## 1 Introduction
In 2021, [PERSON] et al. combined OGSM and serial relay system, and added convolutional coding to form an optical spatial modulation scheme suitable for FSO communication. Research shows that OGSM FSO system with serial relay has better performance [16]. However, they did not investigate more advanced upgrade schemes for optical spatial modulation to achieve lower BER, and the OGSM scheme did not consider the limitation that the system is subjected to in terms of transmission rate. The research gap in the literature survey expresses that modern FSO communication systems still require higher spectral efficiency. At present, the research community has not conducted performance analysis for any superior optical generalized spatial modulation for FSO communication, which motivates us to propose more innovative optical generalized spatial modulation schemes for FSO communication systems. Enhanced fully optical generalized spatial modulation (EFOGSM) scheme transmits information not only by the index of the laser, but also by using the type bearing of the signal constellation. That is, a constellation type is added so that the primary constellation and the secondary constellation are used to increase the number of combinations between the modulation and the laser. At the same time, EFOGSM makes full use of all possible laser combinations when the total number of lasers is unchanged, thus breaking the limit of the logarithmic scale relationship between the transmission rate of OGSM system and the number of lasers. It is known that EFOGSM technology has not yet appeared and has not been reported in FSO communication system. In order to further mitigate fading effects, provide diversity gain and enhance the advantages of optical spatial modulation in FSO systems, the integration of space time block coding (STBC) and OSM has been considered as a particularly promising modulation scheme in recent years. At present, only [PERSON]'s team have studied the STBC-OSM scheme in FSO communication system in 2017. They derived the bit error rate (BER) expressions of the system under lognormal and double generalized gamma fading channels, and the simulation results show that the system has a significant improvement in BER performance compared with OSM [17]. The results show that STBC-OSM can also achieve high diversity gain by transmitting the same data information in different time slots, while maintaining high energy efficiency and low complexity. However, the STBC-OSM scheme can only use two lasers in the transmission cycle and transmit a pair of symbols of the STBC matrix at the same time, which greatly wastes the space resources. The system needs to use an arbitrary number of lasers to achieve higher spectral efficiency, which makes the STBC-OSM system need further optimization, that is, the STBC-EFOGSM scheme appears. However, the selection process of laser combinations becomes complicated when multiple lasers are used in STBC-EFOGSM. Since the information bits are transmitted through an activated laser combination based on different modulation symbols, this complicates the mapping process from bits to modulation symbols. More importantly, when the number of lasers increases, the design of STBC space code-word becomes more complex, and the diversity gain will also decrease. Therefore, this paper proposes a generalized space time block coded enhanced fully optical generalized spatial modulation (GSTBC-EFOGSM) system to overcome this complexity. At present, GSTBC-EFOGSM system has not been reported in FSO communication system. Based on the above facts, this paper proposes the EFOGSM scheme and the GSTBC-EFOGSM system. Under M distribution model, considering the combined effects of path loss, pointing error and atmospheric turbulence on the GSTBC-EFOGSM system, the Meijer's G form closed expression of the average bit error rate (ABER) of the system is derived.