Statistical physics of error-correcting codes

In this thesis we use statistical physics techniques to study the typical performance of four families of error-correcting codes based on very sparse linear transformations: Sourlas codes, Gallager codes, MacKay-Neal codes and Kanter-Saad codes. We map the decoding problem onto an Ising spin system with many-spins interactions. We then employ the replica method to calculate averages over the quenched disorder represented by the code constructions, the arbitrary messages and the random noise vectors. We find, as the noise level increases, a phase transition between successful decoding and failure phases. This phase transition coincides with upper bounds derived in the information theory literature in most of the cases. We connect the practical decoding algorithm known as probability propagation with the task of finding local minima of the related Bethe free-energy. We show that the practical decoding thresholds correspond to noise levels where suboptimal minima of the free-energy emerge. Simulations of practical decoding scenarios using probability propagation agree with theoretical predictions of the replica symmetric theory. The typical performance predicted by the thermodynamic phase transitions is shown to be attainable in computation times that grow exponentially with the system size. We use the insights obtained to design a method to calculate the performance and optimise parameters of the high performance codes proposed by Kanter and Saad.

On the Automorphism Groups of some AG-Codes Based on Ca;b Curves

*Partially supported by NATO.

Decoding stimulus-related information from single-trial EEG responses based on voltage topographies

Neuroimaging studies typically compare experimental conditions using average brain responses, thereby overlooking the stimulus-related information conveyed by distributed spatio-temporal patterns of single-trial responses. Here, we take advantage of this rich information at a single-trial level to decode stimulus-related signals in two event-related potential (ERP) studies. Our method models the statistical distribution of the voltage topographies with a Gaussian Mixture Model (GMM), which reduces the dataset to a number of representative voltage topographies. The degree of presence of these topographies across trials at specific latencies is then used to classify experimental conditions. We tested the algorithm using a cross-validation procedure in two independent EEG datasets. In the first ERP study, we classified left- versus right-hemifield checkerboard stimuli for upper and lower visual hemifields. In a second ERP study, when functional differences cannot be assumed, we classified initial versus repeated presentations of visual objects. With minimal a priori information, the GMM model provides neurophysiologically interpretable features - vis à vis voltage topographies - as well as dynamic information about brain function. This method can in principle be applied to any ERP dataset testing the functional relevance of specific time periods for stimulus processing, the predictability of subject's behavior and cognitive states, and the discrimination between healthy and clinical populations.

Quantifying the time for accurate EEG decoding of single value-based decisions.

BACKGROUND: Recent neuroimaging studies suggest that value-based decision-making may rely on mechanisms of evidence accumulation. However no studies have explicitly investigated the time when single decisions are taken based on such an accumulation process. NEW METHOD: Here, we outline a novel electroencephalography (EEG) decoding technique which is based on accumulating the probability of appearance of prototypical voltage topographies and can be used for predicting subjects' decisions. We use this approach for studying the time-course of single decisions, during a task where subjects were asked to compare reward vs. loss points for accepting or rejecting offers. RESULTS: We show that based on this new method, we can accurately decode decisions for the majority of the subjects. The typical time-period for accurate decoding was modulated by task difficulty on a trial-by-trial basis. Typical latencies of when decisions are made were detected at ∼500ms for 'easy' vs. ∼700ms for 'hard' decisions, well before subjects' response (∼340ms). Importantly, this decision time correlated with the drift rates of a diffusion model, evaluated independently at the behavioral level. COMPARISON WITH EXISTING METHOD(S): We compare the performance of our algorithm with logistic regression and support vector machine and show that we obtain significant results for a higher number of subjects than with these two approaches. We also carry out analyses at the average event-related potential level, for comparison with previous studies on decision-making. CONCLUSIONS: We present a novel approach for studying the timing of value-based decision-making, by accumulating patterns of topographic EEG activity at single-trial level.

Study of the audio coding algorithm of the MPEG-4 AAC standard and comparison among implementations of modules of the algorithm

Audio coding is used to compress digital audio signals, thereby reducing the amount of bits needed to transmit or to store an audio signal. This is useful when network bandwidth or storage capacity is very limited. Audio compression algorithms are based on an encoding and decoding process. In the encoding step, the uncompressed audio signal is transformed into a coded representation, thereby compressing the audio signal. Thereafter, the coded audio signal eventually needs to be restored (e.g. for playing back) through decoding of the coded audio signal. The decoder receives the bitstream and reconverts it into an uncompressed signal. ISO-MPEG is a standard for high-quality, low bit-rate video and audio coding. The audio part of the standard is composed by algorithms for high-quality low-bit-rate audio coding, i.e. algorithms that reduce the original bit-rate, while guaranteeing high quality of the audio signal. The audio coding algorithms consists of MPEG-1 (with three different layers), MPEG-2, MPEG-2 AAC, and MPEG-4. This work presents a study of the MPEG-4 AAC audio coding algorithm. Besides, it presents the implementation of the AAC algorithm on different platforms, and comparisons among implementations. The implementations are in C language, in Assembly of Intel Pentium, in C-language using DSP processor, and in HDL. Since each implementation has its own application niche, each one is valid as a final solution. Moreover, another purpose of this work is the comparison among these implementations, considering estimated costs, execution time, and advantages and disadvantages of each one.

Construction and decoding of BCH codes over finite commutative rings

BCH codes over arbitrary finite commutative rings with identity are derived in terms of their locator vector. The derivation is based on the factorization of xs -1 over the unit ring of an appropriate extension of the finite ring. We present an efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, for these codes. The code construction and the decoding procedures are very similar to the BCH codes over finite integer rings. © 1999 Elsevier B.V. All rights reserved.

Construction and decoding of BCH codes over chain of commutative rings

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A0 ⊂ A1 ⊂···⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0 ⊂ K1 ⊂···⊂ Kt−1 ⊂ Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A∗ i and K∗ i are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A∗ i and K∗ i for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

Goppa codes through polynomials of B[X;(1/2^2)Z_0] and its decoding principle

In this paper, we present a decoding principle for Goppa codes constructed by generalized polynomials, which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight $t\leq 2r$, i.e., whose minimum Hamming distance is $2^{2}r+1$.

Approximate decoding approaches for network coded correlated data

This paper considers a framework where data from correlated sources are transmitted with the help of network coding in ad hoc network topologies. The correlated data are encoded independently at sensors and network coding is employed in the intermediate nodes in order to improve the data delivery performance. In such settings, we focus on the problem of reconstructing the sources at decoder when perfect decoding is not possible due to losses or bandwidth variations. We show that the source data similarity can be used at decoder to permit decoding based on a novel and simple approximate decoding scheme. We analyze the influence of the network coding parameters and in particular the size of finite coding fields on the decoding performance. We further determine the optimal field size that maximizes the expected decoding performance as a trade-off between information loss incurred by limiting the resolution of the source data and the error probability in the reconstructed data. Moreover, we show that the performance of the approximate decoding improves when the accuracy of the source model increases even with simple approximate decoding techniques. We provide illustrative examples showing how the proposed algorithm can be deployed in sensor networks and distributed imaging applications.

Decoding Delay Minimization in Inter-Session Network Coding

Intra-session network coding has been shown to offer significant gains in terms of achievable throughput and delay in settings where one source multicasts data to several clients. In this paper, we consider a more general scenario where multiple sources transmit data to sets of clients over a wireline overlay network. We propose a novel framework for efficient rate allocation in networks where intermediate network nodes have the opportunity to combine packets from different sources using randomized network coding. We formulate the problem as the minimization of the average decoding delay in the client population and solve it with a gradient-based stochastic algorithm. Our optimized inter-session network coding solution is evaluated in different network topologies and is compared with basic intra-session network coding solutions. Our results show the benefits of proper coding decisions and effective rate allocation for lowering the decoding delay when the network is used by concurrent multicast sessions.

Systematic analysis of the decoding delay in multiview video

We present a framework for the analysis of the decoding delay in multiview video coding (MVC). We show that in real-time applications, an accurate estimation of the decoding delay is essential to achieve a minimum communication latency. As opposed to single-view codecs, the complexity of the multiview prediction structure and the parallel decoding of several views requires a systematic analysis of this decoding delay, which we solve using graph theory and a model of the decoder hardware architecture. Our framework assumes a decoder implementation in general purpose multi-core processors with multi-threading capabilities. For this hardware model, we show that frame processing times depend on the computational load of the decoder and we provide an iterative algorithm to compute jointly frame processing times and decoding delay. Finally, we show that decoding delay analysis can be applied to design decoders with the objective of minimizing the communication latency of the MVC system.

The Crystal-t Algorithm: A New Approach To Calculate The Sle Of Lipidic Mixtures Presenting Solid Solutions.

Lipidic mixtures present a particular phase change profile highly affected by their unique crystalline structure. However, classical solid-liquid equilibrium (SLE) thermodynamic modeling approaches, which assume the solid phase to be a pure component, sometimes fail in the correct description of the phase behavior. In addition, their inability increases with the complexity of the system. To overcome some of these problems, this study describes a new procedure to depict the SLE of fatty binary mixtures presenting solid solutions, namely the Crystal-T algorithm. Considering the non-ideality of both liquid and solid phases, this algorithm is aimed at the determination of the temperature in which the first and last crystal of the mixture melts. The evaluation is focused on experimental data measured and reported in this work for systems composed of triacylglycerols and fatty alcohols. The liquidus and solidus lines of the SLE phase diagrams were described by using excess Gibbs energy based equations, and the group contribution UNIFAC model for the calculation of the activity coefficients of both liquid and solid phases. Very low deviations of theoretical and experimental data evidenced the strength of the algorithm, contributing to the enlargement of the scope of the SLE modeling.

Full threshold vs. Swedish interactive threshold algorithm (SITA) em pacientes glaucomatosos submetidos à perimetria computadorizada pela primeira vez

PURPOSE: To compare the Full Threshold (FT) and SITA Standard (SS) strategies in glaucomatous patients undergoing automated perimetry for the first time. METHODS: Thirty-one glaucomatous patients who had never undergone perimetry underwent automated perimetry (Humphrey, program 30-2) with both FT and SS on the same day, with an interval of at least 15 minutes. The order of the examination was randomized, and only one eye per patient was analyzed. Three analyses were performed: a) all the examinations, regardless of the order of application; b) only the first examinations; c) only the second examinations. In order to calculate the sensitivity of both strategies, the following criteria were used to define abnormality: glaucoma hemifield test (GHT) outside normal limits, pattern standard deviation (PSD) <5%, or a cluster of 3 adjacent points with p<5% at the pattern deviation probability plot. RESULTS: When the results of all examinations were analyzed regardless of the order in which they were performed, the number of depressed points with p<0.5% in the pattern deviation probability map was significantly greater with SS (p=0.037), and the sensitivities were 87.1% for SS and 77.4% for FT (p=0.506). When only the first examinations were compared, there were no statistically significant differences regarding the number of depressed points, but the sensitivity of SS (100%) was significantly greater than that obtained with FT (70.6%) (p=0.048). When only the second examinations were compared, there were no statistically significant differences regarding the number of depressed points, and the sensitivities of SS (76.5%) and FT (85.7%) (p=0.664). CONCLUSION: SS may have a higher sensitivity than FT in glaucomatous patients undergoing automated perimetry for the first time. However, this difference tends to disappear in subsequent examinations.

Estimating HIV-1 incidence using the serologic testing algorithm for recent HIV infections at HIV counseling and testing centers in the city of São Paulo, Brazil

The network of HIV counseling and testing centers in São Paulo, Brazil is a major source of data used to build epidemiological profiles of the client population. We examined HIV-1 incidence from November 2000 to April 2001, comparing epidemiological and socio-behavioral data of recently-infected individuals with those with long-standing infection. A less sensitive ELISA was employed to identify recent infection. The overall incidence of HIV-1 infection was 0.53/100/year (95% CI: 0.31-0.85/100/year): 0.77/100/year for males (95% CI: 0.42-1.27/100/year) and 0.22/100/ year (95% CI: 0.05-0.59/100/year) for females. Overall HIV-1 prevalence was 3.2% (95% CI: 2.8-3.7%), being 4.0% among males (95% CI: 3.3-4.7%) and 2.1% among females (95% CI: 1.6-2.8%). Recent infections accounted for 15% of the total (95% CI: 10.2-20.8%). Recent infection correlated with being younger and male (p = 0.019). Therefore, recent infection was more common among younger males and older females.