The problem of data communications through impulsive power-line channels using Reed-Solomon (RS) codes combined with M-ary modulation is treated. It is shown that under the influence of impulsive interference, the seemingly robust algebraic RS decoder can suffer drastic performance degradation even with the aid of an interleaver. We attempt to elucidate the mechanism that leads to the breakdown of such coding schemes. It is found that in impulse channels, excessive random and burst errors can occur prior to RS decoding mainly due to the deficiency of the Gaussian-based demodulator. An impulse mitigation strategy derived from the myriad filtering framework is incorporated into the demodulator for efficient baseband filtering and RS decoding. Even without an interleaver, the proposed solution leads to substantial performance improvements over the conventional interleaved scheme in impulsive channels. Consequently, the delay caused by interleaving can be avoided, which has significant benefits for future broadband power-line communication systems supporting interactive applications.