This paper shows that bit-level reliability can be converted into block-level reliability for binary codes over BMS channels through Forney's concatenation scheme. The construction concatenates an inner code of rate R-o(1) and vanishing bit-error probability εwith an outer code of rate 1-o(1) equipped with a bounded-distance decoder. We show that, when the outer correction radius exceeds εand a suitable large-deviation condition holds, the concatenated code has rate R-o(1) and vanishing block-error probability. Hence, a family with vanishing bit-error probability for rates below capacity can be converted into a capacity-achieving concatenated-code family. We further show that, when binary BCH codes are used as outer codes, inner codes with bit-error probability ε=o(1/\log\log n) can be converted into a capacity-achieving concatenated family.

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