  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.

Combinatorics