Let C be an arbitrary simple-root cyclic code and let G be the subgroup of Aut(C) (the automorphism group of C) generated by the multiplier, the cyclic shift and the scalar multiplications. To the best of our knowledge, the subgroup G is the largest subgroup of Aut(C). In this paper, an explicit formula, in some cases an upper bound, for the number of orbits of G on C\{0} is established. An explicit upper bound on the number of non-zero weights of C is consequently derived and a necessary and sufficient condition for the code 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 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.

翻译：设C为任意简单根循环码，G为C的自同构群Aut(C)中由乘子、循环移位和标量乘法生成的子群。据我们所知，子群G是Aut(C)的最大子群。本文建立了G在C\{0}上轨道数量的显式公式（某些情形下为上界）。由此导出C非零权重数量的显式上界，并给出了该码达到此上界的充要条件。大量实例表明，我们的新上界是紧的，且严格小于Chen与Zhang（IEEE-TIT, 2023）中的上界。此外，针对两类特殊循环码，通过将G替换为其自同构群的更大子群，我们得到了此类码非零权重数量的更小上界。作为副产品，主要结果揭示了一种寻找少权循环码的新方法。