  The question whether RM codes are capacity-achieving is a long-standing open
problem in coding theory that was recently answered in the affirmative for
transmission over erasure channels [1], [2]. Remarkably, the proof does not
rely on specific properties of RM codes, apart from their symmetry. Indeed, the
main technical result consists in showing that any sequence of linear codes,
with doubly-transitive permutation groups, achieves capacity on the memoryless
erasure channel under bit-MAP decoding. Thus, a natural question is what
happens under block-MAP decoding. In [1], [2], by exploiting further symmetries
of the code, the bit-MAP threshold was shown to be sharp enough so that the
block erasure probability also converges to 0. However, this technique relies
heavily on the fact that the transmission is over an erasure channel.
  We present an alternative approach to strengthen results regarding the
bit-MAP threshold to block-MAP thresholds. This approach is based on a careful
analysis of the weight distribution of RM codes. In particular, the flavor of
the main result is the following: assume that the bit-MAP error probability
decays as $N^{-\delta}$, for some $\delta>0$. Then, the block-MAP error
probability also converges to 0. This technique applies to transmission over
any binary memoryless symmetric channel. Thus, it can be thought of as a first
step in extending the proof that RM codes are capacity-achieving to the general
case.
