The symmetry group of ℤ ⁿ _q in the Lee space and the ℤ _q n-linear codes

Abstract

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 ⁿ _q , d _L ), as one of the most interesting spaces for coding applications.We establish the symmetry group of ℤ ⁿ _q 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 ⁿ _q ) acting transitively in ℤ ⁿ _q . Therefore, there exists no ℤ _{q n} -linear code with respect to the cyclic subgroup.

This work has been supported by Fundação de Amparo a Pesquisa do Estado de São Paulo, FAPESP, under grant 95/4720-8, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under grant 301416/85-0, Brazil.