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Publication# Gaussian channel transmission of images and audio files using cryptcoding

Résumé

Random codes based on quasigroups (RCBQ) are cryptcodes, i.e. they are error-correcting codes, which provide information security. Cut-Decoding and 4-Sets-Cut-Decoding algorithms for these codes are defined elsewhere. Also, the performance of these codes for the transmission of text messages is investigated elsewhere. In this study, the authors investigate the RCBQ's performance with Cut-Decoding and 4-Sets-Cut-Decoding algorithms for transmission of images and audio files through a Gaussian channel. They compare experimental results for both coding/decoding algorithms and for different values of signal-to-noise ratio. In all experiments, the differences between the transmitted and decoded image or audio file are considered. Experimentally obtained values for bit-error rate and packet error rate and the decoding speed of both algorithms are compared. Also, two filters for enhancing the quality of the images decoded using RCBQ are proposed.

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This work deals with coding systems based on sparse graph codes. The key issue we address is the relationship between iterative (in particular belief propagation) and maximum a posteriori decoding. We show that between the two there is a fundamental connection, which is reminiscent of the Maxwell construction in thermodynamics. The main objects we consider are EXIT-like functions. EXIT functions were originally introduced as handy tools for the design of iterative coding systems. It gradually became clear that EXIT functions possess several fundamental properties. Many of these properties, however, apply only to the erasure case. This motivates us to introduce GEXIT functions that coincide with EXIT functions over the erasure channel. In many aspects, GEXIT functions over general memoryless output-symmetric channels play the same role as EXIT functions do over the erasure channel. In particular, GEXIT functions are characterized by the general area theorem. As a first consequence, we demonstrate that in order for the rate of an ensemble of codes to approach the capacity under belief propagation decoding, the GEXIT functions of the component codes have to be matched perfectly. This statement was previously known as the matching condition for the erasure case. We then use these GEXIT functions to show that in the limit of large blocklengths a fundamental connection appears between belief propagation and maximum a posteriori decoding. A decoding algorithm, which we call Maxwell decoder, provides an operational interpretation of this relationship for the erasure case. Both the algorithm and the analysis of the decoder are the translation of the Maxwell construction from statistical mechanics to the context of probabilistic decoding. We take the first steps to extend this construction to general memoryless output-symmetric channels. More exactly, a general upper bound on the maximum a posteriori threshold for sparse graph codes is given. It is conjectured that the fundamental connection between belief propagation and maximum a posteriori decoding carries over to the general case.

The invention of Fountain codes is a major advance in the field of error correcting codes. The goal of this work is to study and develop algorithms for source and channel coding using a family of Fountain codes known as Raptor codes. From an asymptotic point of view, the best currently known sum-product decoding algorithm for non binary alphabets has a high complexity that limits its use in practice. For binary channels, sum-product decoding algorithms have been extensively studied and are known to perform well. In the first part of this work, we develop a decoding algorithm for binary codes on non-binary channels based on a combination of sum-product and maximum-likelihood decoding. We apply this algorithm to Raptor codes on both symmetric and non-symmetric channels. Our algorithm shows the best performance in terms of complexity and error rate per symbol for blocks of finite length for symmetric channels. Then, we examine the performance of Raptor codes under sum-product decoding when the transmission is taking place on piecewise stationary memoryless channels and on channels with memory corrupted by noise. We develop algorithms for joint estimation and detection while simultaneously employing expectation maximization to estimate the noise, and sum-product algorithm to correct errors. We also develop a hard decision algorithm for Raptor codes on piecewise stationary memoryless channels. Finally, we generalize our joint LT estimation-decoding algorithms for Markov-modulated channels. In the third part of this work, we develop compression algorithms using Raptor codes. More specifically we introduce a lossless text compression algorithm, obtaining in this way competitive results compared to the existing classical approaches. Moreover, we propose distributed source coding algorithms based on the paradigm proposed by Slepian and Wolf.

The general subject considered in this thesis is a recently discovered coding technique, polar coding, which is used to construct a class of error correction codes with unique properties. In his ground-breaking work, Arikan proved that this class of codes, called polar codes, achieve the symmetric capacity --- the mutual information evaluated at the uniform input distribution ---of any stationary binary discrete memoryless channel with low complexity encoders and decoders requiring in the order of $O(N\log N)$ operations in the block-length $N$. This discovery settled the long standing open problem left by Shannon of finding low complexity codes achieving the channel capacity. Polar codes are not only appealing for being the first to 'close the deal'. In contrast to most of the existing coding schemes, polar codes admit an explicit low complexity construction. In addition, for symmetric channels, the polar code construction is deterministic; the theoretically beautiful but practically limited "average performance of an ensemble of codes is good, so there must exist one particular code in the ensemble at least as good as the average'' formalism of information theory is bypassed. Simulations are thus not necessary in principle for evaluating the error probability which is shown in a study by Telatar and Arikan to scale exponentially in the square root of the block-length. As such, at the time of this writing, polar codes are appealing for being the only class of codes proved, and proved with mathematical elegance, to possess all of these properties. Polar coding settled an open problem in information theory, yet opened plenty of challenging problems that need to be addressed. This novel coding scheme is a promising method from which, in addition to data transmission, problems such as data compression or compressed sensing, which includes all types of measurement processes like the MRI or ultrasound, could benefit in terms of efficiency. To make this technique fulfill its promise, the original theory has been, and should still be, extended in multiple directions. A significant part of this thesis is dedicated to advancing the knowledge about this technique in two directions. The first one provides a better understanding of polar coding by generalizing some of the existing results and discussing their implications, and the second one studies the robustness of the theory over communication models introducing various forms of uncertainty or variations into the probabilistic model of the channel. See the fulltext of the thesis for the complete abstract.