Trying to understand the LPN encryption in this paper: https://eprint.iacr.org/2021/120.pdf. From Definitions 4 and 5, ciphertext is $c=C\cdot s\oplus e\oplus G\cdot m$ and the paper says the LPN encryption scheme is both message homomorphic and key homomorphic, where $s$ is the secret key, $C$ and $G$ are two matrices.
But, I did not understand why it is homomorphic.
Suppose two ciphertexts with same secret key $s$: $c_1=C\cdot s\oplus e_1\oplus G\cdot m_1$ and $c_2=C\cdot s\oplus e_2\oplus G\cdot m_2$. XOR of these two ciphertext gives $c_1\oplus c_2=e_1\oplus e_2\oplus G\cdot (m_1\oplus m_2)$. This is not an encryption of $m_1\oplus m_2$. Also, $e_1\oplus e_2$ cannot guarantee correct decryption, isn't it?