## 1 Introduction  
Visible light communication (VLC) is a promising candidate for future wireless communication [1; 2]. By utilizing the LED as the transmitting end and the PIN as the receiving end, VLC can be low-cost and can realize no electromagnetic interference communication without spectrum authorization [3]. In recent years, the demand for VLC traffic is continuously increasing. Advanced modulation formats, such as carrier-less amplitude phase (CAP) and orthogonal frequency division multiplexing (OFDM), combined with quadrature amplitude modulation (QAM), are widely used in VLC to increase the capacity of VLC systems [4; 5; 6]. However, traditional QAM has two apparent disadvantages: First, traditional QAM has a coarse granularity in spectral efficiency and cannot make full use of the channel capacity [7]. Second, traditional QAM is not the optimal constellation for the channel, which has an apparent gap to the Shannon capacity [8]. Therefore, it is necessary to optimize the transmitted signal to make it more suitable for the distribution of the channel. In recent years, probabilistic shaping (PS) has become a hotspot in high data rate transmission [9; 10]. By changing the probability of each constellation point to make a Gaussian-like constellation, PS could provide additional shaping gain to eliminate the gap between the Shannon capacity and system source entropy (SE). It is proven that in the additive white noise Gaussian (AWGN) channel, the Maxwell-Boltzmann distribution is the optimal probability distribution for the standard QAM constellation, which could provide shaping gain of at most 1.53 dB [11]. Furthermore, PS can realize arbitrary SE and rate adaption transmission according to channel state information [12]. Among the ways to realize a PS system, probabilistic amplitude shaping (PAS) is one of the more popular methods [13; 14]. At the transmitting side, the PAS system utilizes constant composition distribution matching (CCDM) and low-density parity-check (LDPC) codes to implement binary to non-uniform symbol mapping [15; 16]. At the receiver side, inverse CCDM and bit-metric decoding is implemented to recover the non-uniform symbol to binary streams and improve the system performance [17]. As end-to-end binary decoding will introduce an additional computation complexity and latency to the system, generalized mutual information (GMI) and normalized GMI (NGMI) are introduced to measure the system performance more conveniently [18]. In [19], the author has proven that the NGMI is a powerful tool in predicting the bit error rate (BER) of the post forward error correction (FEC) after soft decision bit-metric decoding. Therefore, it is convincible to utilize the GMI and NGMI as criteria. Additionally, the achievable information rate (AIR, also named as net transmission rate) could give out the actual system information rate after removing coding redundancy. It is proven to be a fairer way for different PS constellations and source entropies [20; 21]. The advantages of PS have attracted many scholars to investigate the superior performance in different scenarios. It has been proven that PS is superior to the uniform constellation and geometrical shaping. In [22], PS constellations with static entropies could provide more than 1 dB signal-to-noise ratio (SNR) sensitivity gain for 256 QAM in the AWGN fiber system. In [12; 21] and [23], the PS constellation could realize 15.5-40% more transmission reach in optical systems. However, these papers operate at a fixed QAM order and pay little attention to the QAM order and its corresponding SE, according to different channel state information, which is a fundamental issue to realizing higher system performance. In [24], the author concludes that a 4-probability mass function (PMF) is sufficient for 64 QAM when a mismatch of 0.1 dB SNR is acceptable. However, the shaping factor of PMF is optimized by maximizing the system's mutual information (MI), which cannot reflect the system AIR without coding redundancy.