This thesis investigates channel-coded physical layer network coding, in which the relay directly transforms the noisy superimposed channel-coded packets received from the two end nodes, to the network-coded combination of the source packets. This is in contrast to the traditional multiple-access problem, in which the goal is to obtain each message explicitly at the relay. Here, the end nodes $A$ and $B$ choose their symbols, $S_A$ and $S_B$, from a small non-binary field, $\mathbb{F}$, and use non-binary PSK constellation mapper during the transmission phase. The relay then directly decodes the network-coded combination ${aS_A+bS_B}$ over $\mathbb{F}$ from the noisy superimposed channel-coded packets received from two end nodes. Trying to obtain $S_A$ and $S_B$ explicitly at the relay is overly ambitious when the relay only needs $aS_B+bS_B$. For the binary case, the only possible network-coded combination, ${S_A+S_B}$ over the binary field, does not offer the best performance in several channel conditions. The advantage of working over non-binary fields is that it offers the opportunity to decode according to multiple decoding coefficients $(a,b)$. As only one of the network-coded combinations needs to be successfully decoded, a key advantage is then a reduction in error probability by attempting to decode against all choices of decoding coefficients. In this thesis, we compare different constellation mappers and prove that not all of them have distinct performance in terms of frame error rate. Moreover, we derive a lower bound on the frame error rate performance of decoding the network-coded combinations at the relay. Simulation results show that if we adopt concatenated Reed-Solomon and convolutional coding or low density parity check codes at the two end nodes, our non-binary constellations can outperform the binary case significantly in the sense of minimizing the frame error rate and, in particular, the ternary constellation has the best frame error rate performance among all considered cases.

This thesis investigates channel-coded physical layer network coding, in which the relay directly transforms the noisy superimposed channel-coded packets received from the two end nodes, to the network-coded combination of the source packets. This is in contrast to the traditional multiple-access problem, in which the goal is to obtain each message explicitly at the relay. Here, the end nodes $A$ and $B$ choose their symbols, $S_A$ and $S_B$, from a small non-binary field, $\mathbb{F}$, and use non-binary PSK constellation mapper during the transmission phase. The relay then directly decodes the network-coded combination ${aS_A+bS_B}$ over $\mathbb{F}$ from the noisy superimposed channel-coded packets received from two end nodes. Trying to obtain $S_A$ and $S_B$ explicitly at the relay is overly ambitious when the relay only needs $aS_B+bS_B$. For the binary case, the only possible network-coded combination, ${S_A+S_B}$ over the binary field, does not offer the best performance in several channel conditions. The advantage of working over non-binary fields is that it offers the opportunity to decode according to multiple decoding coefficients $(a,b)$. As only one of the network-coded combinations needs to be successfully decoded, a key advantage is then a reduction in error probability by attempting to decode against all choices of decoding coefficients. In this thesis, we compare different constellation mappers and prove that not all of them have distinct performance in terms of frame error rate. Moreover, we derive a lower bound on the frame error rate performance of decoding the network-coded combinations at the relay. Simulation results show that if we adopt concatenated Reed-Solomon and convolutional coding or low density parity check codes at the two end nodes, our non-binary constellations can outperform the binary case significantly in the sense of minimizing the frame error rate and, in particular, the ternary constellation has the best frame error rate performance among all considered cases.