The symmetry group of ℤqn 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.
KeywordsSymmetry Group Linear Code Cyclic Subgroup Dihedral Group Zero Vector
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