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We show that Reed-Muller codes achieve capacity under maximum a posterioribit decoding for transmission over the binary erasure channel for all rates $0< R < 1$. The proof is generic and applies to other codes with sufficientamount of symmetry as well. The main idea is to combine the followingobservations: (i) monotone functions experience a sharp threshold behavior,(ii) the extrinsic information transfer (EXIT) functions are monotone, (iii)Reed--Muller codes are 2-transitive and thus the EXIT functions associated withtheir codeword bits are all equal, and (iv) therefore the Area Theorem for theaverage EXIT functions implies that RM codes' threshold is at channel capacity.

Reed-Muller Codes Achieve Capacity on the Binary Erasure Channel under MAP Decoding