Two network nodes may exchange messages through a relay using physical-layer network coding combined with forward error correction coding (FEC). The relay determines a prime field order based on the channel condition and communicates the field order to the network nodes. Each network node encodes an outgoing message with linear codes over a field such as finite field of the field order, and transmits a signal carrying the encoded outgoing message. The relay receives a composite signal carrying the summation of the messages from the two network nodes. The relay decodes the composite signal and extracts a composite message with linear codes over finite field of field order, and broadcasts a signal carrying the composite message. Each network node receives the signal from the relay and extracts the message intended for it using linear subtraction over finite field of field order.