  A permutationally invariant n-bit code for quantum error correction can be
realized as a subspace stabilized by the non-Abelian group S_n. The code
corresponds to bases for the trivial representation, and all other irreducible
representations, both those of higher dimension and orthogonal bases for the
trivial representation, are available for error correction.
  A number of new (non-additive) binary codes are obtained, including two new
7-bit codes and a large family of new 9-bit codes. It is shown that the
degeneracy arising from permutational symmetry facilitates the correction of
certain types of two-bit errors. The correction of two-bit errors of the same
type is considered in detail, but is shown not to be compatible with single-bit
error correction using 9-bit codes.
