Improved Upper Bounds on the Number of Non-Zero Weights of Cyclic Codes

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