Multicode CDMA systems convert a high-rate serial data stream into low-rate parallel data streams prior to transmission, but reducing the peak-to-average-power-ratio (PAPR) is a prerequisite. In this paper, we propose constant amplitude coding schemes with forward error correction (EC) capability. The proposed schemes overcome the adverse nonlinear effects of the high power amplifier (HPA) by using the transmitted signal of constant amplitude and parity channel. In the first scheme, we add the EC capability to the previously reported constant-amplitude rate 4/4 (Suil's) scheme, which can transmit data without energy loss. Next, we propose a rate 12/16 decoder with EC capability, which is slightly different from the previous work through the addition of EC capability. Lastly, we propose a new high-rate EC capable 16/16 scheme without energy loss, which makes it superior to the conventional 12/16 scheme which experiences excessive energy loss due to redundancy. Computer simulation results confirm that new 4/4 decoder along with 12/16 decoder and 16/16 encoder/decoder can effectively reduce the inherent problem of high PAPR in the multicode CDMA signal transmission. Our methods also yield better BER performance than other constant amplitude coding schemes.