  A striking feature of quantum error correcting codes is that they can
sometimes be used to correct more errors than they can uniquely identify. Such
degenerate codes have long been known, but have remained poorly understood. We
provide a heuristic for designing degenerate quantum codes for high noise
rates, which is applied to generate codes that can be used to communicate over
almost any Pauli channel at rates that are impossible for a nondegenerate code.
The gap between nondegenerate and degenerate code performance is quite large,
in contrast to the tiny magnitude of the only previous demonstration of this
effect. We also identify a channel for which none of our codes outperform the
best nondegenerate code and show that it is nevertheless quite unlike any
channel for which nondegenerate codes are known to be optimal.
