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 or not the parameters of a degenerate quantum code can violate this bound for nondegenerate quantum codes. In this paper we show that Calderbank-Shor-Steane (CSS) codes with alphabet $q\ge 5$ 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 $\text{floor}(n+1)\mathrm{/}6$ errors to $\text{floor}(n-k+1)\mathrm{/}6$. Additionally, we also show that the general quantum codes $[[n,k,d]{]}_q$ with $k+d\leq {(1-2eq^{-2})n}$ cannot beat the quantum Hamming bound.