A tight upper bound on the number of non-zero weights of a constacyclic code

Abstract

For a simple-root $\lambda$-constacyclic code $\mathcal{C}$ over ${\mathbb{F}}_q$, let $⟨\rho ⟩$ and $⟨\rho ,M⟩$ be the subgroups of the automorphism group of $\mathcal{C}$ generated by the cyclic shift $\rho$, and by the cyclic shift $\rho$ and the scalar multiplication $M$, respectively. Let $N_G({\mathcal{C}}^{\ast })$ be the number of orbits of a subgroup $G$ of automorphism group of $\mathcal{C}$ acting on ${\mathcal{C}}^{\ast }=\mathcal{C}\backslash \{0\}$. In this paper, we establish explicit formulas for $N_{⟨\rho ⟩}({\mathcal{C}}^{\ast })$ and $N_{⟨\rho ,M⟩}({\mathcal{C}}^{\ast })$. Consequently, we derive a upper bound on the number of nonzero weights of $\mathcal{C}$. We present some irreducible and reducible $\lambda$-constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee $N_{⟨\rho ⟩}({\mathcal{C}}^{\ast })=N_{⟨\rho ,M⟩}({\mathcal{C}}^{\ast })$ is presented.