  We present a family of additive quantum error-correcting codes whose
capacities exceeds that of quantum random coding (hashing) for very noisy
channels. These codes provide non-zero capacity in a depolarizing channel for
fidelity parameters $f$ when $f> .80944$. Random coding has non-zero capacity
only for $f>.81071$; by analogy to the classical Shannon coding limit, this
value had previously been conjectured to be a lower bound. We use the method
introduced by Shor and Smolin of concatenating a non-random (cat) code within a
random code to obtain good codes. The cat code with block size five is shown to
be optimal for single concatenation. The best known multiple-concatenated code
we found has a block size of 25. We derive a general relation between the
capacity attainable by these concatenation schemes and the coherent information
of the inner code states.
