The Treewidth of MDS and Reed-Muller Codes

The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parameterization of the maximum-likelihood decoding complexity for linear codes. In this paper, we show the surprising fact that for maximum distance separable codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code, is defined to be the least constraint complexity (or branch complexity) of any of its trellis realizations. From this, we obtain exact expressions for the treewidth of these codes, which constitute the only known explicit expressions for the treewidth of algebraic codes.

Linearization and Reconstruction of Non-linear Diffuse Optical Tomographic Image

Diffuse optical tomography (DOT) is one of the ways to probe highly scattering media such as tissue using low-energy near infra-red light (NIR) to reconstruct a map of the optical property distribution. The interaction of the photons in biological tissue is a non-linear process and the phton transport through the tissue is modelled using diffusion theory. The inversion problem is often solved through iterative methods based on nonlinear optimization for the minimization of a data-model misfit function. The solution of the non-linear problem can be improved by modeling and optimizing the cost functional. The cost functional is f(x) = x(T)Ax - b(T)x + c and after minimization, the cost functional reduces to Ax = b. The spatial distribution of optical parameter can be obtained by solving the above equation iteratively for x. As the problem is non-linear, ill-posed and ill-conditioned, there will be an error or correction term for x at each iteration. A linearization strategy is proposed for the solution of the nonlinear ill-posed inverse problem by linear combination of system matrix and error in solution. By propagating the error (e) information (obtained from previous iteration) to the minimization function f(x), we can rewrite the minimization function as f(x; e) = (x + e)(T) A(x + e) - b(T)(x + e) + c. The revised cost functional is f(x; e) = f(x) + e(T)Ae. The self guided spatial weighted prior (e(T)Ae) error (e, error in estimating x) information along the principal nodes facilitates a well resolved dominant solution over the region of interest. The local minimization reduces the spreading of inclusion and removes the side lobes, thereby improving the contrast, localization and resolution of reconstructed image which has not been possible with conventional linear and regularization algorithm.

Derating Based Hardware Optimizations in Soft Error Tolerant Designs

Ensuring reliable operation over an extended period of time is one of the biggest challenges facing present day electronic systems. The increased vulnerability of the components to atmospheric particle strikes poses a big threat in attaining the reliability required for various mission critical applications. Various soft error mitigation methodologies exist to address this reliability challenge. A general solution to this problem is to arrive at a soft error mitigation methodology with an acceptable implementation overhead and error tolerance level. This implementation overhead can then be reduced by taking advantage of various derating effects like logical derating, electrical derating and timing window derating, and/or making use of application redundancy, e. g. redundancy in firmware/software executing on the so designed robust hardware. In this paper, we analyze the impact of various derating factors and show how they can be profitably employed to reduce the hardware overhead to implement a given level of soft error robustness. This analysis is performed on a set of benchmark circuits using the delayed capture methodology. Experimental results show upto 23% reduction in the hardware overhead when considering individual and combined derating factors.

Stochastic Simulations Suggest that HIV-1 Survives Close to Its Error Threshold

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, mu(c), however, is not known. Application of the quasispecies theory to determine mu(c) poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and mu(c). We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated mu(c) to be 7 x 10(-5) -1 x 10(-4) substitutions/site/replication, similar to 2-6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, mu(c) increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of mu(c) may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.

Efficient Algorithms for Linear Summed Error Structural SVMs

Structural Support Vector Machines (SSVMs) have become a popular tool in machine learning for predicting structured objects like parse trees, Part-of-Speech (POS) label sequences and image segments. Various efficient algorithmic techniques have been proposed for training SSVMs for large datasets. The typical SSVM formulation contains a regularizer term and a composite loss term. The loss term is usually composed of the Linear Maximum Error (LME) associated with the training examples. Other alternatives for the loss term are yet to be explored for SSVMs. We formulate a new SSVM with Linear Summed Error (LSE) loss term and propose efficient algorithms to train the new SSVM formulation using primal cutting-plane method and sequential dual coordinate descent method. Numerical experiments on benchmark datasets demonstrate that the sequential dual coordinate descent method is faster than the cutting-plane method and reaches the steady-state generalization performance faster. It is thus a useful alternative for training SSVMs when linear summed error is used.

A Two Stage Selective Averaging LDPC Decoding

Low density parity-check (LDPC) codes are a class of linear block codes that are decoded by running belief propagation (BP) algorithm or log-likelihood ratio belief propagation (LLR-BP) over the factor graph of the code. One of the disadvantages of LDPC codes is the onset of an error floor at high values of signal to noise ratio caused by trapping sets. In this paper, we propose a two stage decoder to deal with different types of trapping sets. Oscillating trapping sets are taken care by the first stage of the decoder and the elementary trapping sets are handled by the second stage of the decoder. Simulation results on the regular PEG (504,252,3,6) code and the irregular PEG (1024,518,15,8) code shows that the proposed two stage decoder performs significantly better than the standard decoder.

Bubble size measurement and error analysis in a gas liquid ejector

Bubble size in a gas liquid ejector has been measured using the image technique and analysed for estimation of Sauter mean diameter. The individual bubble diameter is estimated by considering the two dimensional contour of the ellipse, for the actual three dimensional ellipsoid in the system by equating the volume of the ellipsoid to that of the sphere. It is observed that the bubbles are of oblate and prolate shaped ellipsoid in this air water system. The bubble diameter is calculated based on this concept and the Sauter mean diameter is estimated. The error between these considerations is reported. The bubble size at different locations from the nozzle of the ejector is presented along with their percentage error which is around 18%.

EXIT Chart Based Design of Irregular LDPC Codes for Large-MIMO Systems

In this letter, we characterize the extrinsic information transfer (EXIT) behavior of a factor graph based message passing algorithm for detection in large multiple-input multiple-output (MIMO) systems with tens to hundreds of antennas. The EXIT curves of a joint detection-decoding receiver are obtained for low density parity check (LDPC) codes of given degree distributions. From the obtained EXIT curves, an optimization of the LDPC code degree profiles is carried out to design irregular LDPC codes matched to the large-MIMO channel and joint message passing receiver. With low complexity joint detection-decoding, these codes are shown to perform better than off-the-shelf irregular codes in the literature by about 1 to 1.5 dB at a coded BER of 10(-5) in 16 x 16, 64 x 64 and 256 x 256 MIMO systems.

An Adaptive Conditional Zero-Forcing Decoder With Full-Diversity, Least Complexity and Essentially-ML Performance for STBCs

A low complexity, essentially-ML decoding technique for the Golden code and the three antenna Perfect code was introduced by Sirianunpiboon, Howard and Calderbank. Though no theoretical analysis of the decoder was given, the simulations showed that this decoding technique has almost maximum-likelihood (ML) performance. Inspired by this technique, in this paper we introduce two new low complexity decoders for Space-Time Block Codes (STBCs)-the Adaptive Conditional Zero-Forcing (ACZF) decoder and the ACZF decoder with successive interference cancellation (ACZF-SIC), which include as a special case the decoding technique of Sirianunpiboon et al. We show that both ACZF and ACZF-SIC decoders are capable of achieving full-diversity, and we give a set of sufficient conditions for an STBC to give full-diversity with these decoders. We then show that the Golden code, the three and four antenna Perfect codes, the three antenna Threaded Algebraic Space-Time code and the four antenna rate 2 code of Srinath and Rajan are all full-diversity ACZF/ACZF-SIC decodable with complexity strictly less than that of their ML decoders. Simulations show that the proposed decoding method performs identical to ML decoding for all these five codes. These STBCs along with the proposed decoding algorithm have the least decoding complexity and best error performance among all known codes for transmit antennas. We further provide a lower bound on the complexity of full-diversity ACZF/ACZF-SIC decoding. All the five codes listed above achieve this lower bound and hence are optimal in terms of minimizing the ACZF/ACZF-SIC decoding complexity. Both ACZF and ACZF-SIC decoders are amenable to sphere decoding implementation.

Information-theoretically secure regenerating codes for distributed storage

Regenerating codes are a class of codes for distributed storage networks that provide reliability and availability of data, and also perform efficient node repair. Another important aspect of a distributed storage network is its security. In this paper, we consider a threat model where an eavesdropper may gain access to the data stored in a subset of the storage nodes, and possibly also, to the data downloaded during repair of some nodes. We provide explicit constructions of regenerating codes that achieve information-theoretic secrecy capacity in this setting.

Error exponent analysis of energy-based bayesian spectrum sensing under fading channels

This paper analyzes the error exponents in Bayesian decentralized spectrum sensing, i.e., the detection of occupancy of the primary spectrum by a cognitive radio, with probability of error as the performance metric. At the individual sensors, the error exponents of a Central Limit Theorem (CLT) based detection scheme are analyzed. At the fusion center, a K-out-of-N rule is employed to arrive at the overall decision. It is shown that, in the presence of fading, for a fixed number of sensors, the error exponents with respect to the number of observations at both the individual sensors as well as at the fusion center are zero. This motivates the development of the error exponent with a certain probability as a novel metric that can be used to compare different detection schemes in the presence of fading. The metric is useful, for example, in answering the question of whether to sense for a pilot tone in a narrow band (and suffer Rayleigh fading) or to sense the entire wide-band signal (and suffer log-normal shadowing), in terms of the error exponent performance. The error exponents with a certain probability at both the individual sensors and at the fusion center are derived, with both Rayleigh as well as log-normal shadow fading. Numerical results are used to illustrate and provide a visual feel for the theoretical expressions obtained.

Field extension code based dispersion matrices for coherently detected space-time shift keying

Motivated by the recent Coherent Space-Time Shift Keying (CSTSK) philosophy, we construct new dispersion matrices for rotationally invariant PSK signaling sets. Given a specific PSK signal constellation, the dispersion matrices of the existing CSTSK scheme were chosen by maximizing the mutual information over randomly generated sets of dispersion matrices. In this contribution we propose a general method for constructing a set of structured dispersion matrices for arbitrary PSK signaling sets using Field Extension (FE) codes and then study the attainable Symbol Error Rate (SER) performance of some example constructions. We demonstrate that the proposed dispersion scheme is capable of outperforming the existing dispersion arrangement at medium to high SNRs.

W/T SPR codes offer higher throughput in OCDMA networks

In this paper optical code-division multiple-access (O-CDMA) packet network is considered, which offers inherent security in the access networks. Two types of random access protocols are proposed for packet transmission. In protocol 1, all distinct codes and in protocol 2, distinct codes as well as shifted versions of all these codes are used. O-CDMA network performance using optical orthogonal codes (OOCs) 1-D and two-dimensional (2-D) wavelength/time single-pulse-per-row (W/T SPR) codes are analyzed. The main advantage of using 2-D codes instead of one-dimensional (1-D) codes is to reduce the errors due to multiple access interference among different users. SPR codes are chosen as they have nearly ideal correlation properties. In this paper, correlation receiver is considered in the analysis. Using analytical model, we compare the OOC and SPR code performances in O-CDMA networks. We compute packet-success probability and throughput for both the types of codes. The analysis shows improved performance with SPR codes as compared to OOC codes.

A nested linear codes approach to distributed function computation over subspaces

In this paper, we consider a distributed function computation setting, where there are m distributed but correlated sources X1,...,Xm and a receiver interested in computing an s-dimensional subspace generated by [X1,...,Xm]Γ for some (m × s) matrix Γ of rank s. We construct a scheme based on nested linear codes and characterize the achievable rates obtained using the scheme. The proposed nested-linear-code approach performs at least as well as the Slepian-Wolf scheme in terms of sum-rate performance for all subspaces and source distributions. In addition, for a large class of distributions and subspaces, the scheme improves upon the Slepian-Wolf approach. The nested-linear-code scheme may be viewed as uniting under a common framework, both the Korner-Marton approach of using a common linear encoder as well as the Slepian-Wolf approach of employing different encoders at each source. Along the way, we prove an interesting and fundamental structural result on the nature of subspaces of an m-dimensional vector space V with respect to a normalized measure of entropy. Here, each element in V corresponds to a distinct linear combination of a set {Xi}im=1 of m random variables whose joint probability distribution function is given.

On the treewidth of MDS and Reed-Muller codes

The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parametrization of the maximum-likelihood decoding complexity for linear codes. In this paper, we compute exact expressions for the treewidth of maximum distance separable codes, and first- and second-order Reed-Muller codes. These results constitute the only known explicit expressions for the treewidth of algebraic codes.