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.

翻译：设$\mathbb{F}_q$上的单根$\lambda$-常循环码$\mathcal{C}$，记$\langle\rho\rangle$和$\langle\rho,M\rangle$分别为由循环移位$\rho$生成的、以及由循环移位$\rho$与标量乘法$M$共同生成的$\mathcal{C}$的自同构群子群。设$N_G(\mathcal{C}^\ast)$为自同构群子群$G$作用在$\mathcal{C}^\ast=\mathcal{C}\backslash\{0\}$上的轨道数。本文建立了$N_{\langle\rho\rangle}(\mathcal{C}^\ast)$和$N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$的显式公式，进而推导出$\mathcal{C}$非零权重数量的一个上界。我们给出若干既约与可约$\lambda$-常循环码实例，表明该上界是紧的。最后给出了保证$N_{\langle\rho\rangle}(\mathcal{C}^\ast)=N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$成立的一个充分条件。