I'm not an expert in quantum computing at all, but recently I've started to learn it (read Shen-Vyalyi-Kitaev's book and looked up some other literature here and there).


There are few remarkable constructions of quantum error-correcting codes in this book. They all use $N$ q-bits to achieve the coding distance $\sqrt{N}$.


What is known in general about the lower bound on the coding distance? I think I have figured out some nice construction which could do better than $c \cdot \sqrt{N}$, but I'm not sure yet and want to check some known bounds first.



Can quantum codes have more than $c \cdot \sqrt{N}$ correction distance for N encoding qbits?
Well, the best lower bound is $d=1$. As far as upper bounds are concerned, [randomly generated code families can achieve linear distance](https://arxiv.org/abs/quant-ph/9512032). However a major open question is whether there exist code families with bounded-weight checks that achieve linear distance. Currently, the best known construction requires some fancy tools, [and only achieves $d = \theta(\sqrt{N \log(N)})$](https://www.researchwithrutgers.com/en/publications/zsub2sub-systolic-freedom-and-quantum-codes).


