Improved 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
${\mathrm{ Aut}}(\mathcal {C})$
(the automorphism group of
$\mathcal {C}$
) generated by the multiplier, the cyclic shift and the scalar multiplications. To the best of our knowledge, the subgroup
$\mathcal {G}$
is the largest subgroup of
${\mathrm{ Aut}}(\mathcal {C})$
. In this paper, an explicit formula, in some cases an upper bound, for the number of orbits of
$\mathcal {G}$
on
$\mathcal {C}\backslash \{\mathbf{ 0\}}$
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.