Probabilities of failure for quantum error correction

We investigate the performance of a quantum error-correcting code when pushed beyond its intended capacity to protect information against errors, presenting formulae for the probability of failure when the errors affect more qudits than that specified by the code's minimum distance. Such formulae provide a means to rank different codes of the same minimum distance. We consider both error detection and error correction, treating explicit examples in the case of stabilizer codes constructed from qubits and encoding a single qubit.