Shor's algorithm with noiseless qubits and gates works in theory.
Shor's algorithm with noisy qubits and gates does not work in theory. It needs exponentially small noise.
Both in theory.
That's it.
I thought that this is what error correction fixes-- with a huge but only polynomial blow up.
If I understand it correctly, the error correction could reduce noise only in linear fashion - QC are analog machines - ten times more resources would reduce error only tenfold at most. For Shor's to work this is not enough at all - it needs exponentially smaller noise.