At bit rates between 4 and 16 kbit/s, many state‐of‐the‐art speech coding algorithms fall into the class of linear‐prediction based analysis‐by‐synthesis (LPAS) speech coders. At the lower bit rates the waveform‐matching, on which LPAS coders rely, constrains the speech quality. To overcome this drawback, we present a coder (RCELP) that uses a generalization of the analysis‐by‐synthesis paradigm. This generalization relaxes the waveform‐matching constraints without affecting speech quality. We describe several implementations at bit rates between 4 and 6 kbit/s. MOS tests show that a 6 kbit/s RCELP has a quality similar or better than the 13 kbit/s GSM full‐rate coder, and a 4.4 kbit/s RCELP has a speech quality significantly better than the 4.8 kbit/s FS1016 standard.