  The increasing interest in using quantum error correcting codes in practical
devices has heightened the need for designing quantum error correcting codes
that can correct against specialized errors, such as that of amplitude damping
errors which model photon loss. Although considerable research has been devoted
to quantum error correcting codes for amplitude damping, not so much attention
has been paid to having these codes simultaneously lie within the decoherence
free subspace of their underlying physical system. One common physical system
comprises of quantum harmonic oscillators, and constant-excitation quantum
codes can be naturally stabilized within them. The purpose of this paper is to
give constant-excitation quantum codes that not only correct amplitude damping
errors, but are also immune against permutations of their underlying modes. To
construct such quantum codes, we use the nullspace of a specially constructed
matrix based on integer partitions.
