On the section 5.4 of the paper Improved Security for a Ring-Based Fully Homomorphic Encryption Scheme, the authors explain how to discard some bits of the ciphertexts to get smaller ciphertexts and smaller evaluation keys.
The function Discard simply takes a ciphertext $c$ and an integer $i$ (number of words to be discarded) and returns
$\lfloor w^{-i}c \rfloor$
where $w$ is the number of bits of each word.
But, since $c$ is a polynomial with integer coefficients (coefficients in $\mathbb{Z}_q$), I don't understand how this function is discarding bits. To me, it seems like it is only zeroing all the coefficients whose absolute values are smaller than $w^i$.
I also checked the reference from where this technique of truncating ciphertext was extracted, named the paper Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP, but its explanation does not make sense to me as well.
So, do you know how this technique works? Could you give more details or show me what point I am interpreting wrong?