Classification of the perfect codes in the ∞ -Lee metric

Claudio M. Qureshi, Sueli I. R. Costa

DOI: 10.14209/sbrt.2015.64

Evento: XXXIII Simpósio Brasileiro de Telecomunicações (SBrT2015)

Keywords: Perfect codes Lee metric p -Lee metric group isomorphism

Abstract

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 .

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