For any simple-root constacyclic code $\mathcal{C}$ over a finite field $\mathbb{F}_q$, as far as we know, the group $\mathcal{G}$ generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group ${\rm Aut}(\mathcal{C})$ of $\mathcal{C}$. In this paper, by calculating the number of $\mathcal{G}$-orbits of $\mathcal{C}\backslash\{\bf 0\}$, we give an explicit upper bound on the number of non-zero weights of $\mathcal{C}$ and present a necessary and sufficient condition for $\mathcal{C}$ to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in [Zhang and Cao, FFA, 2024]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code $\mathcal{C}$ belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of $\mathcal{C}$ by substituting $\mathcal{G}$ with a larger subgroup of ${\rm Aut}(\mathcal{C})$. The results derived in this paper generalize the main results in [Chen, Fu and Liu, IEEE-TIT, 2024]}.

翻译：对于有限域 $\mathbb{F}_q$ 上的任意单根常循环码 $\mathcal{C}$，据我们所知，由乘子、常循环移位和标量乘法生成的群 $\mathcal{G}$ 是 $\mathcal{C}$ 的自同构群 ${\rm Aut}(\mathcal{C})$ 的最大子群。本文通过计算 $\mathcal{C}\backslash\{\bf 0\}$ 的 $\mathcal{G}$-轨道数量，给出了 $\mathcal{C}$ 非零权重数量的显式上界，并提出了 $\mathcal{C}$ 达到该上界的充要条件。文中的若干示例表明，我们的上界是紧的，且优于 [Zhang and Cao, FFA, 2024] 中的上界。特别地，主要结果为构造少权重常循环码提供了新方法。此外，对于两类特殊常循环码 $\mathcal{C}$，通过将 $\mathcal{G}$ 替换为 ${\rm Aut}(\mathcal{C})$ 的更大子群，我们得到了 $\mathcal{C}$ 非零权重数量的更小上界。本文推导的结果推广了 [Chen, Fu and Liu, IEEE-TIT, 2024] 中的主要结论。