Engineering Hardware Realization of Convolutional Encoder with rate R=1/2

[mad mathematician]

Homework Statement

For $R=1/2$ convolutional encoder with a generator matrix $G(x)=[1+x^2+x^3 1+x+x^3]$
1.draw the hardware realization of the encoder.
2. determine the convolutional matrix generator, G.
3. For the input sequence $m=[1011011]$ determine the coded output sequence.

Relevant Equations

In the attachment, there's the drawing where G2 is the output of G(x) second entry above and G1 is the first output in the first output entry above. D is the shift register.
Is my drawing correct?
As for item 2. I don't see how to get the convolutional matrix G from G(X) above, section 3 is just $mG$, i.e multiplication of the message vector by $G$ (multiplication of a vector with a matrix from left to the matrix.)
Thanks in advance!

My attempt at solution says it all.

 

[mad mathematician]

OK, I think I get it now.
$G_0=[11], G_1=[01],G_2=[10],G_3=[11]$
So $$G=\begin{bmatrix}G_0& G_1& G_2& G_3\\
00 & G_0 &G_1 & G_2\\
00 & 00 & G_0 & G_1\\
00 & 00 & 00 & G_0\\\end{bmatrix}$$

Where I got G_i's as the coefficients of the respective polynomials.

 

[mad mathematician]

Well, G is inifinite dimensional so this matrix should repeat itself endlessly.