## 1 Introduction  
Fifth-generation systems (5G) are in the phase of intensive development although some crucial decisions on their structure, architecture, and form of physical, medium access control, and network layers have already been made and have received the form of standards [1, 2]. However, several topics remain open. Among them, non-orthogonal multiple access [3] aiming at increasing the efficiency of radio resource utilization is one of the more interesting ones. Another idea which can be potentially useful in increasing network throughput is network coding (NC). The idea of network coding was introduced by [PERSON], [PERSON], [PERSON], [PERSON] in their seminal paper [4], released in 2000. Since that time, hundreds of papers have been published in this area. Papers [5, 6, 7] are representative examples of fundamental extensions of [4]. In network coding, network nodes not only route information blocks but can also perform some arithmetic operations on the incoming blocks in order to route the outgoing blocks in a modified form. Such operations have been proven to enable the improvement of network throughput. Wireless networks are particularly well-suited to network coding, as they have a natural capability to send signals to several nodes or terminals concurrently. Numerous researchers have studied the performance of network coding in wireless networks, taking into account the properties of omnidirectional transmission and several other transmission arrangements. The representative papers related to these topics are [8, 9, 10]. Network coding has been presented in several monographs, tutorials, and academic books, e.g., [11, 12, 13]. A new idea, particularly suited for wireless network coding, is the application of signal operations in the physical layer of a communication system instead of operations on bit streams, performed in regular network coding. This way, inherent features of propagation in wireless media can be efficiently utilized. The first paper devoted to physical layer network coding (PNC) was presented by [PERSON], [PERSON], and [PERSON] in 2006 [14]. They considered an idealized communication system in which two terminals transmit data to the third one, the latter playing the role of an access point or a relay. The phase of concurrent transmission to the relay is called the multiple access phase. The authors made some simplifying assumptions, such as that the power and phase of the signals received by the relay from both terminals are equal, which is hard to achieve [15]. The relay interprets the superposition of the received signals as a joint signal constellation and, using appropriate mapping, generates a new signal that results from a modulo-2 sum of the bit blocks sent from both terminals by the component signals. This signal, broadcast by the relay, is received by both terminals (broadcast phase), which allows them to decode the other terminal signal, based on the received one and the knowledge of their own. In 2018, the book [16], specifically devoted to physical layer network coding, was published, establishing a solid source of knowledge on this topic. In wireless links, the application of channel coding is necessary. If regular or physical layer network coding is applied, channel coding has to be used in terminals and often in relays. Two basic approaches can be applied in a relay: the generation of the relay packet in standard or physical layer network coding is performed either on the bit-by-bit or symbol-by-symbol basis, or after channel code decoding (Decode-and-Forward (DF) rule). Each approach has advantages and drawbacks. In the first approach, network coding does not introduce a substantial delay and the relay is more energy efficient. On the other hand, performing network coding operations before channel decoding is more error-prone and can result in lower link performance. The second approach results in a higher performance, but it requires more computations and introduces longer delay. We claim that the contributions of this paper are the following:  
1.