon-counting VLC systems. This is because the AWGN channel models fail to capture the correlation between the signal and noise.