  We introduce 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 our method
exploits code symmetry. In particular, the technique applies to any sequence of
linear codes where the blocklengths are strictly increasing, the code rates
converge, and the permutation group of each code is doubly transitive. In other
words, we show that symmetry alone implies near-optimal performance.
  An important consequence of this result is that a sequence of Reed-Muller
codes with increasing blocklength and converging rate achieves capacity. 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 all affine-invariant codes and, thus, to extended
primitive narrow-sense BCH codes. This also resolves, in the affirmative, the
existence question for capacity-achieving sequences of binary cyclic codes. The
primary tools used in the proof are the sharp threshold property for symmetric
monotone boolean functions and the area theorem for extrinsic information
transfer functions.
