A Tight Upper Bound on the Number of Non-Zero Weights of a Cyclic Code

Let
$\mathcal {C}$
be a simple-root cyclic code and let
$\mathcal {G}$
be the subgroup of the automorphism group of
$\mathcal {C}$
generated by the cyclic shift of
$\mathcal {C}$
and the scalar multiplications of
$\mathcal {C}$
. In this paper, we find an explicit formula for the number of orbits of
$\mathcal {G}$
on
$\mathcal {C}\setminus \{\mathbf {0}\}$
. Consequently, an explicit upper bound on the number of non-zero weights of
$\mathcal {C}$
is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. Several reducible and irreducible cyclic codes meeting the bound are presented, revealing that our bound is tight. In particular, we find that some infinite families of irreducible cyclic codes constructed in (Ding, 2009) meet our bound; we then conclude that such known codes enjoy an additional property that any two codewords with the same weight belong to the same
$\mathcal {G}$
-orbit, a fact that may not have been known before. Our main result improves and generalizes some of the results in (Shi et al., 2019).