Low-density parity-check (LDPC) codes have been shown to have good error correcting performance approaching Shannon’s limit. Good error correcting performance enables efficient and reliable communication. However, a LDPC code decoding algorithm needs to be executed efficiently to meet cost, time, power and bandwidth requirements of target applications. The constructed codes should also meet error rate performance requirements of those applications. Since their rediscovery, there has been much research work on LDPC code construction and implementation. LDPC codes can be designed over a wide space with parameters such as girth, rate and length. There is no unique method of constructing LDPC codes. Existing construction methods are limited in some way in producing good error correcting performing and easily implementable codes for a given rate and length. There is a need to develop methods of constructing codes over a wide range of rates and lengths with good performance and ease of hardware implementability. LDPC code hardware design and implementation depend on the structure of target LDPC code and is also as varied as LDPC matrix designs and constructions. There are several factors to be considered including decoding algorithm computations,processing nodes interconnection network, number of processing nodes, amount of memory, number of quantization bits and decoding delay. All of these issues can be handled in several different ways. This thesis is about construction of LDPC codes and their hardware implementation. LDPC code construction and implementation issues mentioned above are too many to be addressed in one thesis. The main contribution of this thesis is the development of LDPC code construction methods for some classes of structured LDPC codes and techniques for reducing decoding time. We introduce two main methods for constructing structured codes. In the first method, column-weight two LDPC codes are derived from distance graphs. A wide range of girths, rates and lengths are obtained compared to existing methods. The performance and implementation complexity of obtained codes depends on the structure of their corresponding distance graphs. In the second method, a search algorithm based on bit-filing and progressive-edge growth algorithms is introduced for constructing quasi-cyclic LDPC codes. The algorithm can be used to form a distance or Tanner graph of a code. This method could also obtain codes over a wide range of parameters. Cycles of length four are avoided by observing the row-column constraint. Row-column connections observing this condition are searched sequentially or randomly. Although the girth conditions are not sufficient beyond six, larger girths codes were easily obtained especially at low rates. The advantage of this algorithm compared to other methods is its flexibility. It could be used to construct codes for a given rate and length with girths of at least six for any sub-matrix configuration or rearrangement. The code size is also easily varied by increasing or decreasing sub-matrix size. Codes obtained using a sequential search criteria show poor performance at low girths (6 and 8) while random searches result in good performing codes. Quasi-cyclic codes could be implemented in a variety of decoder architectures. One of the many options is the choice of processing nodes interconnect. We show how quasi-cyclic codes processing could be scheduled through a multistage network. Although these net-works have more delay than other modes of communication, they offer more flexibility at a reasonable cost. Banyan and Benes networks are suggested as the most suitable networks. Decoding delay is also one of several issues considered in decoder design and implementation. In this thesis, we overlap check and variable node computations to reduce decoding time. Three techniques are discussed, two of which are introduced in this thesis. The techniques are code matrix permutation, matrix space restriction and sub-matrix row-column scheduling. Matrix permutation rearranges the parity-check matrix such that rows and columns that do not have connections in common are separated. This techniques can be applied to any matrix. Its effectiveness largely depends on the structure of the code. We show that its success also depends on the size of row and column weights. Matrix space restriction is another technique that can be applied to any code and has fixed reduction in time or amount of overlap. Its success depends on the amount of restriction and may be traded with performance loss. The third technique already suggested in literature relies on the internal cyclic structure of sub-matrices to achieve overlapping. The technique is limited to LDPC code matrices in which the number of sub-matrices is equal to row and column weights. We show that it can be applied to other codes with a lager number of sub-matrices than code weights. However, in this case maximum overlap is not guaranteed. We calculate the lower bound on the amount of overlapping. Overlapping could be applied to any sub-matrix configuration of quasi-cyclic codes by arbitrarily choosing the starting rows for processing. Overlapping decoding time depends on inter-iteration waiting times. We show that there are upper bounds on waiting times which depend on the code weights. Waiting times could be further reduced by restricting shifts in identity sub-matrices or using smaller sub-matrices. This overlapping technique can reduce the decoding time by up to 50% compared to conventional message and computation scheduling. Techniques of matrix permutation and space restriction results in decoder architectures that are flexible in LDPC code design in terms of code weights and size. This is due to the fact that with these techniques, rows and columns are processed in sequential order to achieve overlapping. However, in the existing technique, all sub-matrices have to be processed in parallel to achieve overlapping. Parallel processing of all code sub-matrices requires the architecture to have the number of processing units at least equal to the number sub-matrices. Processing units and memory space should therefore be distributed among the sub-matrices according to the sub-matrices arrangement. This leads to high complexity or inflexibility in the decoder architecture. We propose a simple, programmable and high throughput decoder architecture based on matrix permutation and space restriction techniques. / Thesis(Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2007

Thesis (Ph.D.)--University of Melbourne, Dept. of Computer Science and Software Engineering, Faculty of Engineering, 2007. / Typescript. Includes bibliographical references (leaves 171-179).

Mass., Massachusetts Inst. of Technology, Diss.--Cambridge, 2001. / Literaturverz. S. [337] - 347.

Thesis (M.Eng.(Electronic Engineering))--University of Pretoria, 2008. / Summaries in Afrikaans and English. Includes bibliographical references (leaves [92]-99).

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 163-177). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 163-177). Also available on the Internet.

Thesis (M.App.Sc.) - Carleton University, 2007. / Includes bibliographical references (p. 101-107). Also available in electronic format on the Internet.

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains xix, 136 p.; also includes graphics (some col.). Includes bibliographical references (p. 129-136). Available online via OhioLINK's ETD Center

Orientador: Marcelo Firer / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T06:58:11Z (GMT). No. of bitstreams: 1 Spreafico_MarcosViniciusPereira_M.pdf: 558857 bytes, checksum: a9d033bd1132fd1bc42fcb1aa2296ef5 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho fazemos uma apresentação dos espaços poset, introduzidos por Brualdi (1995), apresentamos os conceitos necessarios da teoria de conjuntos parcialmente ordenados e da teoria de codigos. Trabalhamos com uma questão de caráter amplo e estrutural deste contexto, o problema da determinação da ordem atraves da distribuição de pesos. A distribuição de pesos é essencialmente o conjunto das cardinalidades das esferas métricas e a pergunta que se coloca é em que medida este invariante determina a métrica em questão. Demonstramos que para as classes de codigos, cadeia, anticadeia, coroa e hierárquico, classes importantes no contexto da teoria de codigos, o problema possui uma resposta positiva e justificamos algumas conjecturas que relacionam este problema ao da reconstrução de grafos. / Abstract: In this work, we introduce the concept of poset codes (Brualdi - 1995) and in this context we study the weight distribution problem, presenting the necessary concepts of the partially ordered set and error correcting codes theory. The weight distribution is the cardinality of metric-spheres in finite dimensional vector space over a finite field endowed with a poset metric. The weight distribution problem asks for conditions to ensure that the weight distribution determines the metric. In this work we show that the weight distribution of some families of posets, namely the classes of anti-chain, chain, crown and hierarchical posets, determines the metric. We also show that the weight distribution determines some known invariants of posets. Finally, we present some conjectures relating the weight distribution problem and the reconstruction problem of graphs. / Mestrado / Mestre em Matemática

Orientadores: Sueli Irene Rodrigues Costa, Simone Maria de Moraes / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T17:40:32Z (GMT). No. of bitstreams: 1 Silva_AndersonTiagoda_M.pdf: 781127 bytes, checksum: 22670fa6bf0a9cc9e4533bcc2ef952d8 (MD5) Previous issue date: 2007 / Resumo: Este trabalho está dividido essencialmente em quatro tópicos. O primeiro capítulo é dedicado a uma introdução à teoria dos códigos corretores de erros com algumas propriedades e exemplos. No segundo capítulo abordamos reticulados e suas propriedades com foco na análise do quociente de reticulados gerando grafos em toros planares, grafos circulantes obtidos através de quociente de reticulados e ladrilhamentos associados. O terceiro capítulo é dedicado a códigos esféricos, com ênfase na obtenção de códigos ótimos. Foram introduzidos alguns limitantes importantes como o de Rankim, e a demonstração de que alguns códigos esféricos como o simplex e biortogonal são ótimos. No capítulo quatro apresentamos uma construção de reticulados através de códigos binários e também a construção de códigos esféricos a partir de reticulados que possuem sub-reticulados com base ortogonal. Analisamos o caso especial do reticulado BCC que é o de melhor densidade no espaço e pode ser gerado por código binário. Mostramos que o quociente deste por um sub reticulado especial produz o melhor código esférico associado ao grupo comutativo Z2 2 ×Z4 . Também identificamos o reticulado que é associado ao melhor código de grupo comutativo de 16 elementos em R6 / Abstract: In this work it is presented through examples a connection between inary codes, lattices and spherical codes. A brief introduction to coding theory, properties and examples is included in the first chapter. In Chapter 2 lattices are approached with focus on the quotient of lattices, graphs on flat tori and connections with circulant graphs. An introduction to spherical codes and some of their bounds, as the Ranking bound, are described in Chapter 3. Finally in Chapter 4 the three topics above are connected. The construction of lattices from linear binary codes and the construction of spherical codes from the lattices which have orthogonal sub-lattices are presented. We analyze specifically the case of the three dimensional BCC lattice, which has the best packing density for this dimension, and show that a quotient of this lattice give rise to the best spherical code associate to the commutative group Z2 2 ×Z4. We also identify the lattice which is associate to the best commutative group code with 16 elements in em R6 / Mestrado / Mestre em Matemática