> 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.
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.