Degeneracy Cannot Violate the Quantum Hamming Bound (opens in new tab)

The quantum Hamming bound is the standard finite-length sphere-packing bound for exact correction of arbitrary qubit errors. Whether degeneracy can evade this bound has remained unresolved in full generality for nearly three decades: distinct correctable errors may act identically on the code space, so the usual disjoint-sphere argument breaks down. We prove that every exact binary quantum subspace code with $K>1$ obeys the bound, without assumi...