## 1 Introduction  
In the 5G era, with the rapid development of communication networks and a surge in the type or scale of communication services, spectrum resources are becoming scarcer. Meanwhile, the users' quality of service (QoS) requirement is also increasing, which requires more perfect communication protocols to meet the actual needs of users. However, uncertainty is the essential characteristic of communication systems in the process of information transmission because the number of terminal users that try to access network resources using the same channel in a slot is random. If more than one user shares a physical link in a slot, the information between different users will conflict. In order to ensure reliable and efficient communication in the case of multiple users sharing a single channel, it is most important to reduce the probability of information collision. Therefore, how to enable each of the active users to send a packet successfully is a key problem to be solved in the development of a multiple access protocol. Among traditional random-access protocols, once a conflict occurs, all packets involved in the conflict are unrecoverable and discarded directly. For example, the slotted-ALOHA protocol, as a traditional random-access approach, is used extensively [1, 2, 3]. Regarding the slotted-ALOHA protocol, the packet transmission is slot-based. The width of a slot is equal to the transmission time of a packet. Each node sends a packet synchronously at the beginning of a slot. The goal of the slotted-ALOHA protocol is to enable each of the active users to send its packet without conflict. Forward error correction (FEC) technology is adopted to recover conflicting packets; that is, after the information sequence is sent, the sender adds redundant information codes for error detection and correction according to certain rules to form a complete codeword and then sends this to the channel [4, 5]. [PERSON] et al. [6] proposed a novel random-access protocol. The principle is that a conflict is regarded as the sum of several packets. Conflicting packets are buffered by the receiver rather than discarded directly. Then, these packets will be reused in decoding, which is based on successive interference cancellation (SIC) [7]. Recall that in the process of recovering conflicting packets by SIC, the biggest problem is that the receiver has no knowledge about the set of transmitting users [8]. However, the research of [9] solves this problem. In this work, the sender adds a user specific signature code to the header of each user's packet [10]. If the packets conflict, the receiver can identify all of the transmitting users with signatures. The authors of [11] combined signature codes and SIC to solve the conflict where there are no more than two users. However, the authors of [11] only studied the performance of the slotted-ALOHA system under the general condition of taking two as the maximum number of detectable conflicting users. Based on the M (total number of users) out of K (number of active users) multi-user coding in [12], a hybrid random-access solution is proposed in [13]. It is concluded that the throughput of the hybrid random-access system reaches the maximum when the number of active users is three. Hence, in this work, we consider extending the general case of [11]; that is, taking three as the maximum number of detectable conflicting users. Recently, several researches employ the slotted-ALOHA protocol with non-orthogonal multiple access (NOMA), which is a promising multiple access approach applied in a 5G network to improve the throughput performance [14; 15; 16; 17]. For instance, NOMA based on the coded slotted-ALOHA for machine-type communications (MTC) is proposed in [14]. Based on the slotted-ALOHA and uplink NOMA, two random-access protocols are proposed and applied to wireless sensor networks [15].