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This paper proposes a novel modulation format that results in a constant envelope when orthogonal multicode spread-spectrum signals are linearly combined. This feature is desirable for the use of a low-cost nonlinear amplifier in a high-data-rate multiple access system, especially on the uplink of cellular systems. To allow coherent demodulation, per-symbol detection is introduced to the constant envelope modulation (CEM), which increases receiver complexity modestly compared to conventional bit-by-bit detection. Analysis of the multiuser performance reveals superiority to an ideal system without nonlinear distortion, and significant improvement when nonlinearity is taken into account.