The aim of coding theory is to design codes which can achieve the fundamental limits of communication [52] and yet are simple to implement. On average randomly constructed codes can achieve this goal, but with a decoding complexity that is impractical. Consequently, highly structured codes with practical decoding algorithms have been extensively studied. Unfortunately the vast majority of these codes do not approach capacity. Recent advances involving simple 'random like' codes with practical iterative decoding algorithms have closely approached capacity as the blocklength increases. This thesis investigates single parity check (SPC) product codes and introduces the class of randomly interleaved (RI) SPC product codes. It will be shown that RI SPC product codes can asymptotically force the probability of error to zero, at code rates up to capacity, for almost all codewords. Furthermore the structure of these codes allows a very simple, sub-optimal, iterative decoding algorithm to be used. This thesis also derives an asymptotic analysis on SPC product codes from the decoding point of view. It is shown that the probability of error can be driven to zero, as the blocklength increases, for signal to noise ratios within 2dB of capacity on the additive white Gaussian noise (AWGN) channel. Simulation results for both SPC and RI SPC product codes in an AWGN channel are presented. These results indicate that RI SPC product codes perform very well, typically within 1.5dB of capacity over a wide range of blocklengths and code rates. Further analysis on the weight enumerator of finite length RI SPC product codes is used to confirm the error floor of these codes. Extensions to parallel and serially concatenated SPC product codes are also investigated. Simulation results show an advantageous trade-off between code rate, blocklength and performance for three dimensional parallel concatenated SPC product codes. The design of irregular SPC product codes is also considered, and some simulation results are presented.

International Telemetering Conference Proceedings / October 22-25, 2001 / Riviera Hotel and Convention Center, Las Vegas, Nevada / Telemetry for Earth-Observation missions is characterized by very high data rates and stringent requirements. Channel codes both power and bandwidth efficient must be used to improve downlink performance and to achieve the very low values of error rates needed at the received side. In this paper, we review and analyzed three codes of possible interest for these applications: turbo codes, serial turbo codes and product codes. These schemes are evaluated and compared both by simulation and analytical techniques. A particular attention is devoted to complexity, a key issue for practical implementation at high data rates.

Turbo códigos têm se tornado um importante ramo na pesquisa de codificação de canal e já foram adotados como padrão para a terceira geração de comunicação móvel. Devido ao seu alto ganho de codificação, os turbo códigos são vistos como fortes candidatos a serem adotados como padrão das futuras gerações de redes sem fio. Esse esquema de codificação é baseado na decodificação iterativa, onde decodificadores de entrada e saída suaves produzem refinamento da informação a cada iteração. Essa dissertação apresenta resultados de um estudo comparativo entre dois esquemas de codificação: turbo códigos de bloco e turbo códigos convolucionais. Os resultados mostram que os dois esquemas de codificação têm desempenho funcional complementar, sendo importante a especificação de um alvo em termos de relação sinal/ruído ou taxa de erro de bits para a escolha do esquema de codificação mais adequado. Com o mesmo modelo em linguagem de programação C foi feita uma exploração do algoritmo visando diminuição do consumo de potência. Essa exploração em parte foi feita segundo uma metodologia de exploração sistemática das possibilidades de transferência e armazenamento de dados (DTSE). Com a exploração, a redução total de consumo de potência para o armazenamento de dados foi estimada em 34%. / Turbo codes have become an important branch on channel coding research and have been adopted as standard in the third generation of mobile communication systems. Due to their high coding gain, turbo codes are expected to be part of the next generations of wireless networks standards. This coding scheme is based on iterative decoding, as soft input/soft output decoders produce an information refinement in each iteration. This dissertation shows the results of a comparative performance study of two different turbo coding schemes: block turbo codes and convolutional turbo codes. The results obtained show that the two schemes have complementary performance. It is necessary to specify a target in terms of bit error rate or signal/noise ratio. With the same C model an exploration aiming at reducing power consumption was done. Part of this exploration was done following a systematic methodology of data transfer and storage exploration (DTSE). With this exploration, a reduction of 34% on power consumption was estimated.

Turbo códigos têm se tornado um importante ramo na pesquisa de codificação de canal e já foram adotados como padrão para a terceira geração de comunicação móvel. Devido ao seu alto ganho de codificação, os turbo códigos são vistos como fortes candidatos a serem adotados como padrão das futuras gerações de redes sem fio. Esse esquema de codificação é baseado na decodificação iterativa, onde decodificadores de entrada e saída suaves produzem refinamento da informação a cada iteração. Essa dissertação apresenta resultados de um estudo comparativo entre dois esquemas de codificação: turbo códigos de bloco e turbo códigos convolucionais. Os resultados mostram que os dois esquemas de codificação têm desempenho funcional complementar, sendo importante a especificação de um alvo em termos de relação sinal/ruído ou taxa de erro de bits para a escolha do esquema de codificação mais adequado. Com o mesmo modelo em linguagem de programação C foi feita uma exploração do algoritmo visando diminuição do consumo de potência. Essa exploração em parte foi feita segundo uma metodologia de exploração sistemática das possibilidades de transferência e armazenamento de dados (DTSE). Com a exploração, a redução total de consumo de potência para o armazenamento de dados foi estimada em 34%. / Turbo codes have become an important branch on channel coding research and have been adopted as standard in the third generation of mobile communication systems. Due to their high coding gain, turbo codes are expected to be part of the next generations of wireless networks standards. This coding scheme is based on iterative decoding, as soft input/soft output decoders produce an information refinement in each iteration. This dissertation shows the results of a comparative performance study of two different turbo coding schemes: block turbo codes and convolutional turbo codes. The results obtained show that the two schemes have complementary performance. It is necessary to specify a target in terms of bit error rate or signal/noise ratio. With the same C model an exploration aiming at reducing power consumption was done. Part of this exploration was done following a systematic methodology of data transfer and storage exploration (DTSE). With this exploration, a reduction of 34% on power consumption was estimated.

The growing popularity of a class of linear block codes called the low-density parity-check (LDPC) codes can be attributed to the low complexity of the iterative decoders, and their potential to achieve performance very close to the Shannon capacity. This makes them an attractive candidate for ECC applications in communication systems. This report proposes methods to systematically construct regular and irregular LDPC codes.A class of regular LDPC codes are constructed from incidence structures in finite geometries like projective geometry and affine geometry. A class of irregular LDPC codes are constructed by systematically splitting blocks of balanced incomplete block designs to achieve desired weight distributions. These codes are decoded iteratively using message-passing algorithms, and the performance of these codes for various channels are presented in this report.The application of iterative decoders is generally limited to a class of codes whose graph representations are free of small cycles. Unfortunately, the large class of conventional algebraic codes, like RS codes, has several four cycles in their graph representations. This report proposes an algorithm that aims to alleviate this drawback by constructing an equivalent graph representation that is free of four cycles. It is theoretically shown that the four-cycle free representation is better suited to iterative erasure decoding than the conventional representation. Also, the new representation is exploited to realize, with limited success, iterative decoding of Reed-Solomon codes over the additive white Gaussian noise channel.Wiberg, Forney, Richardson, Koetter, and Vontobel have made significant contributions in developing theoretical frameworks that facilitate finite length analysis of codes. With an exception of Richardson's, most of the other frameworks are much suited for the analysis of short codes. In this report, we further the understanding of the failures in iterative decoders for the binary symmetric channel. The failures of the decoder are classified into two categories by defining trapping sets and propagating sets. Such a classification leads to a successful estimation of the performance of codes under the Gallager B decoder. Especially, the estimation techniques show great promise in the high signal-to-noise ratio regime where the simulation techniques are less feasible.

Orientador: Jaime Portugheis / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-03T15:25:23Z (GMT). No. of bitstreams: 1 Cunha_DanielCarvalhoda_M.pdf: 3453050 bytes, checksum: 13fe48dc7912e447d190ba1ba03e67dd (MD5) Previous issue date: 2003 / Resumo: Neste trabalho desenvolvemos um estudo sobre a aplicação de um algoritmo de de-codificação iterativa (turbo) de códigos produto em sistemas de transmissão que utilizam dois tipos de canais não-gaussianos: o canal CDMA de salto rápido em freqüência e o canal com ruído impulsivo juntamente com modulação BPSK. Para ambos os casos, propusemos uma maneira de calcular a confiabilidade das decisões tomadas após cada iteração, que leva em conta as densidades de probabilidade dos canais considerados. No caso do canal CDMA, os resultados mostraram que, para códigos de taxas baixa, média e alta, o desempenho do sistema melhora com o aumento do número de iterações de decodificação, o que comprova a validade da proposta. No caso de canais com ruído impulsivo, além da validade da proposta, ve-rificamos que o conhecimento da relação sinal-ruído do canal pelo decodificador pode melhorar o desempenho do sistema. Entretanto, para alguns valores dos parâmetros do ruído impulsivo, verificamos que uma escolha antecipada dos fatores a e ? do algoritmo iterativo que garanta a convergência do mesmo, pode se tornar uma tarefa bastante árdua / Abstract: In this work we develop a study of iterative (turbo) algorithms for product codes applied to transmission systems which utilize two types of non-Gaussian channels: a fast frequency hopping CDMA channel and impulsive noise channels with BPSK modulation. In both cases, we propose a way to calculate the reliability of decisions made after each iteration taking into account the probability density functions of the considered channels. In the case of a CDMA channel, for low, medium and high rate codes, the results showed that the performance of the systems becomes better as the number of decoding iterations increases, a result that validates the proposal. In the case of impulsive noise channels, we verified not only the validity of the proposal but also that the decoder knowledge of channel signal-to-noise ratios can improve the performance of the system. However, for some specific values of impulsive noise parameters, we verified that to give a predefined value to a and ? factors for the iterative algorithm which guarantees its convergence, can become a considerable difficult task / Mestrado / Mestre em Engenharia Elétrica

ITC/USA 2005 Conference Proceedings / The Forty-First Annual International Telemetering Conference and Technical Exhibition / October 24-27, 2005 / Riviera Hotel & Convention Center, Las Vegas, Nevada / Hardware implementing forward error correction (FEC) is currently available for utilization by the airborne telemetry system designer. This paper will discuss the potential benefits along with drawbacks when using this technology. Laboratory testing is supplemented with real-world flight testing. Performance results comparing FEC and non-FEC systems are presented for both IRIG-106 Pulse Code Modulation/Frequency Modulation, PCM/FM, (or Continuous Phase Frequency Shift Keying, CPFSK, with filtering, or ARTM Tier 0) and Shaped Offset Quadrature Phase Shift Keying, Telemetry Group version (SOQPSK-TG or ARTM Tier I) waveforms.

Cette thèse étudie les performances d'un réseau de capteurs sans-fil coopératif, basé sur un codage réseau algébrique linéaire appliqué au relais. On considère un schéma coopératif basé sur le code produit en bloc où un grand nombre de sources transmettent des données indépendantes codées par un premier code en bloc vers un seul destinataire avec l'aide du relais. Dans ce schéma, le relais applique le codage réseau algébrique linéaire en utilisant un code correcteur d'erreur systématique linéaire en bloc sur les mots de code source détectés par le relais. Seule, la redondance générée par le relais est transférée vers le destinataire. Le destinataire observe un mot de code produit en bloc en combinant les observations des sources et du relais. Premièrement, on aborde la coopération en mode time-division multiple-access (TDMA) et suppose un canal source-relais bruité. On analyse les probabilités théoriques à l'entrée et à la sortie du relais pour différente stratégies de détection au relais. On établit aussi une borne théorique sur la probabilité d'erreur de trame pour le schéma coopératif proposé. Puis on évalue la coopération multi-relais afin de traiter la corrélation des erreurs dans la redondance générée par le relais. Différents configurations de coopération (mono ou multi-relais avec différentes stratégies au relais) sont comparées. On montre que la liaison source-relais est le maillon faible du réseau. On évalue ensuite la capacité du réseau sous la condition de taille finie du code. Ensuite, on étudie la coopération basée sur la technique code-division multiple-access (CDMA) appliqué au relais de telle sorte que le signal du relais est avec ceux des sources dans la même bande de fréquence radio. Pour simplifier l'analyse, on suppose un canal source-relais sans erreur. On propose une procédure de décodage itératif avec la neutralisation de l'interférence. On formule deux cas de coopération basé sur CDMA: TDMA-CDMA avec répartition orthogonale dans le temps entre les sources et FDMA-CDMA avec allocation de sous-bandes de fréquence disjointes pour les sources. Le ratio d'allocation d'énergie entre les sources et le relais est évalué en utilisant les simulations.

We study the Hamming distance and the structure of repeated-root cyclic codes, and their generalizations to constacyclic and polycyclic codes, over finite fields and Galois rings. We develop a method to compute the Hamming distance of these codes. Our computation gives the Hamming distance of constacyclic codes of length $np^s$ in many cases. In particular, we determine the Hamming distance of all constacyclic, and therefore cyclic and negacyclic, codes of lengths p^s and 2p^s over a finite field of characteristic $p$. It turns out that the generating sets for the ambient space obtained by torsional degrees and strong Groebner basis for the ambient space are essentially the same and one can be obtained from the other. In the second part of the thesis, we study matrix product codes. We show that using nested constituent codes and a non-constant matrix in the construction of matrix product codes with polynomial units is a crucial part of the construction. We prove a lower bound on the Hamming distance of matrix product codes with polynomial units when the constituent codes are nested. This generalizes the technique used to construct the record-breaking examples of Hernando and Ruano. Contrary to a similar construction previously introduced, this bound is not sharp and need not hold when the constituent codes are not nested. We give a comparison of this construction with a previous one. We also construct new binary codes having the same parameters, of the examples of Hernando and Ruano, but non-equivalent to them.

This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed. A new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD. The optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution.