Constant amplitude transforms like discrete Fourier transform (DFT), Walsh transform, nonlinear phase Walsh-like transforms and Gold codes have been successfully used in many wire-line and wireless communications technologies including code division multiple access (CDMA), discrete multi-tone (DMT), and orthogonal frequency division multiplexing (OFDM) types. In this paper, we present a generalized framework for DFT called Generalized DFT (GDFT) with nonlinear phase by exploiting the phase space. It is shown that GDFT offers sizable performance improvements over Walsh, Gold and DFT codes in multi-carrier communications scenarios considered. We also highlight that known constant modulus code families are special solutions of the proposed GDFT framework. Moreover, we introduce practical design methods offering computationally efficient implementations for GDFT. We expect performance improvements in future communications systems employing GDFT intelligently.