  An operator sum representation is derived for a decoherence-free subspace
(DFS) and used to (i) show that DFSs are the class of quantum error correcting
codes (QECCs) with fixed, unitary recovery operators, and (ii) find explicit
representations for the Kraus operators of collective decoherence. We
demonstrate how this can be used to construct a concatenated DFS-QECC code
which protects against collective decoherence perturbed by independent
decoherence. The code yields an error threshold which depends only on the
perturbing independent decoherence rate.
