Minimal secure index of quasi-cyclic codes for code-based cryptography

It is well-known that keys (generator matrices) of code-based cryptography are very large for its practical usage. To reduce them scientists propose to use families of quasi-cyclic codes. I remind that a linear code $$C \subset \mathbb{F}_q^n$$ is called $$\textit{quasi-cyclic}$$ of index $$m \mid n$$ if for any codeword $$c = (c_1, c_2, \cdots\!, c_n) \in C$$ the shift by $$m$$ positions gives another codeword, that is, $$(c_{m+1}, c_{m+2}, \cdots\!, c_{m}) \in C$$. It is clear that for cyclic codes (when $$m=1$$) we obtain the most compact representation of the key. However cyclic codes seem to be attacked, i.e., insecure.

Help, please. What is the minimal value $$m$$ now considered to be secure?