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

For any simple-root constacyclic code C over a finite field Fq, 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 Aut (C) of C. In this paper, by calculating the number of G-orbits of C\ 0\, 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, FFA, 2024. 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 Aut (C). The results derived in this paper generalize the main results in Chen, Fu and Liu, IEEE-TIT, 2024}.