Can the McEliece cryptosystem be used as an additively homomorphic encryption scheme? Since the McEliece cryptosystem is linear, if matrix G is kept constant for different plaintexts, it can be used for linearly combining the corresponding ciphertexts. In that case, what are the advantages of McEliece over Paillier encryption?
In that case, what are the advantages of McEliece over Paillier encryption?


It's not that easy to think of any specific advantages; the public key will be huge (because you would need to expand the code to allow the relatively small initial error vectors to be secure), and those keys are already large enough to begin with.

About the only thing that springs to mind is the addition operation; for Pallier, the addition operation is addition modulo $\mathbb{Z}_n$, with the homomorphic McEliece, it would be addition in some field $GF(2^n)$, for some relatively large $n$, and perhaps some application would prefer that addition.

On the other hand, there are other cryptosystems (such as Goldwasser-Micali) which are homomorphic over addition in $GF(2)$, and running $n$ parallel copies would give you addition in $GF(2^n)$. It would appear likely that the homomorphic operation in McEliece is cheaper than the homomophic in $n$-ways parallel Goldwasser-Micali; however, to me, that sounds like a minor advantage (given the other disadvantages, such as Goldwasser-Micali has no bound on the number of homomorphic operations you can do before the ciphertext becomes undecryptable)