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 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 $\floor{(n+1)/6}$ errors to $\floor{(n-k+1)/6}$. Additionally, we also show that the general quantum codes $[[n,k,d]{]}_q$ with $k+d\le (1-2e{q}^{-2})n$ cannot beat the quantum Hamming bound.

Reformulated one of the results, corrected an erroneous remark and added a few more results

References in corpus (1)

• On optimal quantum codes

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• Minimal degenerate CSS quantum code with low cost circuit