Give a complete list of ternary cyclic code of length 3.
For $x^3-1\in\mathbb{F}_3[x]$, $x^3-1=(x-1)(x^2+x+1)$. Generator polynomials of ternary code of length $3$ are $\{1,x-1,x^2+x+1,x^3-1\}$, these generator polynomials generates \begin{align*} \langle\langle 1\rangle\rangle&=\mathbb{F}_3^3\\ \langle\langle x-1\rangle\rangle&=\{(0,0,0),(2,1,0),(0,2,1),(1,0,2)\}\\ \langle\langle x^2+x+1\rangle\rangle&=\{(0,0,0),(1,1,1)\}\\ \langle\langle x^3-1\rangle\rangle&=\{0,0,0\} \end{align*}
Can anyone check this solution? I am not sure this is right or not?
EDIT:
$\langle\langle x-1\rangle\rangle=\{(0,0,0),(2,1,0),(0,2,1),(2,0,2),(0,2,2),(2,2,0),(1,2,0),(0,1,2),(2,0,1)\}$
I got 9 codewords, but I don't have $(1,0,2)$.
$\langle\langle x^2+x+1\rangle\rangle=\{(0,0,0),(1,1,1),(2,2,2)\}$