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

Abstract

For a simple-root λ-constacyclic code C over Fq, 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 group M, respectively. Let NG(C⁎) be the number of orbits of a subgroup G of the 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 non-zero 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.