  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 $q\geq 5$ 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 $\floor{(n+1)/6}$ errors to $\floor{(n-k+1)/6}$. 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.

Quantum Physics