Upper bounds on the number of non-zero weights of cyclic codes

Let $\mathcal{C}$ be an arbitrary simple-root cyclic code and let $\mathcal{G}$ be the subgroup of ${\rm Aut}(\mathcal{C})$ (the automorphism group of $\mathcal{C}$) generated by the multiplier, the cyclic shift and the scalar multiplications.In this talk, an explicit formula, in some cases an upper bound, for the number of orbits of $\mathcal{G}$ on $\mathcal{C}\backslash {\bf 0}$ is established. An explicit upper bound on the number of non-zero weights of $\mathcal{C}$ is consequently derived and a necessary and sufficient condition for the code $\mathcal{C}$ meeting the bound is exhibited. Many examples are presented to show that our new upper bounds are tight and are strictly less than the upper bounds in [Chen and Zhang, IEEE-TIT, 2023]. In addition, for two special classes of cyclic codes, smaller upper bounds on the number of non-zero weights of such codes are obtained by replacing $\mathcal{G}$ with larger subgroups of the automorphism groups of these codes. As a byproduct, our main results suggest a new way to find few-weight cyclic codes. This talk is based on joint works with Yuqing Fu, Hongwei Liu and Guanghui Zhang.