## 1 Introduction
### Related Work  
In recent years, wireless cooperative communication has attracted extensive attention from academia and industry insiders. Moreover, wireless cooperative communication has huge industrial and academic potential and has application prospects in improving spectral efficiency [7; 8; 9]. From the cooperative diversity transmission between initial users and relay nodes [10; 11], wireless cooperative communication has gradually developed into a more generalized network cooperative transmission with cooperative transmission between users or relays [12; 13; 14]. Wireless cooperative communication makes full use of the broadcast characteristics of wireless media and the potential relay nodes in wireless networks. By effectively improving the system spectrum utilization in traditional wireless communication systems, MIMO has become one of the milestone technologies in the development of satellite mobile communication [15; 16; 17; 18]. MIMO technology can turn the adverse multipath influence factors in traditional wireless communication systems into favorable enhancement factors for communication performance. MIMO effectively uses random fading and possible multiple propagation paths to multiply the transmission rate of services to improve system capacity. In traditional point-to-point research theory, it has been proved that if MIMO technology is adopted, the system spectral efficiency can increase linearly with the number of antennas [19; 20; 21]. In the field of cooperative communication, technologies such as cooperative diversity [22] and cooperative virtual MIMO [23] have also verified the effectiveness of MIMO in different communication systems. However, MIMO communication theory and key technologies for physical layer network coding (PNC) need to be studied further, and expansion and application based on basic theories and key technologies need to be explored. At present, MIMO-based advanced PNC technology mostly uses the spatial freedom provided by the MIMO Two-way Relay Network (TWRC) to provide higher spectrum utilization and transmission reliability for Two-way Relay transmission. Research involving PNC transmission methods for MIMO multidirectional relay systems is currently on the rise. In recent years, Interference Alignment (IA) [24] has been introduced into PNC-based MIMO Multi-way Relay Networks (MWRC). The Signal Space Alignment (SSA) theory for network coding looks at interference from a new perspective; it is not blind to suppress interference directly, but intelligent to use interference and achieve the effect of interference suppression through the joint signal processing algorithm of the source node and relay node [25]. This provides a new direction for future interference utilization, interference coordination, and interference game research. Based on this theory, ref. [26] proposed a basic MIMO multidirectional relay channel, namely the MIMO-Y channel. Subsequently, ref. [27] expanded on the work of [26] and discussed the feasibility conditions and achievable degrees of freedom of SSA for users in MIMO-Y channels. It is worth noting that feasibility analysis is a basic problem in IA and SSA [28] and is an important prerequisite for further discussion of the reachable degrees of freedom for the system. However, most of the current studies on the basic MMO-Y channel only consider the reachability of the channel degrees of freedom, while there are few theoretical results on the transmission reliability of this channel. As shown in [29], SSA can be implemented in multiple directions by configuring more antennas at the source node, so that transmission precoding can be selected, and transmission reliability of the system can be improved. In fact, in view of the diversity of antenna configurations in MIMO-Y channels, other methods to improve the transmission reliability of MIMO-Y channels are still worthy of in-depth study. There are two kinds of transmission models in MIMO-Y channels: the multichannel unicast transmission model [30] and the multicasting transmission model [31].