I am given $\mathcal{C}$ the following code over $\mathbb{F}_2$ with generator matrix
$$\begin{bmatrix} 0 & 1 & 0 & 1 & 1 &0 \\ 1 & 0 & 0 & 1 & 0 &1 \\ 0 & 0 & 1 & 0 & 1 &1 \end{bmatrix}$$
and I have to decide whether or not the following code is cyclic. I don't have the solution, so I need a check on my procedure
My attempt:
Let $c=(c_1,ldots,c_6$ a word in $\mathcal{C}$. Then $c = [a_1,a_2,a_3] G$, so I have $$c = (a_2,a_1,a_3,a_1+a_2,a_1+a_3,a_2+a_3)$$
If I consider the shift of one position, I obtain $$c[1] = (a_2 + a_3,a_2,a_1,a_3,a_1+a_2,a_1+a_3)$$
and I have to see if this word is in $\mathcal{C}$. So, I want to determine if there exists $(b_1,b_2,b_3)$ such that $$c[1] = b_1 r_1 + b_2 r_2 + b_3 r_3 = (b_2,b_1,b_3,b_1+b_2,b_1+b_3,b_2+b_3)$$
Hence, I find
$\begin{cases} b_1 = a_2 \\ b_3 = a_1 \\ b_1 + b_2 = a_3 \\ \end{cases}$
from which I determine $b_2 = a_3+ a_2$
But, for instance $$b_2 +b_3 = a_3 +a_2 + a_1 \ne a_1+a_3$$
and so since this condition is not fulfilled, it's not cyclic.
Is eveytihng right?