  This paper considers the performance of Reed-Muller (RM) codes transmitted
over binary memoryless symmetric (BMS) channels under bitwise
maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed
BMS channel, the family of binary RM codes can achieve a vanishing bit-error
probability at rates approaching the channel capacity. This partially resolves
a long-standing open problem that connects information theory and
error-correcting codes. In contrast with the earlier result for the binary
erasure channel, the new proof does not rely on hypercontractivity. Instead, it
combines a nesting property of RM codes with new information inequalities
relating the generalized extrinsic information transfer function and the
extrinsic minimum mean-squared error.
