The Structure of Cube Tilings Under Symmetry Conditions

Kisielewicz, Andrzej; Przesławski, Krzysztof

doi:10.1007/s00454-012-9438-0

Cited by 5 publications

(3 citation statements)

References 35 publications

(15 reference statements)

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“…It is likely to be possible to extend some of our results to non-linear codes for which the permutation associated to the code is well defined. We also could study isometry classes of perfect non-linear codes ( [17,24] could be helpful). It should be interesting to obtain for higher dimensions a result analogous to the parametrization theorem (Theorem 3.22), that is, in such a way that isometry and isomorphism classes correspond with certain generalized cosets (Theorems 5.27 and 5.29 provide a partial answer for the maximal case).…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

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

Qureshi,

Costa

2015

Preprint

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“…A new approach to Minkowski's conjecture can be found in Kolountzakis's paper [13] and in [12]. The Hajos proof of Minkowski's conjecture stimulates the development work on the factorization of abelian groups.…”

Section: Letmentioning

confidence: 99%

Rigid polyboxes and keller's conjecture

Kisielewicz

2017

Advances in Geometry

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A cube tiling of R d is a family of pairwise disjoint cubes [0, 1) , and let L(T, x, i) be the set of all ith coordinates ti of vectors t ∈ T such that ([0, 1)It is known that Keller's conjecture is true in dimension seven for cube tilings [0,1) 7 + T for which r − (T ) ≤ 2. In the present paper we show that it is also true for d = 7 if r + (T ) ≥ 6. Thus, if [0, 1) d + T is a counterexample to Keller's conjecture in dimension seven, then r − (T ), r + (T ) ∈ {3, 4, 5}.

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“…It is likely to be possible to extend some of our results from the linear case to non-linear codes for which the permutation associated to the code is well defined. We can also consider isometries acting on perfect non-linear codes aiming to classify isometry classes and [KP12c,MC03] could be helpful in this sense. In Section 4.4 we obtain a parametrization for linear perfect codes in such a way that isometry classes and isomorphism classes correspond to certain generalized cosets (Theorem 4.2.46), so it would be interesting to obtain an analogous result for higher dimensions (Theorems 4.4.27 and 4.4.29 provide a partial answer for the maximal case).…”

Section: Chapter 6 Conclusion and Perspectivesmentioning

confidence: 99%

Códigos perfeitos nas métricas de Lee e Chebyshev e iterações de funções de Rédei

Valdez¹

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To the Dalipaj family who have greatly supported me during my staying in Canada and for their friendship, also to the family of Daniel Panario.To all of them I dedicate this thesis. Special thanks:This work was supported by FAPESP foundation, through scholarship grants 2012/10600-2, BEPE 2014/04096-5 and the thematic project 2013/25977-7 "Security and reliability of informtaion: theory and practice", and also by CAPES (PhD scholarship from 01/03/2012 to 31/10/2012). Without this support this thesis would not have been possible. RESUMOO conteúdo desta tese insere-se dentro de duas áreas de pesquisa muito ativas: a teoria de códigos corretores de erros e sistemas dinâmicos sobre corpos finitos. Para abordar problemas em ambos os tópicos introduzimos um tipo de sequência finita que chamamos v-séries. No conjunto destas definimos uma métrica que induz uma estrutura de poset usada no estudo das possíveis estruturas de grupo abeliano representadas por códigos perfeitos na métrica de Chebyshev. Por outro lado, cada v-série é associada a uma árvore com raiz, a qual terá um papel importante em resultados relacionados à estrutura dinâmica de iterações de funções de Rédei. Na teoria de códigos corretores de erros, estudamos códigos perfeitos na métrica de Lee e na métrica de Chebyshev (correspondentes à métrica ℓ p para p = 1 e p = ∞, respetivamente). Os principais resultados obtidos estão relacionados com a descrição dos códigos q-ários n-dimensionais com raio de empacotamento e, que sejam perfeitos nestas métricas, a obtenção de suas matrizes geradoras e a classificação destes, a menos de isometrias e a menos de isomorfismos. Várias construções de códigos perfeitos e famílias interessantes destes códigos com respeito à métrica de Chebyshev são apresentadas. Em sistemas dinâmicos sobre corpos finitos centramos nossa atenção em iterações de funções de Rédei, sendo o principal resultado um teorema estrutural para estas funções, o qual permite estender vários resultados sobre funções de Rédei. Este teorema pode também ser aplicado para outras classes de funções permitindo obter provas alternativas mais simples de alguns resultados conhecidos como o número de componentes conexas, o número de pontos periódicos e o valor esperado para o período e preperíodo da aplicação exponencial sobre corpos finitos.Palavras-chave: Códigos perfeitos, métrica de Lee, conjectura de Minkowski, ladrilhamento por cubos, funções de Rédei.

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The Structure of Cube Tilings Under Symmetry Conditions

Cited by 5 publications

References 35 publications

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

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

Rigid polyboxes and keller's conjecture

Códigos perfeitos nas métricas de Lee e Chebyshev e iterações de funções de Rédei

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