## 1 Introduction  
Cooperative communication is an important paradigm for improving the operation and performance of wireless telecommunication networks [1, 2, 3, 4, 5]. Relaying techniques are principal enablers for cooperative communication, which are commonly used in wireless systems [6, 7, 8, 9]. In such systems, user terminal (source) nodes transmit information carrying signals to the relays and the destination nodes. The relay nodes then forward information related signals to the destination following specific protocols. Finally, destination nodes decode the information from user terminal based on the signals received from these terminals and relay nodes. With the aid of relays that provide additional paths for information transmission, the error performance and power efficiency of such systems is improved compared to non-cooperative schemes [10]. Common protocols for relaying systems are broadly classified into amplify-and-forward (AF) and decode-and-forward (DF) methods [11]. With the AF protocol, the relays amplify the received signals from the terminalsand forward them to the destination. In the DF protocol, the relays first decode the signals from the terminals and extract the information, which is then re-encoded and forwarded to the destination. Detect-and-forward (DetF) relaying, considered in Zhang and Leib [7] as well as in Hou and Leib [12], is closely related to DF, except that each relay only detects (demodulates) the signal, without decoding. These protocols are further classified into selective AF and selective DF techniques [13], where the relays operate selectively on the signals received from the terminals according to specific criteria. Using selective techniques, the performance is improved over conventional all-participation relaying [14]. Network coding, first proposed by [PERSON] [15], requires computation within the network. In a network-coded system, intermediate nodes form and transmit functions of the information they receive; a technique that can improve throughput. Network coding has found many applications in modern communication networks [16] as well as in cooperative communication [17, 18]. Furthermore, network coding serves as a basis for many relaying protocols [19, 20]. Three-node two-way relay networks (TWRNs) are a simple model of a cooperative communication system. In TWRNs, two terminals exchange information with the aid of one relay. In traditional TWRNs, a full information exchange cycle requires four phases, where each terminal consumes one phase to transmit to the relay and other terminal, then the relay consumes two phases to transmit to each of the two terminals. With network coding, the transmission is reduced to three phases, where the relay requires only one phase to transmit the properly combined data [7, 21] to both terminals. This model can be further generalized to multi-way relay networks (MWRNs) [10, 22], where multiple terminals exchange information with the aid of multiple relays. A linear block-coding framework for network-coded MWRNs with differential modulation that is based on systematic codes has been proposed in [12], which addresses the use of low-density parity check (LDPC) codes in systematic form as network codes. Decoding has been performed using the Belief Propagation (BP) algorithm. A simpler alternative is offered by systematic equivalent polar codes used as network codes for MWRNs, which is the main subject of the present paper. Polar codes were first proposed by [PERSON] [23] and have drawn increasing research interest [24]. Polar codes are linear block codes that can achieve capacity over binary-input discrete memoryless channels (B-DMC). Compared with other advanced codes such as Turbo codes [25] and LDPC codes [26], polar codes have the advantage of lower encoding and decoding complexities with successive cancellation (SC) decoding. Furthermore, polar codes are more flexible in rate-adaptation, meaning that changing the number of information bits given a fixed codeword length will not affect the encoder structure.