Degenerate quantum codes and the quantum Hamming bound

TL;DR

This paper investigates the limitations of degenerate quantum codes, proving that certain classes cannot violate the quantum Hamming bound and establishing bounds on their error correction capabilities.

Contribution

It demonstrates that CSS codes with alphabet size $q extgreater 4$ and general quantum codes with specific parameters cannot surpass the quantum Hamming bound.

Findings

CSS codes with $q extgreater 4$ cannot beat the quantum Hamming bound

A quantum Griesmer bound is established for CSS codes

Bounds on error correction for general quantum codes are derived

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 $q\geq 5$ 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.