## 1 Introduction  
With the exploration of marine resources and the increase of human underwater activities, the demand for efficient underwater communication has increased significantly. At present, the most widely applied means of underwater wireless communication is acoustic technology, which still suffers from low data rate, high signal attenuation, high latency and severe multipath effect. As for radio frequency (RF) which is commonly used in our daily life, it is not very suitable for underwater communication due to the high attenuation of RF waves in seawater. Another way is utilizing fiber optic to implement long-range and high-speed underwater communication. However, it is not flexible enough for increasingly diverse underwater applications because of the requirement of a physical cable connection between transmitters and receivers. Therefore, underwater visible light communication (UVLC) has aroused wild attention and is considered an alternative or complement to the underwater communication technologies mentioned above [1; 2; 3]. Visible light communication (VLC) is an emerging wireless communication technology with several advantages such as large transmission capacity, high security, low cost, anti-electromagnetic interference and license-free [4; 5; 6]. Considering the relatively low attenuation window of seawater in the blue-green portion of the electromagnetic spectrum, VLC has great potential to provide a high data rate and low latency within a short range [1; 2]. There are two kinds of commonly used VLC transmitters, which are based on laser diodes (LDs) and light-emitting diodes (LEDs). LDs enable longer-range transmission in point-to-point scenarios due to their high power density but require strict alignment between transmitters and receivers. Compared with LDs, LEDs have a larger divergence angle, which allows for more short-range applications, including point-to-point and point-to-multipoint scenarios [7]. In this paper, we focus on UVLC systems based on LEDs which have become a promising candidate for underwater communication. Despite the many significant advantages of UVLC systems based on LEDs mentioned above, the bottleneck problems it faces cannot be ignored. The limited bandwidth of LEDs, various noises, and linear and nonlinear distortions largely restrict the communication performance of UVLC systems [6]. The modulation bandwidth of commercially available LEDs is usually several megahertz (MHz) [8], which has become one of the main factors limiting the development of high-speed VLC systems. Due to the bandwidth limitation of LEDs and other optoelectronic devices, the received signals suffer from serious linear distortion, thus interfering with the correct demodulation and decoding. When the bandwidth of the transmission signals exceeds the effective bandwidth of VLC systems, the bandwidth limitation behaves as a strong low-pass filter, which will compress the signal spectrum, resulting in serious inter-symbol interference (ISI). Furthermore, the signals' nonlinear effects will severely deteriorate communication performance, which results from the nonlinearity of devices' optoelectronic characteristics, distortions caused by amplifiers and so on. When the transmitting power is very high and the channel is complex, nonlinearity will become the main challenge restricting the systems' performance, especially for underwater applications, which usually require considerable transmitting power. To alleviate these bottleneck problems, advanced modulation formats are proposed, including carrier-less amplitude and phase (CAP) modulation [9], adaptive bit loading Orthogonal Frequency Division Multiplexing (OFDM) [10] and Nyquist single carrier (N-SC) modulation [11]. To further improve communication performance, Geometrically-shaping (GS) technology [12] was proposed to increase Euclidean distance between the constellation points and decrease Peak-to-Average Power Ratio (PAPR) by optimizing the constellation distribution.