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