The Symmetry Group of Znq in the Lee Space and the Zqn-Linear Codes

The ℤ4-linearity is a construction technique of good binary codes. Motivated by this property, we address the problem of extending the ℤ4-linearity to ℤ q n -linearity. In this direction, we consider the n-dimensional Lee space of order q, that is, (ℤ q n , d L ), as one of the most interesting spaces for coding applications.We establish the symmetry group of ℤ q n for any n and q by determining its isometries.We also show that there is no cyclic subgroup of order q n in Γ(ℤ q n ) acting transitively in ℤ q n . Therefore, there exists no ℤ q n -linear code with respect to the cyclic subgroup.