2015/12/18 by Claudio Qureshi, Qureshi, Claudio, Sueli I. R. Costa +1
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Cellular Automata and Applications
paper · pdf · doi:10.48550/arxiv.1512.05908
We describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of Rn. 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.