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 [(n+1)/6] errors to [(n-k+1)/6]. Additionally, we also show that any [[n,k,d]]q quantum code with k+d≤(1-2eq-2)n cannot beat the quantum Hamming bound.