In this paper we propose a novel power adaptive network coding(PANC)for a non-orthogonal multiple-access relay channel(MARC),where two sources transmit their information simultaneously to the destination with the help of a relay.In contrast to the conventional XOR-based network coding(CXNC),the relay in PANC generates network coded symbols by considering the coefficients of the source-to-relay channels,and forwards each symbol with a pre-optimized power level.Specifically,by defining a symbol pair as two symbols from the two sources,we first derive the expression of symbol pair error rate(SPER)for the system.We also propose a coordinate transform(CT)method on the received constellation to simplify the derivations of the SPER.Next,we obtain the optimal power level by decomposing it as a multiplication of a power scaling factor and a power adaptation factor.We prove that with the power scaling factor at the relay,our PANC scheme can achieve a full diversity gain,while the CXNC can achieve only an order of one diversity gain.In addition,we optimize the power adaptation factor at the relay to minimize the SPER at the destination by considering of the relationship between SPER and minimum Euclidean distance of the received constellation,resulting in an improved coding gain.