# Violation of the Quantum Hamming bound

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

$$2^{N-k}\geq\sum_{n=0}^{\lfloor d/2\rfloor}3^n\begin{pmatrix}N \\ n\end{pmatrix}.$$ 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.