A tight upper bound on the number of non-zero weights of a quasi-cyclic code

Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard cyclic shift. In this paper, we find an explicit formula of orbits of $G$ on $\mathcal{C}\setminus \{\mathbf{0}\}$. Consequently, an explicit upper bound on the number of non-zero weights of $\mathcal{C}$ is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. In particular, we list some examples to show the bounds are tight. Our main result improves and generalizes some of the results in \cite{M2}.