Classification of the perfect codes in the ∞ -Lee metric

This paper is concerned with perfect codes in the ∞ -Lee metric. A complete classification and description of all (linear and non-linear) two-dimensional perfect codes in the ∞ - Lee metric is presented. Moreover, in the linear case we construct a generator matrix for these codes which induces an isomorphism between the code and an abelian group of the form Z a × Z b with a | b and we determine all possible group structures that can be represented for such codes. We also present an algorithm to obtain a q -ary perfect code with a given group structure and two methods of construction of perfect codes in the ∞ -Lee metric from codes of smaller dimension. In particular, these methods allow us to construct interesting families of perfect ∞ -Lee codes including n -dimensional cyclic perfect codes for all n .