On Cube Tilings of Tori and Classification of Perfect Codes in the Maximum Metric

TL;DR

This paper explores cube tilings of tori and their relation to perfect codes in the maximum metric, providing classifications, constructions, and a new matrix-based framework for understanding these codes in various dimensions.

Contribution

It introduces a comprehensive classification of two-dimensional tilings, describes perfect matrices for linear codes, and offers methods to construct perfect codes from lower dimensions or sections.

Findings

Complete characterization of 2D tilings

Introduction of perfect matrices for code generation

Parametrization of maximal perfect codes

Abstract

We describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of R^n. In the language of coding theory these tilings correspond to perfect codes with respect to the maximum metric. A complete characterization of the two-dimensional tillings is presented and in the linear case, a description of general matrices, isometry and isomorphism classes is provided. Several methods to construct perfect codes from codes of smaller dimension or via sections are derived. We introduce a special type of matrices (perfect matrices) which are in correspondence with generator matrices for linear perfect codes in arbitrary dimensions. For maximal perfect codes, a parametrization obtained allows to describe isomorphism classes of such codes. We also approach the problem of what isomorphism classes of abelian groups can be represented by q-ary n-dimensional perfect codes of a given cardinality N.