Degenerate quantum codes and the quantum Hamming bound

The parameters of a nondegenerate quantum code must obey the Hamming bound. An important open problem in quantum coding theory is whether the parameters of a degenerate quantum code can violate this bound for nondegenerate quantum codes. In this article we show that Calderbank-Shor-Steane (CSS) codes, over a prime power alphabet $q⩾5$, cannot beat the quantum Hamming bound. We prove a quantum version of the Griesmer bound for the CSS codes, which allows us to strengthen the Rains’ bound that an $[[n,k,d]{]}_2$ code cannot correct more than $\lfloor (n+1)/6\rfloor$ errors to $\lfloor (n-k+1)/6\rfloor$. Additionally, we also show that any $[[n,k,d]{]}_q$ quantum code with $k+d⩽(1-2{\mathit{eq}}^{-2})n$ cannot beat the quantum Hamming bound.