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The question whether RM codes are capacity-achieving is a long-standing openproblem in coding theory that was recently answered in the affirmative fortransmission over erasure channels [1], [2]. Remarkably, the proof does notrely on specific properties of RM codes, apart from their symmetry. Indeed, themain technical result consists in showing that any sequence of linear codes,with doubly-transitive permutation groups, achieves capacity on the memorylesserasure channel under bit-MAP decoding. Thus, a natural question is whathappens under block-MAP decoding. In [1], [2], by exploiting further symmetriesof the code, the bit-MAP threshold was shown to be sharp enough so that theblock erasure probability also converges to 0. However, this technique reliesheavily on the fact that the transmission is over an erasure channel.  We present an alternative approach to strengthen results regarding thebit-MAP threshold to block-MAP thresholds. This approach is based on a carefulanalysis of the weight distribution of RM codes. In particular, the flavor ofthe main result is the following: assume that the bit-MAP error probabilitydecays as $N^{-\delta}$, for some $\delta>0$. Then, the block-MAP errorprobability also converges to 0. This technique applies to transmission overany binary memoryless symmetric channel. Thus, it can be thought of as a firststep in extending the proof that RM codes are capacity-achieving to the generalcase.

Comparing the Bit-MAP and Block-MAP Decoding Thresholds of Reed-Muller Codes on BMS Channels