Στην εργασία αυτή, μελετώνται διαφορετικοί αποκωδικοποιητές για κώδικες Reed-Solomon. Αφού γίνει μια εισαγωγή στο σύστημα επικοινωνίας, στα πεπερασμένα σώματα και στη θεωρία κωδίκων, παρουσιάζονται αλγόριθμοι που υλοποιούν αποκωδικοποιητές περιορισμένης απόστασης, λίστας και soft decoders. Συγκεκριμένα, μελετώνται κλασσικοί αποκωδικοποιητές, αποκωδικοποιητές βασισμένοι στον αλγόριθμο Guruswami-Sudan, και αποκωδικοποιητές βασισμένοι στον αλγόριθμο Koetter-Vardy. Η σύγκριση γίνεται ως προς τη διορθωτική ικανότητα και τη χρονική πολυπλοκότητα. Στα πλαίσια της εργασίας αυτής, σε συνέχεια της μελέτης υπαρχόντων αποκωδικοποιητών προτείνεται ένα είδος προσαρμοστικού αποκωδικοποιητή: φέρει την ίδια διορθωτική ικανότητα με έναν αποκωδικοποιητή λίστας, αλλά σε ορισμένες περιπτώσεις έχει ικανοποιητικά μικρότερη χρονική πολυπλοκότητα. Επίσης γίνεται μία ποιοτική διερεύνηση για το πότε πρέπει να προτιμάται ο προσαρμοστικός αποκωδικοποιητής. Προτείνονται δύο διαφορετικοί προσαρμοστικοί αποκωδικοποιητές. Η πρώτη εκδοχή, είναι μία διάταξη με δύο αποκωδικοποιητές. Αρχικά το ληφθέν διάνυσμα από το κανάλι, εισέρχεται ως είσοδος σε έναν κλασσικό αποκωδικοποιητή. Αν ο κλασσικός παρουσιάσει αδυναμία αποκωδικοποίησης, τότε επιχειρεί να διορθώσει το ίδιο διάνυσμα ένας αποκωδικοποιητής λίστας. Η δέυτερη εκδοχή, μοιάζει με την πρώτη, με τη διαφορά ότι ο δεύτερος αποκωδικοποιητής χρησιμοποιεί αποτελέσματα που έχουν ήδη υπολογιστεί από τον πρώτο, μειώνοντας έτσι την συνολική χρονική πολυπλοκότητα, σχετικά με την πρώτη εκδοχή. / In this thesis we study a family of linear block codes, the Reed-Solomon(RS) codes. RS codes are q-ary codes over some finite field GF(q). Therefore, they have strong burst-error correction capability, because they deal with groups of bits instead of single bits. Initially, we describe the employed communication system model, and introduce basic from finite field theory (construction and identities) and coding theory. After describing the RS encoding process, we describe different RS decoders (bounded distance decoder, list decoder and soft-input decoder). Specifically, in addition to traditional approaches, we focus on decoders based on Guruswami-Sudan and Koetter – Vardy algorithms. We compare them according to their complexity and performance, both in theory and experimentally. Furthermore, in this thesis, we propose an adaptive decoder, which has the same performance as a list decoder but in some cases it achieves much lower average time complexity. We present the experimental results, highlighting the cases where the adaptive decoder outperforms conventional decoders. The adaptive decoder comes in two different types. The first type is a system with two different decoders: the low complexity decoder attempts to decode a received vector and in case of decoding failure the subsequent decoder tries to decode the same received vector. The second type resembles the general organization of the above system; the second decoder re-uses intermediate results, previously computed by the first one.

Este trabalho investiga a decodificação turbo de códigos produto construídos a partir de códigos de Reed-Solomon sobre constelações QAM. Por meio da geometria euclidiana da constelação utilizada e das relações dos elementos de campo de Galois utilizados na codificação Reed-Solomon com os símbolos pertencentes ao alfabeto de modulação, é proposto um algoritmo de decodificação que utiliza a localização geométrica dos símbolos recebidos no processo de decodificação turbo. Tanto a primeira etapa da decodificação SISO baseada no algoritmo de Chase, como a extração da informação extrínseca baseada no algoritmo proposto por Pyndiah, são tratadas pelo uso de uma abordagem geométrica sobre o espaço euclidiano. Os resultados de simulação utilizando o algoritmo proposto coincidem com aqueles usando a decomposição e análise pragmática binária, sendo que a abordagem apresentada conduz a simplificações e otimizações em relação à metodologia binária no que ser refere à implementação dos possíveis esquemas de decodificação. / This work investigates the turbo decoding of product codes built using Reed- Solomon codes in QAM constellations. Using the euclidian geometry of the constellation and the relations of the Galois field elements used in the Reed-Solomon code and the modulation alphabet, a decoding algorithm for the turbo decoding process is proposed using the geometric localization of the received symbol. Both the first stage of the SISO decoding based on the Chase algorithm, as the calculation of the extrinsic information based on the Pyndiah algorithm, are treated using the geometric approach in the euclidian space. The simulations show that this algorithm gives similar performance results as the pragmatic binary decomposition approach, and leads to simplifications and optimizations in decoding implementations schemes compared to the former.

[pt] Há, nos dias atuais, uma crescente demanda pelo transporte de video sobre IP, i.e., para distribuição de conteúdo pela Internet, por serviços de IPTV em definição padrão e em alta definição e, mesmo para uso interno nas redes de emissoras tradicionais de televisão, que transportam contribuições de elevada qualidade para seus programas. Em tais aplicações, o conteúdo dos programas é transportado usando MPEG-2 ou MPEG-4, sob a forma de MPEG-2 Transport Streams, encapsulados com protocolos tais como RTP, UDP e IP. As redes IP, que são modelizadas como Redes com Apagamento de Pacotes (PEC) não foram, no entanto, concebidas para o transporte de mídias em tempo real, esbarra portanto em problemas comuns como perdas de pacotes e jitter, gerando perturbações que se refletem na recepção do conteúdo. Os métodos tradicionais para superar estas dificuldades, como por exemplo, os que se baseiam em retransmissões usando protocolos ARQ (Automatic Repeat on Request), não são uma solução eficiente para proteger a transmissão de multimídia em tempo real. A proteção de multimídia transmitida em tempo real via IP recorre, neste caso, aos códigos para canal. Há códigos para canal recomendados em RFC s e Padrões, usados amplamente pelos fabricantes de equipamento. Os modernos Códigos Fontanais, possuem características atraentes para o transporte de conteúdos multimídia em tempo real. Neste trabalho, simulações são realizadas, onde o conteúdo encapsulado em Transport Stream, é protegido com Códigos Fontanais antes do encapsulamento para o envio através da rede. A título de comparação, o experimento é realizado também usando outros códigos para canal recomendados. Para realizar a comparação são usadas medições padronizadas do Transport Stream, medições objetivas como artefatos de blocagem e finalmente uma análise subjetiva do conteúdo recebido é usada. O trabalho conclui com a proposta de um Codificador de canal adaptável para Transport Stream. / [en] There is a growing demand for the transport of video over IP today, i.e., for content distribution over the Internet, IPTV services in Standard and High Definition, or even inside traditional broadcasters networks, transporting broadcast quality contributions to the main program. In such applications, the source encoded MPEG-2 or -4 content is transported in the form of MPEG-2 Transport Streams, encapsulated over network protocols. However, IP networks, which can be modeled as Packet Erasure Networks (PEC), were not originally designed for the transport of real time media. There are problems, such as packet drops and jitter, which generate severe impairments in the content that is decoded at the reception. Traditional methods for overcoming these problems, as for example retransmissions performed by Automatic Repeat Request (ARQ) protocols, are not suitable for real-time multimedia protection. Channel coding is the solution of choice for protecting real-time multimedia over IP. There are channel coding schemes specified in open recommendations and Standards, widely adopted by equipment vendors today. Fountain Codes present very attractive characteristics for the transport of real-time multimedia. In the present work, simulations with a Fountain code, protecting Transport Stream contents prior to network encapsulation, are presented. The experiment if repeated with other channel coding techniques commonly employed today. In order to analyze the decoded contents and obtain comparative results, standardized Transport Stream measurements, objective Blocking Artifacts measurements and subjective analysis of the decoded samples are employed. This work is concluded with the proposal of a Transport Stream Adaptive channel encoder, that is explained in Appendix-B.

Orientadores: Dalton Soares Arantes, Júlio César Rodrigues Fernandes de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-27T18:41:53Z (GMT). No. of bitstreams: 1 Salvador_ArleyHenrique_M.pdf: 3542397 bytes, checksum: 0847a4269c07c0969394ba0b40491987 (MD5) Previous issue date: 2015 / Resumo: Este trabalho apresenta a implementação de uma arquitetura paralela de um código corretor de erros para aplicações em redes ópticas que utilizam a técnica Foward Error Correction (FEC). O algoritmo FEC especificamente tratado neste trabalho é o Reed-Solomon, que é destinado principalmente a sistemas que sofrem influência causada por erros em rajadas somados ao sinal durante a transmissão, o que o torna adequado para transmissões ópticas. São expostas estruturas seriais de codificador e decodificador FEC e as etapas para convertê-las para uma estrutura paralela. Após descrever as etapas de conversão de uma estrutura serial para paralela é apresentada a estrutura do codificador/decodficador FEC Reed-Solomon RS(255,239) com estrutura paralela para operar em redes ópticas a uma taxa de 100 Gbit/s. Esta descrição exemplificativa é o objetivo principal deste trabalho. A implementação paralela do FEC oferece como vantagem a capacidade de processar os dados de forma rápida, permitindo o emprego desta solução em sistemas com altas taxas de dados. Foi elaborado um ambiente de testes com uma aplicação em redes de transporte óptico, ou Optical Transport Network (OTN). Esta funcionalidade consiste de um Transponder, que tem a função de mapear um cliente de 100 Gigabits Ethernet dentro de uma estrutura de quadro destinado a transmissão de dados em redes ópticas. Deste modo, pôde-se comprovar os resultados e o desempenho da estrutura proposta / Abstract: This paper presents the implementation of a parallel architecture error-correcting code for optical applications that uses the Forward Error Correction (FEC) technique. The FEC algorithm specifically addressed in this work is applied to the Reed-Solomon, which is mainly intended for systems that are harmed by burst errors, which makes it suitable for optical transmissions. A serial FEC encoder/decoder structure and the steps to convert it to a parallel approach are addressed in this work. An example of method that generates an encoder/decoder for a RS(255,239) Reed-Solomon code with parallel structure, able to operate at 100 Gbit/s data rate in optical networks, is also presented. The parallel implementation offers higher FEC processing speeds to handle higher throughputs. A test environment was designed with an application in optical transport networks (OTN). This feature consists a transponder which maps a 100 Gigabit Ethernet client inside an OTN frame structure. With this setup the expected results for the proposed FEC circuitry could be experimentally verified / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

<p>Predmet istraživanja doktorske disertacije je faktorizacija polinoma dve promenljive sa celobrojnim koeficijentima pomoću njima pridruženih Newton-ovih poligona. Formalizacija potrebnog i dovoljnog uslova za postojanje netrivijalne faktorizacije polinoma dve promenljive sa celobrojnim koeficijentima omogućava konstrukciju efektivnog algoritma za faktorizaciju. Konačno, dobijeni teorijski rezultati su primenjeni na dekodiranje jedne klase Reed &ndash; Solomon kodova, miksa dve kodne reči.</p> / <p>The research subject of the thesis is factorization of bivariate polynomials with integer coefficients via associated Newton polygons. Formalization of the necessary and sufficient condition for the existence of a non &ndash; trivial factorization of an arbitrary bivariate polynomial with integer coefficients obtains theoretical basis for construction of an effective factorization algorithm. Finally, these theoretical results are applied in decoding special class of Reed &ndash; Solomon codewords, mixture of two codewords.</p>

International Telemetering Conference Proceedings / October 23-26, 2000 / Town & Country Hotel and Conference Center, San Diego, California / This paper describes the new Telemetry Processor Hardware (TPH) that Avtec Systems has developed for the Deep Space Network (DSN) Telemetry Processor (TLP) system. Avtec is providing the Telemetry Processor Hardware to RTLogic! for integration into the Telemetry Processor system. The Deep Space Network (DSN) is an international network of antennas that supports interplanetary spacecraft missions for exploration of the solar system and the universe. The Jet Propulsion Laboratory manages the DSN for NASA. The TLP system provides the capability to acquire, process, decode and distribute deep space probe and Earth orbiter telemetry data. The new TLP systems will be deployed at each of the three deep-space communications facilities placed approximately 120 degrees apart around the world: at Goldstone, California; near Madrid, Spain; and near Canberra, Australia. The Telemetry Processor Hardware (TPH) supports both CCSDS and TDM telemetry data formats. The TPH performs the following processing steps: soft-symbol input selection and measurement; convolutional decoding; routing to external decoders; time tagging; frame synchronization; derandomization; and Reed-Solomon decoding. The TPH consists of a VME Viterbi Decoder/MCD III Interface board (VM-7001) and a PCI-mezzanine Frame Synchronizer/Reed-Solomon Decoder (PMC- 6130-J) board. The new Telemetry Processor Hardware is implemented using the latest Field Programmable Gate Array (FPGA) technology to provide the density and speed to meet the current requirements as well as the flexibility to accommodate processing enhancements in the future.

ITC/USA 2011 Conference Proceedings / The Forty-Seventh Annual International Telemetering Conference and Technical Exhibition / October 24-27, 2011 / Bally's Las Vegas, Las Vegas, Nevada / This paper proposes a telemetering method consisting of delayed frame time diversity (DFTD) as the inner code and Reed-Solomon (RS) code as the outer code. DFTD is used to transmit a real-time frame together with a time-delayed frame which was saved in the memory during a defined period. The RS code is serially concatenated with DFTD. This method was applied to the design of telemetry units that have been used for over ten flight tests. The data results of the flight test for four cases with no applied code, with DFTD only, with the RS code only, and with both DFTD and the RS code are used to compare the number of error frames. The results also show that the proposed method is very useful and applicable to telemetry applications in a communication environment with a deep fade.

Please download dissertation to view abstract.

M.Ing. (Electrical and Electronic Engineering) / In this study, the problem of correcting burst of adjacent deletions by adapting product codes was investigated. The first step in any digital transmission is to establish synchronization between the sending and receiving nodes. This initial synchronization ensures that the receiver samples the information bits at the correct interval. Unfortunately synchronization is not guaranteed to last for the entire duration of data transmission. Though synchronization errors rarely occur, it has disastrous effects at the receiving end of transmission. These synchronization errors are modelled as either insertions or deletions in the transmitted data. In the best case scenario, these errors are restricted to single bit errors. In the worst case scenario, these errors lead to bursts of bits being incorrect. If these synchronization errors are not detected and corrected, it can cause a shift in the transmitted sequence which in turn leads to loss of synchronization. When a signal is subjected to synchronization errors it is difficult accurately recover the original data signal. In addition to the loss of synchronization, the information transmitted over the channel is also subjected to noise. This noise in the channel causes inversion errors within the signal. The objective of this dissertation is to investigate if an error correction scheme can be designed that has the ability to detect and correct adjacent bursts of deletions and random inversion errors. This error correction scheme needed to make use of a product code matrix structure. This product matrix needed to incorporate both an error correction and synchronization technique. The chosen error correcting techniques were Hamming and Reed-Solomon codes. The chosen synchronization techniques for this project were the marker technique or an adaptation of the Hamming code technique. In order to find an effective model, combinations of these models were simulated and compared. From the research obtained and analyzed in this document it was found that, depending on the desired performance, complexity and code rate, an error correction scheme can be used in the efficient correction of bursts of adjacent deletions by adapting product codes.

Masters Report A report submitted in ful llment of the requirements for the degree of Master of Science (50/50) in the Centre for Telecommunication Access and Services (CeTAS) School of Electrical and Information Engineering Faculty of Engineering and the Built Environment February 2017 / Bounded distance decoding of Reed Solomon (RS) codes involves nding a unique codeword if there is at least one codeword within the given distance. A corrupted message having errors that is less than or equal to half the minimum distance cor- responds to a unique codeword, and therefore will decode errors correctly using the minimum distance decoder. However, increasing the decoding radius to be slightly higher than half of the minimum distance may result in multiple codewords within the Hamming sphere. The list decoding and syndrome extension methods provide a maximum error correcting capability whereby the radius of the Hamming ball can be extended for low rate RS codes. In this research, we study the probability of having unique codewords for (7; k) RS codes when the decoding radius is increased from the error correcting capability t to t + 1. Simulation results show a signi cant e ect of the code rates on the probability of having unique codewords. It also shows that the probability of having unique codeword for low rate codes is close to one. / MT2017