CPA distinguisher for matrix multiplication in GF(256) with randomized padding What is the best CPA distinguisher for the function $F_k:\{0,1\}^{8n}\to\{0,1\}^{16n}$ described below?

Let $E_k$ be a $2n\times2n$ matrix with elements in $GF(2^8)$, selected by generating bit strings of length $32n^2$ using a predefined function $PRF_k$, until one is found that is invertible e.g. using Gaussian elimination.

To encode an input bit string $M$, generate random bit string $Pad$ of length $8n$. Let the $2n$ element vector $T$ correspond to the concatenation $M|Pad$. Calculate the vector $C$ using the formula $E_kT^T=C^T$.

The decoding function is given by the $n\times2n$ matrix corresponding to the first $n$ rows of $E_k^{-1}$. 

Obviously, you need no more than $4n$ chosen cipher texts and access to a decryption oracle, to be able to derive the decoding matrix. But is there a better CPA distinguisher than statistical analysis of the skew caused by a constant $M$ for a relatively large number of $Pad$ values?
While Ilmari answered the specific question you asked (Chosen Plaintext Distinguishers), I would like to note that the attack can be sharpened into a Known Plaintext Key Recovery attack (where, by key recovery, I don't mean recovering the $E_k$ matrix, but instead allowing an attacker to reconstruct enough information to decrypt arbitrary texts).

One straightforward way to see this is to observe that the decryption process is entirely linear; that is, if we know that:

$Decrypt(C_1) = P_1$ and $Decrypt(C_2) = P_2$

then

$Decrypt(aC_1 + bC_2) = aP_1 + bP_2$  for $a, b \in GF(2^8)$

So, all the attacker needs to do is collect $2n$ linearly independent ciphertexts with corresponding known plaintexts:

$Decrypt(C_1) = P_1$

$Decrypt(C_2) = P_2$

...

$Decrypt(C_{2n}) = P_{2n}$

Then, do decrypt an arbitrary ciphertext $C_x$, he solves the linear equation for $a_1, a_2, \ldots a_{2n}$:

$C_x = \sum_i a_i C_i$

and then he knows that

$Decrypt(C_x) = Decrypt( \sum_i a_i C_i) = \sum_i a_i P_i$