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