Symmetry is at the heart of coding theory. Codes with symmetry, especially
cyclic codes, play an essential role in both theory and practical applications
of classical error-correcting codes. Here we examine symmetry properties for
codeword stabilized (CWS) quantum codes, which is the most general framework
for constructing quantum error-correcting codes known to date. A CWS code Q can
be represented by a self-dual additive code S and a classical code C, i.,e.,
Q=(S,C), however this representation is in general not unique. We show that for
any CWS code Q with certain permutation symmetry, one can always find a
self-dual additive code S with the same permutation symmetry as Q such that
Q=(S,C). As many good CWS codes have been found by starting from a chosen S,
this ensures that when trying to find CWS codes with certain permutation
symmetry, the choice of S with the same symmetry will suffice. A key step for
this result is a new canonical representation for CWS codes, which is given in
terms of a unique decomposition as union stabilizer codes. For CWS codes, so
far mainly the standard form (G,C) has been considered, where G is a graph
state. We analyze the symmetry of the corresponding graph of G, which in
general cannot possess the same permutation symmetry as Q. We show that it is
indeed the case for the toric code on a square lattice with translational
symmetry, even if its encoding graph can be chosen to be translational
invariant.
