I'm trying to solve the following, but have no clue:
Let $$\sigma: \{0, \ldots, n-1\} \rightarrow \{0,\ldots,n-1 \}$$ $$x \mapsto x+a \text{ mod} n$$ a permutation with $(a,n)=1$. Assume to have $\mathcal{C}$ cyclic code of length $n$ over $\mathbb{F}_q$.
Show that $\sigma(\mathcal{C}) = \{(c_{\sigma(0),\ldots,c_\sigma(n-1)}) | (c_0,\ldots,c_{n-1}) \in \mathcal{C} \}$ is a cyclic code equivalent to $\mathcal{C}$
Of course it's equivalent because is just a permutation of the components of the code $\mathcal{C}$. That's precisely the definition of equivalent codes.
The problem is in the cyclic structure: I'd like to show that if I take a vector in $\sigma(\mathcal{C})$, then its shifting by one position towards right is again in $\sigma(C)$. I need to use the specific structure of $\sigma$ and the fact that $C$ is cyclic, but I really can't figure out how to move.