A family of codes for low-complexity quasi-synchronous code division multiple access (CDMA) systems is developed in order to eliminate multiuser interference (MUI) completely in the presence of unknown and even rapidly varying multipath. Judiciously designed precomputable symbol-periodic user codes, which we term Lagrange or Vandermonde, and the corresponding linear receivers offer a generalization of orthogonal frequency division multiplexing (OFDM), which are especially valuable when deep-fading, carrier frequency errors, and Doppler effects are present. The flexibility inherent to the designed transceivers is exploited to derive transmission strategies that cope with major impairments of wireless CDMA channels. The symbol-periodic code design is also generalized to include the class of aperiodic spreading and orthogonal multirate codes for variable bit rate users. Performance analysis and simulations results illustrate the advantages of the proposed scheme over competing alternatives.