Degenerate quantum codes and the quantum Hamming bound

Abstract

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 $q\ensuremath{\geqslant}5$, 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 $\ensuremath{\lfloor}(n+1)/6\ensuremath{\rfloor}$ errors to $\ensuremath{\lfloor}(n\ensuremath{-}k+1)/6\ensuremath{\rfloor}$. Additionally, we also show that any $[[n,k,d]{]}_{q}$ quantum code with $k+d\ensuremath{\leqslant}(1\ensuremath{-}2{\mathit{eq}}^{\ensuremath{-}2})n$ cannot beat the quantum Hamming bound.