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This paper introduces a new approach to proving that a sequence ofdeterministic linear codes achieves capacity on an erasure channel undermaximum a posteriori decoding. Rather than relying on the precise structure ofthe codes, this method requires only that the codes are highly symmetric. Inparticular, the technique applies to any sequence of linear codes where theblocklengths are strictly increasing, the code rates converge to a numberbetween 0 and 1, and the permutation group of each code is doubly transitive.This also provides a rare example in information theory where symmetry aloneimplies near-optimal performance.  An important consequence of this result is that a sequence of Reed-Mullercodes with increasing blocklength achieves capacity if its code rate convergesto a number between 0 and 1. This possibility has been suggested previously inthe literature but it has only been proven for cases where the limiting coderate is 0 or 1. Moreover, these results extend naturally to affine-invariantcodes and, thus, to all extended primitive narrow-sense BCH codes. The primarytools used in the proof are the sharp threshold property for monotone booleanfunctions and the area theorem for extrinsic information transfer functions.

Reed-Muller Codes Achieve Capacity on Erasure Channels