Violation of the generalized quantum Hamming bound in quantum error correction with correlated noise

The question of whether the quantum Hamming bound, which quantifies the minimum resources needed in order to achieve successful quantum error correction for non-degenerate codes, can be violated with degenerate codes has not been answered for more than a decade. We present a generalization of the quantum Hamming bound, which was originally formulated for the case of noise acting independently on the encoding systems, to a generic noise process. We show in particular that in the presence of correlated noise the bound is violated by employing degenerate codes.