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

For a simple-root lambda-constacyclic code Cover F-q, let <rho > and <rho,M > be the subgroups of the automorphism group of C generated by the cyclic shift rho, and by the cyclic shift rho and the scalar multiplication group rho, respectively. Let N-G(C*) be the number of orbits of a subgroup Gof the automorphism group of C acting on C* = C\{0}. In this paper, we establish explicit formulas for N-<rho > (C*) and N-<rho,N-M > (C*). Consequently, we derive a upper bound on the number of non-zero weights of 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 > (C*) = N-<rho,N-M > (C*) is presented. (c) 2023 Elsevier Inc. All rights reserved.