design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

rate distortion efficient piecewise planar 3 d scene representation from 2 d images in any practical application of the 2 d to 3 d conversion that involves storage and transmission representation efficiency has an undisputable importance that is not reflected in the attention the topic received in order to address this problem a novel algorithm which yields efficient 3 d representations in the rate distortion sense is proposed the algorithm utilizes two views of a scene to build a mesh based representation incrementally via adding new vertices while minimizing a distortion measure the experimental results indicate that in scenes that can be approximated by planes the proposed algorithm is superior to the dense depth map and in some practical situations to the block motion vector based representations in the rate distortion sense

spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

switch type hybrid hard decision decoding algorithms for regular low density parity check codes this correspondence presents the switch strategies for switch type hybrid hard decision decoding algorithms for regular low density parity check ldpc codes after the piecewise analysis of extrinsic information transfer functions for gallager decoding algorithm b gb the normalized switch scheme nss for which the majority based algorithms and gb are two examples is proposed then several other examples of nsss are presented and their convergence properties are analyzed based on the extrinsic information transfer exit functions in simulations the proposed nsss show meaningful performance improvements and less sensitivity to channel parameter underestimations compared with gb for small and moderate block length codes

self compensation technique for simplified belief propagation algorithm the min sum algorithm is the most common method to simplify the belief propagation algorithm for decoding low density parity check ldpc codes however there exists a performance gap between the min sum and belief propagation algorithms due to nonlinear approximation in this paper a self compensation technique using dynamic normalization is thus proposed to improve the approximation accuracy the proposed scheme scales the min sum algorithm by a dynamic factor that can be derived theoretically from order statistics moreover applying the proposed technique to several ldpc codes for dvb s2 system the average signal to noise ratio degradation which results from approximation inaccuracy and quantization error is reduced to 0 2 db not only does it enhance the error correcting capability of the min sum algorithm but the proposed self compensation technique also preserves a modest hardware cost after realized with 0 13 mum standard cell library the dynamic normalization requires about 100 additional gates for each check node unit in the min sum algorithm

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

estimation of depth fields suitable for video compression based on 3 d structure and motion of objects intensity prediction along motion trajectories removes temporal redundancy considerably in video compression algorithms in three dimensional 3 d object based video coding both 3 d motion and depth values are required for temporal prediction the required 3 d motion parameters for each object are found by the correspondence based e matrix method the estimation of the correspondences two dimensional 2 d motion field between the frames and segmentation of the scene into objects are achieved simultaneously by minimizing a gibbs energy the depth field is estimated by jointly minimizing a defined distortion and bit rate criterion using the 3 d motion parameters the resulting depth field is efficient in the rate distortion sense bit rate values corresponding to the lossless encoding of the resultant depth fields are obtained using predictive coding prediction errors are encoded by a lempel ziv algorithm the results are satisfactory for real life video scenes

an implementation friendly binary ldpc decoding algorithm we introduce an implementation friendly binary message passing decoding method for low density parity check ldpc codes that does not require the degree information of variable nodes or degree dependent parameters for hard decision decoding given its low complexity the implementation cost for variable node degree information is an important consideration we develop an estimation method for the extrinsic error probability eep as well as its analysis the proposed method offers similar performance as the existing methods for time invariant decoding in most cases while it facilitates efficient circuit implementations of the ldpc decoder

construction of nonbinary cyclic quasi cyclic and regular ldpc codes a finite geometry approach this paper presents five methods for constructing nonbinary ldpc codes based on finite geometries these methods result in five classes of nonbinary ldpc codes one class of cyclic ldpc codes three classes of quasi cyclic ldpc codes and one class of structured regular ldpc codes experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the awgn channel and they achieve significant coding gains over reed solomon codes of the same lengths and rates with either algebraic hard decision decoding or kotter vardy algebraic soft decision decoding at the expense of a larger decoding computational complexity

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

approximating the llr distribution for a class of soft output mimo detectors we present approximations of the llr distribution for a class of fixed complexity soft output mimo detectors such as the optimal soft detector and the soft output via partial marginalization detector more specifically in a mimo awgn setting we approximate the llr distribution conditioned on the transmitted signal and the channel matrix with a gaussian mixture model gmm our main results consist of an analytical expression of the gmm model including the number of modes and their corresponding parameters and a proof that in the limit of high snr this llr distribution converges in probability towards a unique gaussian distribution

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

3d planar representation of stereo depth images for 3dtv applications the depth modality of the multiview video plus depth mvd format is an active research area whose main objective is to develop depth image based rendering friendly efficient compression methods as a part of this research a novel 3d planar based depth representation is proposed the planar approximation of multiple depth images are formulated as an energy based co segmentation problem by a markov random field model the energy terms of this problem are designed to mimic the rate distortion tradeoff for a depth compression application a novel algorithm is developed for practical utilization of the proposed planar approximations in stereo depth compression the co segmented regions are also represented as layered planar structures forming a novel single reference mvd format the ability of the proposed layered planar mvd representation in decoupling the texture and geometric distortions make it a promising approach proposed 3d planar depth compression approaches are compared against the state of the art image video coding standards by objective and visual evaluation and yielded competitive performance

analysis of the iterative decoding of ldpc and product codes using the gaussian approximation we propose a novel density evolution approach to analyze the iterative decoding algorithms of low density parity check ldpc codes and product codes based on gaussian densities namely for these classes of codes we derive a one dimensional 1d map whose iterates directly represent the error probability both for the additive white gaussian noise awgn and the rayleigh fading channel these simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm as an application we compute the decoding thresholds and show that they are consistent with the simulation results available in the literature

maximizing the spectral efficiency of coded cdma under successive decoding we investigate the spectral efficiency achievable by random synchronous code division multiple access cdma with quaternary phase shift keying qpsk modulation and binary error control codes in the large system limit where the number of users the spreading factor and the code block length go to infinity for given codes we maximize spectral efficiency assuming a minimum mean square error mmse successive stripping decoder for the cases of equal rate and equal power users in both cases the maximization of spectral efficiency can be formulated as a linear program and admits a simple closed form solution that can be readily interpreted in terms of power and rate control we provide examples of the proposed optimization methods based on off the shelf low density parity check ldpc codes and we investigate by simulation the performance of practical systems with finite code block length

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

properties of optimum binary message passing decoders we consider a class of message passing decoders for low density parity check ldpc codes whose messages are binary valued we prove that if the channel is symmetric and all codewords are equally likely to be transmitted an optimum decoding rule in the sense of minimizing message error rate should satisfy certain symmetry and isotropy conditions using this result we prove that gallager s algorithm b achieves the optimum decoding threshold among all binary message passing decoding algorithms for regular codes for irregular codes we argue that when the nodes of the message passing decoder do not exploit knowledge of their decoding neighborhood optimality of gallager s algorithm b is preserved we also consider the problem of designing irregular ldpc codes and find a bound on the achievable rates with gallager s algorithm b using this bound we study the case of low error rate channels and analytically find good degree distributions for them

design of regular 2 d sub c ldpc codes over gf q using their binary images in this paper a method to design regular 2 dc ldpc codes over gf q with both good waterfall and error floor properties is presented based on the algebraic properties of their binary image first the algebraic properties of rows of the parity check matrix h associated with a code are characterized and optimized to improve the waterfall then the algebraic properties of cycles and stopping sets associated with the underlying tanner graph are studied and linked to the global binary minimum distance of the code finally simulations are presented to illustrate the excellent performance of the designed codes

hybrid hard decision iterative decoding of irregular low density parity check codes time invariant hybrid hscrti decoding of irregular low density parity check ldpc codes is studied focusing on hscrti algorithms with majority based mb binary message passing constituents we use density evolution de and finite length simulation to analyze the performance and the convergence properties of these algorithms over memoryless binary symmetric channels to apply de we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them a tight upper bound on the threshold of mb hscrti algorithms is derived and it is proven that the asymptotic error probability for these algorithms tends to zero at least exponentially with the number of iterations we devise optimal mb hscrti algorithms for irregular ldpc codes and show that these algorithms outperform gallager s algorithm a applied to optimized irregular ldpc codes we also show that compared to switch type algorithms such as gallager s algorithm b where a comparable improvement is obtained by switching between different mb algorithms mb hscrti algorithms are more robust and can better cope with unknown channel conditions and thus can be practically more attractive

dissipative mechanism of a semilinear higher order parabolic equation in r n it is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems in this paper we investigate the dissipative mechanism for some semilinear fourth order parabolic equations in the spaces of bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor while for second order reaction diffusion equations the dissipativeness mechanism has already been satisfactorily understood see arrieta et al 2004 doi 10 1142 s0218202504003234 for higher order problems in unbounded domains it has not yet been fully developed as shown throughout the paper one of the main differences from the case of reaction diffusion equations stems from the lack of a maximum principle thus we have to rely here on suitable energy estimates for the solutions as in the case of second order reaction diffusion equations we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness thus the dissipative mechanisms in second order and fourth order equations are similar although the lack of a maximum principle makes the proofs more difficult and the results not as complete finally we make essential use of the sharp results of cholewa and rodriguez bernal 2012 doi 10 1016 j na 2011 08 022 on linear fourth order equations with a very large class of linear potentials

an analysis of belief propagation on the turbo decoding graph with gaussian densities motivated by its success in decoding turbo codes we provide an analysis of the belief propagation algorithm on the turbo decoding graph with gaussian densities in this context we are able to show that under certain conditions the algorithm converges and that somewhat surprisingly though the density generated by belief propagation may differ significantly from the desired posterior density the means of these two densities coincide since computation of posterior distributions is tractable when densities are gaussian use of belief propagation in such a setting may appear unwarranted indeed our primary motivation for studying belief propagation in this context stems from a desire to enhance our understanding of the algorithm s dynamics in a non gaussian setting and to gain insights into its excellent performance in turbo codes nevertheless even when the densities are gaussian belief propagation may sometimes provide a more efficient alternative to traditional inference methods

systematic design of low density parity check code ensembles for binary erasure channels we propose a systematic method to design irregular low density parity check ldpc codes for binary erasure channels bec compared to the existing methods which are based on the application of asymptotic analysis tools such as density evolution or extrinsic information transfer exit charts in an optimization process the proposed method is much simpler and faster through a number of examples we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization an important property of the proposed designs is the flexibility to select the number of constituent variable node degrees p the proposed designs include existing systematic designs as a special case with p n 1 where n is the maximum variable node degree compared to the existing systematic designs for a given rate and a given 0 the designed ensembles can have a threshold in neighborhood of the capacity upper bound with smaller values of p and n they can also achieve the capacity of the bec as n and correspondingly p and the maximum check node degree tend to infinity

lp decoding of regular ldpc codes in memoryless channels we study error bounds for linear programming decoding of regular low density parity check ldpc codes for memoryless binary input output symmetric channels we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph for memoryless binary input awgn channel we prove lower bounds on the threshold for regular ldpc codes whose factor graphs have logarithmic girth under lp decoding specifically we prove a lower bound of 0 735 upper bound of eb n0 2 67 db on the threshold of 3 6 regular ldpc codes whose factor graphs have logarithmic girth our proof is an extension of a recent paper of arora daskalakis and steurer stoc 2009 who presented a novel probabilistic analysis of lp decoding over a binary symmetric channel their analysis is based on the primal lp representation and has an explicit connection to message passing algorithms we extend this analysis to any mbios channel

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

on the cahn hilliard equation in h 1 r n in this paper we exhibit the dissipative mechanism of the cahn hilliard equation in h 1 r n we show a weak form of dissipativity by showing that each individual solution is attracted in some sense by the set of equilibria we also indicate that strong dissipativity that is asymptotic compactness in h 1 r n cannot be in general expected then we consider two types of perturbations a nonlinear perturbation and a small linear perturbation in both cases we show that for the resulting equations the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor

polar coded modulation a framework is proposed that allows for a joint description and optimization of both binary polar coding and 2m ary digital pulse amplitude modulation pam schemes for the latter the multilevel coding mlc approach as well as bit interleaved coded modulation bicm are considered the conceptual equivalence of polar coding and multilevel coding is covered in detail based on an alternative characterization of the channel polarization phenomenon rules for the optimum choice of the labeling in coded modulation schemes employing polar codes are developed simulation results regarding the error performance of the proposed schemes on the awgn channel are included

iterative multiuser joint decoding optimal power allocation and low complexity implementation we consider a canonical model for coded code division multiple access cdma with random spreading where the receiver makes use of iterative belief propagation bp joint decoding we provide simple density evolution analysis in the large system limit large number of users of the performance of the bp decoder and of some suboptimal approximations based on interference cancellation ic based on this analysis we optimize the received user signal to noise ratio snr distribution in order to maximize the system spectral efficiency for given user channel codes channel load users per chip and target user bit error rate ber the optimization of the received snr distribution is obtained by solving a simple linear program and can be easily incorporated into practical power control algorithms remarkably under the optimized snr assignment the suboptimal minimum mean square error mmse ic based decoder performs almost as well as the more complex bp decoder moreover for a large class of commonly used convolutional codes we observe that the optimized snr distribution consists of a finite number of discrete snr levels based on this observation we provide a low complexity approximation of the mmse ic decoder that suffers from very small performance degradation while attaining considerable savings in complexity as by products of this work we obtain a closed form expression of the multiuser efficiency me of power mismatched mmse filters in the large system limit and we extend the analysis of the symbol by symbol maximum a posteriori probability map multiuser detector in the large system limit to the case of nonconstant user powers and nonuniform symbol prior probabilities

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

maximum likelihood laplacian correlation channel estimation in layered wyner ziv coding the distributed source coding theory teaches that efficient compression can be achieved by exploiting the correlation between encoded sources at the decoder only a critical issue in this coding paradigm is how to accurately estimate the correlation statistics at the decoder this paper proposes a novel correlation channel estimation method designed for generic layered wyner ziv wz coding driven by real life applications i e video coding the correlation noise is assumed to be memoryless zero mean laplacian with a block stationary behavior unlike existing methods the proposed technique derives a novel maximum likelihood estimate of the noise scaling parameter per stationarity block adhering to layered coding the derived estimate is successively refined as more stages are decoded the cramer rao lower bound crlb for our estimator is derived and the mean squared error performance of the proposed successive refinement algorithm is evaluated through simulations apart from achieving an accuracy close to the derived crlb the proposed algorithm is shown to yield a wz rate distortion performance approaching the ideal but unrealistic scenario where the decoder knows a priori the correlation statistics moreover the proposed technique has been integrated into our latest wz video codec for wireless capsule endoscopy experimentation reveals that the proposed method delivers improved compression performance compared to state of the art techniques while simultaneously reducing the overall decoding complexity

analysis of the viscous cahn hilliard equation in r n abstract solvability of cauchy s problem in r n for an extended viscous cahn hilliard equation is studied the problem is considered first in a standard sobolev space h 1 r n next a notion of the h solution is introduced that is well adapted to the structure of the viscous cahn hilliard equation several properties of the solutions are reported in particular those connected with their asymptotic behavior we collect here also known properties of the unbounded operator 1 in r n that are needed in our considerations

extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations abstract we consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors we then show some applications of the abstract scheme to various evolutionary problems from odes and retarded functional differential equations to parabolic and hyperbolic pdes in particular we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in r n with nonlinearities depending on the gradient of the solution we consider as well systems of reaction diffusion equations in r n and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in r n we further discuss some nonlocal and quasilinear problems as well as the fourth order cahn hilliard equation

an mse based transfer chart for analyzing iterative decoding schemes using a gaussian approximation an alternative to extrinsic information transfer exit charts called mean square error mse charts that use a measure related to the mse instead of mutual information is proposed using the relationship between mutual information and minimum mean square error mmse for the additive white gaussian noise awgn channel a relationship between the rate of any code and the area under a plot of mmse versus signal to noise ratio snr is obtained when the a priori log likelihood ratio llr is from a binary input gaussian channel using this result a justification is provided for designing concatenated codes by matching the exit curves of the inner and outer decoder when the llrs are assumed to be gaussian which is also the typical assumption used for code design using exit charts even though the gaussian assumption is almost never true the results presented in this paper represent a step toward the analysis of iterative decoding schemes using a single parameter finally for the special case of awgn channel it is shown that any capacity achieving code has an exit curve that is a step function

semistable extremal ground states for nonlinear evolution equations in unbounded domains in this paper we show that dissipative reaction diffusion equations in unbounded domains posses extremal semistable ground states equilibria which bound asymptotically the global dynamics uniqueness of such positive ground state and their approximation by extremal equilibria in bounded domains is also studied the results are then applied to the important case of logistic equations

threshold values and convergence properties of majority based algorithms for decoding regular low density parity check codes this work presents a detailed study of a family of binary message passing decoding algorithms for low density parity check ldpc codes referred to as majority based algorithms both gallager s algorithm a g sub a and the standard majority decoding algorithm belong to this family these algorithms which are in fact the building blocks of gallager s algorithm b g sub b work based on a generalized majority decision rule and are particularly attractive for their remarkably simple implementation we investigate the dynamics of these algorithms using density evolution and compute their noise threshold values for regular ldpc codes over the binary symmetric channel for certain ensembles of codes and certain orders of majority based algorithms we show that the threshold value can be characterized as the smallest positive root of a polynomial and thus can be determined analytically we also study the convergence properties of majority based algorithms including their convergence speed our analysis shows that the stand alone version of some of these algorithms provides significantly better performance and or convergence speed compared with g sub a in particular it is shown that for channel parameters below threshold while for g sub a the error probability converges to zero exponentially with iteration number this convergence for other majority based algorithms is super exponential

low density parity check codes over gaussian channels with erasures we consider low density parity check code ldpcc design for additive white gaussian noise awgn channels with erasures this model for example represents a common situation in magnetic and optical recording where defects or thermal asperities in the system are detected and presented to the decoder as erasures we give thresholds of regular and irregular ldpccs and discuss practical code design over the mixed gaussian erasures channel the analysis is an extension of the gaussian approximation work of chung et al in the two limiting cases of no erasures and large signal to noise ratio snr the analysis tends to the results of chung et al see ibid vol 47 p 657 670 feb 2001 and luby et al 1997 respectively giving a general tool for a class of mixed channels we derive a steady state equation which gives a graphical interpretation of decoder convergence this allows one to estimate the maximum erasure capability on the mixture channel or conversely to estimate the additional signal power required to compensate for the loss due to erasures we see that a good capacity approaching ldpcc over an awgn channel is also good over the mixed channel up to a moderate erasure probability we also investigate practical issues such as the maximum number of iterations of message passing decoders the coded block length and types of erasure patterns random block erasures finally we design an optimized ldpcc for the mixed channel which shows better performance if the erasure probability is larger than a certain value 0 1 in our simulation at the expense of performance degradation at unerased awgn channel and lower erasure probability regions less than 0 1 in our simulation

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

performance analysis of iterative decoding algorithms with memory over memoryless channels density evolution is often used to determine the performance of an ensemble of low density parity check ldpc codes under iterative message passing algorithms conventional density evolution techniques over memoryless channels are based on the assumption that messages at iteration l are only a function of the messages at iteration l 1 and possibly the channel output this assumption is valid for many algorithms such as standard belief propagation bp and min sum ms algorithms however there are other important iterative algorithms such as successive relaxation sr versions of bp and ms and differential decoding with binary message passing dd bmp algorithm of mobini et al for which this assumption is not valid the reason is the introduction of memory in these algorithms in this work we propose a model for iterative decoding algorithms with memory which covers sr and dd bmp algorithms as special cases based on this model we derive a bayesian network for iterative algorithms with memory over memoryless channels and use this representation to analyze the performance of the algorithms using density evolution the density evolution technique is developed based on truncating the memory of the decoding process and approximating it with a finite order markov process and can be implemented efficiently as an example we apply our technique to analyze the performance of dd bmp on regular ldpc code ensembles and make a number of interesting observations with regard to the performance complexity tradeoff of dd bmp in comparison with bp and ms algorithms the model presented in this paper is based on certain simplifying assumptions about the memory structure of iterative algorithms such as the existence of memory only at the output of variable nodes in the code s tanner graph rather than at both outputs of variable and check nodes the bayesian network framework introduced here however can still be used to analyze the more general scenarios