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The quantum Hamming bound for a non-degenerate $[[N,k,d]]$ quantum error correction code is defined as:

\begin{equation} 2^{N-k}\geq\sum_{n=0}^{\lfloor d/2\rfloor}3^n\begin{pmatrix}N \\ n\end{pmatrix}. \end{equation} However, there is no proof stating that degenerate codes should obey such bound. I wonder if there exists any example of a degenerate code violating the quantum Hamming bound, or if there have been some advances in proving similar bounds for degenerate codes.

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You may be interested in the answers to this question. One example of a degenerate code beating the quantum Hamming bound is here. I also have a numerical example of a small violation in my own work, here. In Figure two, you will see a zoomed in section. Essentially, the black line is the quantum Hamming bound (that may not be entirely obvious from what is written!), and the grey line is an approximation of what can be achieved with something related to the Toric code. There will be other examples as well!

There appear to be a number of results about classes of degenerate codes that do not violate the quantum Hamming bound (e.g. here and here). I haven't read them, so don't know how useful they are, but the abstracts suggest that they provide a nice counter-point, conveying the rarity of good degenerate codes.

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  • $\begingroup$ Thanks for the reference, I will check it up in detail to see what's going on there. Please expand your answer if you find something interesting related with the topic. $\endgroup$ – Josu Etxezarreta Martinez May 15 '18 at 15:39

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