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In the symmetric version of McEliece cryptosystem, we are adding intentional error vector during encryption. During decryption, first, we cancel the effect of error vector and then multiply the obtained term with the inverse of G matrix to retrieve the plaintext. In some papers, it is mentioned that weight of error vector has to be half of codeword length. So in this case, what is the relation between the error correcting capability of code used and weight of intentional error vector?

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The weight of the synthetic error vector is $n/2$ where $n$ is the block length, this corresponds to maximum entropy error patterns, with independent probability of error $1/2$ for each bit.

The error correcting capability of the code should be much lower for good security.

In the original proposal, $n=1024$ and $t=50$ so this should give you some idea. There is a very readable paper here about this cryptosystem. Look for papers by Sendrier, for updated information on parameters and attacks.

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