For the first time, we present photoconductive, continuous wave (cw) terahertz (THz) detectors for 1550 nm excitation based on rhodium- (Rh) doped indium gallium arsenide (InGaAs) grown by molecular beam epitaxy. Compared to iron- (Fe) doped material, the Rh-doped InGaAs shows higher carrier mobilities with similar carrier lifetimes. Therefore, these photoconductive antennas outperform InGaAs:Fe-based detectors by a factor of 10 in terms of responsivity and noise-equivalent-power (NEP) while maintaining the same bandwidth. In a homodyne spectrometer configuration, we achieve a record peak dynamic range (DNR) of 132 dB, which constitutes an improvement of 20 dB.