  The low-energy subspace of a conformal field theory (CFT) can serve as a
quantum error correcting code, with important consequences in holography and
quantum gravity. We consider generic 1+1D CFT codes under extensive local
dephasing channels and analyze their error correctability in the thermodynamic
limit. We show that (i) there is a finite decoding threshold if and only if the
minimal nonzero scaling dimension in the fusion algebra generated by the jump
operator of the channel is larger than $1/2$ and (ii) the number of protected
logical qubits $k \geq \Omega( \log \log n)$, where $n$ is the number of
physical qubits. As an application, we show that the one-dimensional quantum
critical Ising model has a finite threshold for certain types of dephasing
noise. Our general results also imply that a CFT code with continuous symmetry
saturates a bound on the recovery fidelity for covariant codes.
