Modified belief propagation decoders applied to non-CSS QLDGM codes.

Quantum technology is becoming increasingly popular, and big companies are starting to invest huge amounts of money to ensure they do not get left behind in this technological race. Presently, qubits and operational quantum channels may be thought of as far-fetched ideas, but in the future, quantum computing will be of critical importance. In this project, it is provided a concise overview of the basics of coding theory and how they can be used in the design of quantum computers. Specifically, Low Density Parity Check (LDPC) codes are focused, as they can be integrated within the stabilizer construction to build effective quantum codes. Following this, it is introduced the specifics of the quantum paradigm and present the most common family of quantum codes: stabilizer codes. Finally, it is explained the codes that have been used in this project, discussing what type of code they are and how they are designed. In this last section, it is also presented the ultimate goal of the project: using modified belief propagation decoders that had previously been tested for QLDPCs, for the proposed non-CSS QLDGM codes of this project

Decoding error-correcting codes via linear programming

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (p. 147-151).Error-correcting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full error-correcting potential of the system is often very complex, especially for modern codes that approach the theoretical limits of the communication channel. In this thesis we investigate the application of linear programming (LP) relaxation to the problem of decoding an error-correcting code. Linear programming relaxation is a standard technique in approximation algorithms and operations research, and is central to the study of efficient algorithms to find good (albeit suboptimal) solutions to very difficult optimization problems. Our new "LP decoders" have tight combinatorial characterizations of decoding success that can be used to analyze error-correcting performance. Furthermore, LP decoders have the desirable (and rare) property that whenever they output a result, it is guaranteed to be the optimal result: the most likely (ML) information sent over the channel. We refer to this property as the ML certificate property. We provide specific LP decoders for two major families of codes: turbo codes and low-density parity-check (LDPC) codes. These codes have received a great deal of attention recently due to their unprecedented error-correcting performance.(cont.) Our decoder is particularly attractive for analysis of these codes because the standard message-passing algorithms used for decoding are often difficult to analyze. For turbo codes, we give a relaxation very close to min-cost flow, and show that the success of the decoder depends on the costs in a certain residual graph. For the case of rate-1/2 repeat-accumulate codes (a certain type of turbo code), we give an inverse polynomial upper bound on the probability of decoding failure. For LDPC codes (or any binary linear code), we give a relaxation based on the factor graph representation of the code. We introduce the concept of fractional distance, which is a function of the relaxation, and show that LP decoding always corrects a number of errors up to half the fractional distance. We show that the fractional distance is exponential in the girth of the factor graph. Furthermore, we give an efficient algorithm to compute this fractional distance. We provide experiments showing that the performance of our decoders are comparable to the standard message-passing decoders. We also give new provably convergent message-passing decoders based on linear programming duality that have the ML certificate property.by Jon Feldman.Ph.D

Análisis de las codificaciones de canal: LDPC y Turbo código, utilizando modulaciones no uniformes sobre canales AWGN con desvanecimiento.

En este trabajo se presenta un análisis de los códigos de canal LDPC y Turbo, utilizandoal utilizar modulaciones digitales con constelaciones no uniformes NUC-QAM de orden M=64, 256, 1024 y 4096, en canales con ruido aditivo blanco Gaussiano (AWGN) con desvanecimiento plano. Para estimar el comportamiento del flujo de bits durante la codificación de canal se utilizaron dos métricas: la relación señal a ruido (SNR), y la tasa de error de bit (BER). Se consideran tres posibles escenarios para el Canal de Comunicaciones. El primero caso corresponde a un canal de comunicación sin aplicar codificación de canal, el segundo caso se aplica la codificación de canal LDPC, y el tercer caso se aplica la codificación de canal Turbo. La simulación del canal se realiza con base al modelo estadístico de distribución de probabilidad Rayleigh. Los resultados para cada caso se obtienen a través de funciones programadas en el software Matlab. Los resultados del desempeño de la codificación de canal LDPC y Turbo se comparan con los resultados obtenidos sin aplicar codificación de canal, y se determina cuál de ellos se comporta mejor con un rango amplio de valores de SNR. .Finalmente se obtiene que la mejor codificación es la LDPC ya que el BER se estabiliza en el orden de (〖10〗^(-6)) mucho menor comparada con la codificación Turbo.This research it presents an analysis of LDPC and Turbo channel codes, using digital modulations with non-uniform NUC-QAM constellations of order M=64, 256, 1024 and 4096, in channels with additive white Gaussian noise (AWGN) with flat fading. Two metrics were used to estimate the bitstream behavior during channel coding: the signal to noise ratio (SNR), and the bit error rate (BER). Three possible scenarios are considered for the Communications Channel. The first case corresponds to a communication channel that’s applied without channel coding, the second case applies LDPC channel coding, and the third case it applied Turbo channel coding. The channel simulation is performed based on the Rayleigh probability distribution statistical model. The results for each case are obtained through functions programmed in Matlab software. The performance results of LDPC and Turbo channel coding are compared with the results obtained without applying channel coding, and it is determined which of them performs better with a wide range of SNR values. Finally, it is obtained that the best coding is LDPC as the BER stabilizes in the order of (10^(-6)) much lower compared to Turbo coding

Optical Communication

Optical communication is very much useful in telecommunication systems, data processing and networking. It consists of a transmitter that encodes a message into an optical signal, a channel that carries the signal to its desired destination, and a receiver that reproduces the message from the received optical signal. It presents up to date results on communication systems, along with the explanations of their relevance, from leading researchers in this field. The chapters cover general concepts of optical communication, components, systems, networks, signal processing and MIMO systems. In recent years, optical components and other enhanced signal processing functions are also considered in depth for optical communications systems. The researcher has also concentrated on optical devices, networking, signal processing, and MIMO systems and other enhanced functions for optical communication. This book is targeted at research, development and design engineers from the teams in manufacturing industry, academia and telecommunication industries