## 1 Introduction  
With the rapid development of underwater communication technology and the underwater Internet of Things (IoT), there is a notable escalation in the number of autonomous underwater vehicles (AUV) and sensor node devices [1]. This proliferation engenders a consequential surge in data traffic requirements, compelling the exploration of wireless connectivity technologies capable of affording elevated data rates. Underwater wireless optical communication (UWOC) technology is being used by both academia and industry to support high-speed underwater wireless communication systems due to its advantages of high bandwidth, high speed, and low latency [2]. The next generation (beyond 5 G) wireless networks exhibit heightened demands for system capacity, service quality, and the capacity to accommodate a substantial number of interconnected devices. Multi-beam array communication systems are proficient in generating concurrent and independent high-gain directional beams [3], covering predetermined angular ranges. This capability overcomes the limitation of individual antenna beams with narrow coverage, facilitating a higher degree of concurrent communication and enabling the simultaneous servicing of multiple users. This is very important for improving the access capacity of the system and meeting the needs of multi-user communication. However, despite the advantages brought by an increased number of transmitted beams, each antenna can serve only one user at a time, and the number of access users will be largely limited by the number of transmitting antennas [4], which is not able to satisfy the demand for large-scale equipment access. To address this issue, power domain non-orthogonal multiple access (PD-NOMA) technology can be employed to enhance the number of access users [5]. As a key technology for the next generation of wireless communication, PD-NOMA exhibits advantages over traditional orthogonal multiple access (OMA) in terms of connection number and sum capacity [6]. NOMA superimposes the multi-user signals in the power domain using the same resources and implements successive interference cancellation (SIC) to detect multi-user signals [7]. In this way, the system overcomes the limitation that a single antenna can only access one user and increases the number of accesses. Therefore, multi-user communication based on multi-beam NOMA has gained significant attention. Recently, a hybrid SDMA/NOMA scheme has been employed in UWOC systems to accommodate multiple users [8]. In this scheme, NOMA serves to cater to users covered by each beam. However, multi-beam systems and multiple access users cause inter-beam interference and intra-beam interference problems, which seriously impair system performance. In [9], a thorough study was conducted on the unique patterns of intra-beam and inter-beam interference within multi-beam NOMA systems, comprehensively analyzing their distinct impacts on system performance. In [10], a linear precoding scheme with minimum mean square error (MMSE) was used to reduce inter-beam interference, but this scheme has high computational complexity in multi-user scenarios and is difficult to implement in hardware. Intra-beam and inter-beam interference can be minimized by designing a dynamic power allocation scheme [11]. However, the optimization problem for NOMA user power allocation is non-convex and difficult to solve. It is noteworthy that the aforementioned efforts predominantly focus on enhancing the system sum rate or max-min rate. However, in practical communication scenarios, the bit error rate (BER) performance of multi-beam NOMA systems also stands as a pivotal metric for assessing communication reliability. Spatial modulation (SM) activates only one LED to transmit information at any time [12; 13], which decreases high inter-beam interference at the expense of spatial multiplexing.