Non-orthogonal multiple access (NOMA) has become one of the desirable schemes for the 5G cellular network standard due to its better cell coverage capability, higher data rate, and massive connectivity. Orthogonal frequency division multiplexing (OFDM) can be combined with NOMA to get the higher spectral efficiency. The main drawback of OFDM-based NOMA (OFDM-NOMA) scheme is high peak-to-average power ratio (PAPR). Therefore, this paper presents a new discrete-cosine transform matrix precoding-based uplink OFDM-NOMA scheme for PAPR reduction. Additionally, the proposed precoding-based uplink multicarrier NOMA scheme takes advantage of information spreading over the entire signal spectrum; thus, bit-error rate is also reduced. Simulation results show that the proposed precodingbased NOMA scheme outperforms as compared with the non-precoding-based NOMA schemes available in the literature.