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

For a simple-root $\lambda$-constacyclic code $\mathcal{C}$ over $\mathbb{F}_q$, let $\langle\rho\rangle$ and $\langle\rho,M\rangle$ be the subgroups of the automorphism group of $\mathcal{C}$ generated by the cyclic shift $\rho$, and by the cyclic shift $\rho$ and the scalar multiplication $M$, respectively. Let $N_G(\mathcal{C}^\ast)$ be the number of orbits of a subgroup $G$ of automorphism group of $\mathcal{C}$ acting on $\mathcal{C}^\ast=\mathcal{C}\backslash\{0\}$. In this paper, we establish explicit formulas for $N_{\langle\rho\rangle}(\mathcal{C}^\ast)$ and $N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$. Consequently, we derive a upper bound on the number of nonzero weights of $\mathcal{C}$. We present some irreducible and reducible $\lambda$-constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee $N_{\langle\rho\rangle}(\mathcal{C}^\ast)=N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$ is presented.