> can we make an error correction scheme so good that a message encoded with it can be error corrected back into the original regardless of how the encoded mess

> can we make an error correction scheme so good that a message encoded with it can be error corrected back into the original regardless of how the encoded message is encrypted? No. Preventing this is called IND-CPA (indistiguishability under chosen plaintext attack) security and is basically table stakes for any modern symmetric encryption algorithm. In fact this is even weaker than IND-CPA, since in IND-CPA the attacker can first observe arbitrarily many other plaintexts and use the resulting information to choose two (non-yet-seen) plaintexts specific to the particular encryption algorithm and key to try to distinguish, and they don't have to be sure which is which, only to do significantly better than chance.