We introduce a rate-adaptive system in which the receiver demodulates only those bits that have a high probability of being correct, treating nondemodulated bits as erasures. Several sets of decision regions, derived using composite hypothesis testing, are proposed for 16-QAM and 16-phase-shift keying, which allow for the simple implementation of this demodulation strategy. We demonstrate that pre-encoding the data with a Raptor code allows for simple reconstruction of the message, regardless of the erasure pattern introduced from the nondemodulated bits. We prove the optimality of the proposed decision regions in selecting the most likely subset of bits from any received symbol in moderate-to-high signal-to-noise ratios, and we analyze the performance of demodulating with these decision regions over an additive white Gaussian noise channel. Also demonstrated is the strong performance of 16-QAM for this application, compared with other power-efficient constellations and the near-optimality of using Gray mapping, even under the proposed alternate sets of decision regions