An improvement to the Minkowski‐Hiawka bound for packing superballs

Rush, J. A.; Sloane, N. J. A.

doi:10.1112/s0025579300013231

Cited by 44 publications

(44 citation statements)

References 7 publications

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“…Clearly, the Euclidean weight of c is equivalent to the 2-norm of c defined in [57]. Let w min E (C) be the minimum Euclidean weight of nonzero codewords in C, i.e.,…”

Section: Nested Lattices Via Construction Amentioning

confidence: 99%

An algebraic approach to physical-layer network coding

Feng

Silva

Kschischang

2010

2010 IEEE International Symposium on Information Theory

132

202

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This dissertation studies a family of the state-of-the-art PNC schemes called compute-and-forward (C&F). C&F was originally proposed and studied from an information-theoretic perspective. As such, it typically relies on several strong assumptions: very long block length, almost unbounded complexity, perfect channel state information, and no decoding errors at the relays; its benefit is often analyzed for simple network configurations. The aim of this dissertation is two-fold: first, to relax the above assumptions while preserving the performance of C&F, and second, to understand the benefit of C&F in more realistic network scenarios, such as random-access wireless networks.There are four main results in this dissertation. First, an algebraic framework is developed, which establishes a direct connection between C&F and module theory. This connection allows us to systematically design lattice codes for C&F with controlled block length and complexity. In particular, explicit design criteria are derived, concrete design examples are provided, and it is shown that nominal coding gains from 3 to 7.5 dB can be obtained with relatively short block length and reasonable decoding complexity. Second, a new C&F scheme is proposed, which, unlike conventional C&F schemes, does not require any channel state information (CSI). It is shown that this CSI-free scheme achieves, for a certain class of lattice codes, almost the same throughput as its CSI-enabled counterpart. Third, an end-to-end error control mechanism is designed, which effectively mitigates decoding errors introduced at wireless relays. In particular, the end-to-end error control problem is modeled as a finite-ring matrix channel problem, for which tight capacity bounds and capacity-approaching schemes are provided. The final part of this dissertation studies the benefit of C&F in random-access wireless networks. In particular, it is shown that C&F significantly improves the network throughput and delay performance of slotted-ALOHA-based random-access protocols.ii Acknowledgements This thesis is the result of the support, help, and advice that I have received from an amazing group of scholars and students at the University of Toronto. Without them this dissertation wouldn't have been possible. First, I feel very fortunate to have two great supervisors: Prof. Frank R. Kschischang and Prof.Baochun Li. As an advisor, Frank has consistently exceeded my expectation. He is not only a brilliant researcher and excellent teacher, but also truly a kind and caring person. He has given me a tremendous amount in the past six years, I can only thank him for a subset. In particular, I have benefited greatly from his focus on fundamental problems, his taste for beautiful mathematics, his emphasis on clarity and grace, and his dedication in developing his students to their full potential. I am very grateful to Frank for making my PhD journey so exciting and rewarding! Frank also has a great sense of humor.I will always remember how much fun we had together during our endless discus...

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“…Clearly, the Euclidean weight of c is equivalent to the 2-norm of c defined in [57]. Let w min E (C) be the minimum Euclidean weight of nonzero codewords in C, i.e.,…”

Section: Nested Lattices Via Construction Amentioning

confidence: 99%

An algebraic approach to physical-layer network coding

Feng

Silva

Kschischang

2010

2010 IEEE International Symposium on Information Theory

132

202

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“…In this line of development, Rush and Sloane [14], [15] observed that applying a version of the Varshamov-Gilbert bound to certain linear codes over GF used with "Construction A" [16] proves the existence of lattices with the same asymptotic sphere-packing density as those known earlier from the Minkowski-Hlawka theorem. For large , this is still the best existence result known for such packings.…”

Section: Introductionmentioning

confidence: 99%

“…Then, for any , for all sufficiently small , and all sufficiently large primes , the arithmetic average of over all lattices is bounded by (14) where is any balanced set of linear codes over and where is the fundamental volume of the lattices . Proof: (See also the analogous proof of Theorem 9 in the Appendix.)…”

Section: A Ramdom Coding Theorem For Latticesmentioning

confidence: 99%

Averaging bounds for lattices and linear codes

Loeliger

1997

IEEE Trans. Inform. Theory

222

309

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Abstract-General random coding theorems for lattices are derived from the Minkowski-Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski-Hlawka theorem itself is obtained as the limit, for p ! 1, of a simple lemma for linear codes over GF (p) used with p-level amplitude modulation. The relation between the combinatorial packing of solid bodies and the information-theoretic "soft packing" with arbitrarily small, but positive, overlap is illuminated. The "softpacking" results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda-Poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1=2 log 2 (P=N ).

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“…Conjuntos K de interesse na literatura incluem poliominós [4,11], politopos-cruz [3] (istó e K = {x ∈ R n : x 1 = R}) e bolas na norma l p (K = B n p (R)) [10]. Bolas na norma l p não ladrilham R n , exceto nos casos p = 1 (n = 2) e p = ∞ (qualquer n), portanto os problemas associados a esses conjuntos dizem respeito ao arranjo mais eficiente, ou com maior densidade de empacotamento, no espaço.…”

Section: Preliminares E Resultados Anterioresunclassified

Ladrilhamentos por poliominós na norma lp

Campello¹,

Jorge²,

Strapasson³

et al. 2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo: O objetivo deste trabalhoé estudar ladrilhamentos de R n por poliominós associados a norma l p . No caso p = 1, a Conjectura de Golomb-Welch, em aberto por cerca de 44 anos, propõe que não existe ladrilhamento por tais poliominós, exceto em casos muito especiais. Aqui estamos interessados em discutir a existência de ladrilhamentos no caso geral 1 ≤ p ≤ ∞. Apresentamos resultados não-existenciais de ladrilhamentos para diferentes valores de p, bem como um estudo da forma e das propriedades combinatórias dos poliominós associados.Palavras-chave: Reticulados, Poliominós, Códigos Perfeitos, Métrica de Lee IntroduçãoUm poliominó em R né uma figura conexa formada pela união finita de cubos de lado unitário de tal maneira que pares de cubo são disjuntos ou compartilham uma face (de dimensão k < n). Um poliominó pode ser construído através da união de cubos centrados em pontos de Z n contidos em uma bola na norma l p , como mostrado na Fig. 1.Além do interesse puramente combinatório, o estudo de ladrilhamentos de R n por poliominós surge naturalmente no contexto de códigos corretores de erros. A conexão mais antigaé através de códigos na métrica de Lee, proposta por Golomb e Welch em [4]. No artigo em questão, os autores mostram que códigos perfeitos na métrica de Lee em Z n estão associados a ladrilhamentos de R n por poliominós na norma l 1 . A partir daí,é enunciada a seguinte conjectura: não existem códigos perfeitos na métrica de Lee para raio R > 1 e n > 2. Apesar de diversas tentativas e * Este trabalho teve o apoio da FAPESP proc.

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An improvement to the Minkowski‐Hiawka bound for packing superballs

Cited by 44 publications

References 7 publications

An algebraic approach to physical-layer network coding

An algebraic approach to physical-layer network coding

Averaging bounds for lattices and linear codes

Ladrilhamentos por poliominós na norma lp

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