Using error-correcting codes for the bootstrapping procedure of Fully Homomorphic Schemes In the context of Fully Homomorphic Schemes, we use a technique called "bootstrapping" to refresh the ciphertext, by evaluating homomorphically the decryption circuit with an encrypted version of the private key.

I was wondering if it was possible to consider this approach, of reducing the error size, by using an error-control code instead (eventually designed for our FHE scheme). 

I am not concerned about the efficiency of such a scheme right now, but would it be feasible to design a scheme such that we can apply an "error-reducing code" reducing errors instead of correcting all of them. I don't think there have been researches to this idea so far so it is maybe completely unthinkable but I wanted to discuss it with other people..
I think not, because the two types of error correction are doing fundamentally different things. 

Conventional error correction schemes, for instance Reed-Solomon codes, expect the value to be near some codeword.  To correct an erroneous value, they change it to the nearest codeword.  You can picture it as a space where some points in the space correspond to correct values, and each of those points is encircled by a region of incorrect values that belong to that point.  The assumption of error correction is that value started at a correct point and drifted a little, but stayed in the point's own region. 

With LWE, you are mixing the correct value with a ton of error, and secretly you know some way to unmix it later (bootstrapping).  You can't use less error and still have security.  In terms of the point analogy above, the incorrect values have to be all over the space.  They can't stay close to their correct value in any obvious way.  Otherwise anybody could see what the correct value is - it's the nearest correct point.  I doubt that you can use any conventional error correction without having obvious regions of some kind.

Error correction doesn't do much for the bootstrapping either, I think, because once you unmix the value, you can just throw away the error part.