New upper bounds on the number of non-zero weights of constacyclic codes

For any simple-root constacyclic code $C$ over a finite field $F_q$, as far as we know, the group $G$ generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group $\mathrm{Aut}\left(C\right)$ of $C$. In this paper, by calculating the number of $G$-orbits of $C﹨\left\{0\right\}$, we give an explicit upper bound on the number of non-zero weights of $C$ and present a necessary and sufficient condition for $C$ 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 (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code $C$ belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of $C$ by substituting $G$ with a larger subgroup of $\mathrm{Aut}\left(C\right)$. The results derived in this paper generalize the main results in Chen et al. (2024) [9].