Rigidity and the chessboard theorem for cube packings

Kisielewicz, Andrzej; Przesławski, Krzysztof

doi:10.1016/j.ejc.2012.01.006

Cited by 1 publication

(1 citation statement)

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“…A similar conjecture was proposed by Keller [10] removing the restriction that the centers of the cubes have to form a lattice, however this more stronger version was shown to be true in dimensions n ≤ 6 [25], false in dimensions n ≥ 8 [19,21] and it remains open in dimension n = 7. A lot of variants and problems related to cube tilings have been considered [14,15,16,29,30] as well as application to other areas such as combinatorics, graph Work partially supported by FAPESP grants 2012/10600-2 and 2013/25977-7 and by CNPq grants 312926/2013-8 and 158670/2015-9.…”

Section: Introductionmentioning

confidence: 99%

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

Qureshi,

Costa

2015

Preprint

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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 tilings is presented and a description of general matrices, isometry and isomorphism classes is provided in the linear case. 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 .

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