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For a simple-root $\lambda$-constacyclic code $\mathcal{C}$ over$\mathbb{F}_q$, let $\langle\rho\rangle$ and $\langle\rho,M\rangle$ be thesubgroups of the automorphism group of $\mathcal{C}$ generated by the cyclicshift $\rho$, 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 establishexplicit formulas for $N_{\langle\rho\rangle}(\mathcal{C}^\ast)$ and$N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$. Consequently, we derive a upperbound on the number of nonzero weights of $\mathcal{C}$. We present someirreducible and reducible $\lambda$-constacyclic codes, which show that theupper 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.

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