What is the largest noise rate $\eta(n)$ for which learning parities with noise is easy?

Learning Parity with Noise (LPN) is usually stated with constant noise rate $\eta < 1/2$ on the labels, and it is believed to be hard to learn because of the high statistical dimension of the problem (they are not SQ-learnable even if the parity is assumed to be sparse).

I'm interested in a noise rate which decays with $n$, the number of features, and I want to know where the known easiness threshold is for this problem.

Specific notes:

• LPN is easy with noise rate $\eta = O(1/n)$, because drawing $O(n)$ samples will with good probability produce $O(1)$ noisy examples.
• It seems like it should still be hard for $\eta = O(1/\log n)$ and even $O(1/\sqrt{n})$.
• Is it known to be easy for any $\eta = \omega(1/n)$?
• LPN is equivalent to decoding random linear codes, but I have not heard of any work with sublinear noise rates for coding theory.