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We strengthen the results from a recent work by the second author, achieving bounds on the weight distribution of binary linear codes that are successful under block-MAP (as well as bit-MAP) decoding on the BEC. We conclude that a linear code that is successful on the BEC can also decode over a range of binary memoryless symmetric (BMS) channels. In particular, applying the result of Kudekar, Kumar, Mondelli, Pfister, Şaşoğlu and Urbanke from STOC 2016, we prove that a Reed–Muller code of positive rate R decodes errors on the p with high probability if p < 1/2 − √2−R(1−2−R).

On codes decoding a constant fraction of errors on the BSC