<s>[INST] Code is not cyclic for any q I have code $C$ over $F_p$ with generator matrix which looks like 
$G = 
\begin{pmatrix} 
0 &0& 0& 1& 0& 1& 1 &1\\
1& 0 &0& 0 &1 &0 &1& 1\\
1& 1& 0& 0& 0& 1& 0& 1\\
1 &1& 1& 0 &0 &0& 1 &0\end{pmatrix}$
I need to show that this code is not cyclic for any $p$.
I constructed vector which looks like 
$$(c_2+c_3+c_4, c_3+c_4, c_4, c_1, c_2, c_1+c_3, c_1+c_2+c_4, c_1+c_2+c_3)$$ and its shifts but it didn't help me a lot. 
Does somebody have any ideas how it can be proven?
 [/INST] A base for your code is a base for $\ker G$. So the code has dimension 4, and a base is
$$
v_1=\begin{pmatrix}0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ -1\end{pmatrix}
\quad v_2=\begin{pmatrix}-1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0\end{pmatrix}
\quad v_3=\begin{pmatrix}0 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0\end{pmatrix}
\quad v_4=\begin{pmatrix}0 \\ 0 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0\end{pmatrix}
$$
The code was cyclic if and only if the base, shifted by one, is still a base of the code, but 
$$
v_2=\begin{pmatrix}-1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0\end{pmatrix}
\quad v_3=\begin{pmatrix}0 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0\end{pmatrix}
\quad v_4=\begin{pmatrix}0 \\ 0 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0\end{pmatrix}
\quad v_1=\begin{pmatrix}0 \\ 0 \\ 0 \\ -1 \\ 0 \\ 0 \\ 0 \\ 1\end{pmatrix}
$$ 
Is a base, so it is cyclic for every p
</s>