Linear Physical-Layer Network Coding and Information Combining for the $K$ -User Fading Multiple-Access Relay Network

We propose a new linear physical-layer network coding (LPNC) and information combining scheme for the $K$ -user fading multiple-access relay network (MARN), which consists of $K$ users, one relay, and one destination. The relay and the destination are connected by a rate-constraint wired or wireless backhaul. In the proposed scheme, the $K$ users transmit signals simultaneously. The relay and the destination receive the superimposed signals distorted by fading and noise. The relay reconstructs $L$ linear combinations of the $K$ users’ messages, referred to as $L$ network-coded (NC) messages, and forwards them to the destination. The destination then attempts to recover all $K$ users’ messages by combining its received signals and the NC messages obtained from the relay. We develop an explicit expression on the selection of the coefficients of the NC messages at the relay that minimizes the end-to-end error probability at a high signal-to-noise ratio. We develop a channel-coded LPNC scheme by using an irregular repeat-accumulate modulation code over GF( $q$ ). An iterative belief-propagation algorithm is employed to compute the NC messages at the relay, while a new algorithm is proposed for the information combining decoding at the destination. We demonstrate that our proposed scheme outperforms benchmark schemes significantly in both un-channel-coded and channel-coded MARNs.