New upper bounds on the number of non-zero weights of constacyclic codes
<|reference_start|>New upper bounds on the number of non-zero weights of constacyclic codes: 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]}.<|reference_end|>
@article{chen2024new,
    title={New upper bounds on the number of non-zero weights of constacyclic codes},
    author={Li Chen, Yuqing Fu and Hongwei Liu},
    journal={arXiv preprint arXiv:2405.00309},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.00309},
    primaryClass={cs.IT math.IT}
}