# What is the relation between LWE and coding based cryptography?

I've recently heard about coding based cryptography and it seems very close to the LWE assumption in that it is based on the idea that the error is hard to identify. They are both post-quantum schemes too.

Is there a direct link between the two? (e.g., coding based can be seen as an instance of LWE under certain assumptions) or is there absolutely no relation?

1. Code based cryptography (take McEliece as an example) is based on randomizing a structured error correcting code, adding an error vector which is of weight exactly equal to the code correction capability and using the structure of the code as the trapdoor information. This is typically over the binary field, $$n$$ is the codeword length, $$t$$ is the error correction capability and the rate $$k/n$$ of the code is fixed (the code has $$2^k$$ codewords in the space $$\mathbb{F}_2^n$$ with $$2^n$$ vectors.
2. Most closely related is LPN (learning parity with noise) based cryptography where corresponds to the decoding problem where $$n$$ is the number of samples, $$k$$ the length of the secret while the error is sampled according to a Bernoulli distribution of fixed rate $$t/n.$$ As the number of samples in LPN is unlimited, this problem actually corresponds to decoding a random code of rate arbitrarily close to $$0.$$