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

deriving good ldpc convolutional codes from ldpc block codes low density parity check ldpc convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity in this paper we discuss several graph cover based methods for deriving families of time invariant and time varying ldpc convolutional codes from ldpc block codes and show how earlier proposed ldpc convolutional code constructions can be presented within this framework some of the constructed convolutional codes significantly outperform the underlying ldpc block codes we investigate some possible reasons for this convolutional gain and we also discuss the mostly moderate decoder cost increase that is incurred by going from ldpc block to ldpc convolutional codes

computation of acoustic propagation in two dimensional sheared ducted flows most aeroacoustic noise prediction methods rely on an acoustic analogy featuring a propagation equation associated with source terms these models were mainly applied to computation of acoustic far fields radiated by simple free flows like jets the assumption is made in many formulations that the radiated acoustic field is not perturbed by the shear flow giving rise to the noise sources these acoustic analogies thus do not provide a full description of acoustic flow interactions the lilley equation was introduced to account to a certain extent for mean shear effects on propagation more recently this problem has been treated by making use of the linearized euler equations which are more flexible and more adequate for numerical simulations as several types of modes are supported by the euler equations problems linked to their coupling have to be considered it is then necessary to investigate acoustic field computations in complex flows our aim in the present article is to validate the wave operator associated with linearized euler equations numerical tests deal with propagation in two dimensional sheared ducted flows results are compared with other solutions deduced from analytical developments and direct numerical simulations this study shows that the linearized euler operator may be used to account for mean effects on wave propagation in the presence of sheared ducted flows processes that are specifically considered are 1 convection effects on axial disturbances 2 refraction effects on oblique wave generation and 3 source radiation effects on propagation in sheared flows

a simplified min sum decoding algorithm for non binary ldpc codes non binary low density parity check codes are robust to various channel impairments however based on the existing decoding algorithms the decoder implementations are expensive because of their excessive computational complexity and memory usage based on the combinatorial optimization we present an approximation method for the check node processing the simulation results demonstrate that our scheme has small performance loss over the additive white gaussian noise channel and independent rayleigh fading channel furthermore the proposed reduced complexity realization provides significant savings on hardware so it yields a good performance complexity tradeoff and can be efficiently implemented

weight distributions of non binary multi edge type ldpc code ensembles analysis and efficient evaluation non binary multi edge type ensembles of low density parity check codes are analyzed in terms of non binary codeword weight distribution and its growth rate in particular an exact expression of the growth rate for small weights is developed as a side result the stopping set distributions of these ensembles are developed examples of weight distributions are provided showing that the derived closed form expressions can be easily evaluated the obtained results can thus be exploited to analyze and design non binary low density parity check codes that fall within the multi edge type framework such as but not limited to protograph based codes

a class of nonbinary ldpc codes with fast encoding and decoding algorithms this letter is concerned with a class of nonbinary low density parity check ldpc codes referred to as column scaled ldpc cs ldpc codes whose parity check matrices have a property that each column is a scaled binary vector the cs ldpc codes which include algebraically constructed nonbinary ldpc codes as subclasses admit fast encoding and decoding algorithms specifically for a code over the finite field f2p the encoder can be implemented with p parallel binary ldpc encoders followed by a series of bijective mappers while the decoder can be implemented with an iterative decoder in which no message permutations are required during the iterations in addition there exist low complexity iterative multistage decoders that can be utilized to trade off the performance against the complexity simulation results show that the performance degradation caused by the iterative multistage decoding algorithms is relevant to the code structure

practical codes for photon communication in a recent paper pierce studied the problems of communicating at optical frequencies using photon counting techniques and concluded that at low temperatures we encounter insuperable problems of encoding long before we approach channel capacity in this paper it is shown that even assuming a noiseless model for photon communication for which capacity measured in nats photon is infinite it is unlikely that a signaling efficiency of even 10 nats photon could be achieved practically on the positive side it is shown that pulse position modulation plus reed solomon coding yields practical results in the range of 2 to 3 nats photon

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

approaching capacity by equiprobable signaling on the gaussian channel for the additive white gaussian noise channel there is a gap between the channel capacity and the highest achievable rate of equiprobable uniformly spaced one dimensional signaling it is commonly believed that approaching channel capacity requires the constituent one dimensional signal points to have a gaussian probability distribution it is shown that the channel capacity can be achieved by equiprobable signaling with geometrical gaussian like signal sets construction of these signal constellations is explicitly given this result implicates that it is possible to approach channel capacity without using any kind of shaping codes

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

construction of quasi cyclic ldpc codes for awgn and binary erasure channels a finite field approach in the late 1950s and early 1960s finite fields were successfully used to construct linear block codes especially cyclic codes with large minimum distances for hard decision algebraic decoding such as bose chaudhuri hocquenghem bch and reed solomon rs codes this paper shows that finite fields can also be successfully used to construct algebraic low density parity check ldpc codes for iterative soft decision decoding methods of construction are presented ldpc codes constructed by these methods are quasi cyclic qc and they perform very well over the additive white gaussian noise awgn binary random and burst erasure channels with iterative decoding in terms of bit error probability block error probability error floor and rate of decoding convergence collectively particularly they have low error floors since the codes are qc they can be encoded using simple shift registers with linear 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

distance bounds for periodically time varying and tail biting ldpc convolutional codes existence type lower bounds on the free distance of periodically time varying ldpc convolutional codes and on the minimum distance of tail biting ldpc convolutional codes are derived it is demonstrated that the bound on free distance of periodically time varying ldpc convolutional codes approaches the bound on free distance of general nonperiodic time varying ldpc convolutional codes as the period increases the proof of the bound is based on lower bounding the minimum distance of corresponding tail biting ldpc convolutional codes which is of interest in its own right

multiresolution representation of data a general framework in this paper we present a general framework for a multiresolution representation of data which is obtained by the discretization of mappings this framework which can be viewed as a generalization of the theory of wavelets includes discretization corresponding to unstructured grids in several space dimensions and thus is general enough to enable us to embed most numerical problems in a multiresolution setting furthermore this framework allows for nonlinear data dependent multiresolution representation schemes and thus enables us to design adaptive data compression algorithms in this paper we also study the stability of linear schemes for a multiresolution representation and derive sufficient conditions for existence of a multiresolution basis

performance comparison of ldpc block and spatially coupled codes over gf q in this paper we compare the finite length performance of protograph based spatially coupled low density paritycheck sc ldpc codes and ldpc block codes ldpc bcs over gf q to reduce computational complexity and latency a sliding window decoder with a stopping rule based on a soft belief propagation bp estimate is used for the q ary sc ldpc codes two regimes are considered one when the constraint length of q ary sc ldpc codes is equal to the block length of q ary ldpc bcs and the other when the two decoding latencies are equal simulation results confirm that in both regimes 3 6 3 9 and 3 12 regular non binary sc ldpc codes can significantly outperform both binary and non binary ldpc bcs and binary sc ldpc codes finally we present a computational complexity comparison of q ary sc ldpc codes and q ary ldpc bcs under equal decoding latency and equal decoding performance assumptions

non binary protograph based ldpc codes enumerators analysis and designs this paper provides a comprehensive analysis of nonbinary low density parity check ldpc codes built out of protographs we consider both random and constrained edge weight labeling and refer to the former as the unconstrained nonbinary protograph based ldpc codes u nbpb codes and to the latter as the constrained nonbinary protograph based ldpc codes c nbpb codes equipped with combinatorial definitions extended to the nonbinary domain ensemble enumerators of codewords trapping sets stopping sets and pseudocodewords are calculated the exact enumerators are presented in the finite length regime and the corresponding growth rates are calculated in the asymptotic regime an exit chart tool for computing the iterative decoding thresholds of protograph based ldpc codes is presented followed by several examples of finite length u nbpb and c nbpb codes with high performance throughout this paper we provide accompanying examples which demonstrate the advantage of nonbinary protograph based ldpc codes over their binary counterparts and over random constructions the results presented in this paper advance the analytical toolbox of nonbinary graph based codes

enumerators for protograph based ensembles of ldpc and generalized ldpc codes protograph based ldpc and generalized ldpc g ldpc codes have the advantages of a simple design procedure and highly structured encoders and decoders the design of such protograph based codes relies on what is effectively a computer based search as such following gallager it is prudent to restrict the search to a good ensemble for example an ensemble whose minimum distance grows linearly with codeword length a good ensemble can also mean one with good stopping set trapping set or pseudocodeword properties in this paper ensemble codeword weight enumerators for finite length ldpc and g ldpc codes based on protographs were derived and then the asymptotic case was considered the asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length then the codeword weight enumerator technique is adapted to yield ensemble stopping set trapping set and pseudocodeword enumerators for protograph ldpc and g ldpc codes in this case the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size trapping set size and pseudoweight grows linearly with codeword length trapping set enumerators for g ldpc code ensembles represent a more complex problem which we do not consider here

low complexity x ems algorithms for nonbinary ldpc codes the extended min sum ems algorithm is redescribed as a reduced search trellis algorithm called m ems algorithm two variants of the m ems algorithm called t ems algorithm and d ems algorithm are presented simulation results show that these three algorithms referred to as x ems algorithms for convenience combined with factor correction techniques perform almost as well as the q ary sum product algorithm qspa but with a much lower complexity

structural analysis of array based non binary ldpc codes structural properties of array based non binary low density parity check nbldpc codes are studied in this paper first we characterize graphical substructures induced by codewords of symbol weight six in array based nbldpc codes defined by parity check matrices with column weight three we also reveal necessary conditions for these graphical substructures to incur weight 6 codewords such conditions can be used to select nonzero elements for avoiding weight 6 codewords or reducing the multiplicity of weight 6 codewords second we show that there exist weight 7 codewords in array based nbldpc codes defined by parity check matrices with column weight three as a byproduct we find that the graphical substructure induced by a weight 7 codeword takes the graphical substructure induced by the related weight 6 codewords as a subgraph third we show that there may exist codewords with symbol weight four six and seven in array based nbldpc codes defined by parity check matrices with column weight two these results enrich the structural analysis of array based ldpc codes in addition simulation results show the performance advantage of array based nbldpc codes

near shannon limit quasi cyclic low density parity check codes this letter presents two classes of quasi cyclic low density parity check codes that perform close to the shannon limit

mathematical modelling of surface vibration with volume constraint and its analysis abstract a hyperbolic problem with volume constraint is investigated by means of a minimizing method called the discrete morse flow a weak solution is introduced in a constructive way and results of numerical computations are shown

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

pseudocodeword performance analysis for ldpc convolutional codes message passing iterative decoders for low density parity check ldpc block codes are known to be subject to decoding failures due to so called pseudocodewords these failures can cause the large signal to noise ratio snr performance of message passing iterative decoding to be worse than that predicted by the maximum likelihood ml decoding union bound in this paper we address the pseudocodeword problem from the convolutional code perspective in particular we compare the performance of ldpc convolutional codes with that of their wrapped quasi cyclic block versions and we show that the minimum pseudoweight of an ldpc convolutional code is at least as large as the minimum pseudoweight of an underlying quasi cyclic code this result which parallels a well known relationship between the minimum hamming weight of convolutional codes and the minimum hamming weight of their quasi cyclic counterparts is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code s pseudocodeword this difference in the weight spectra leads to improved performance at low to moderate snrs for the convolutional code a conclusion supported by simulation results

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

threshold saturation for nonbinary sc ldpc codes on the binary erasure channel we analyze the asymptotic performance of nonbinary spatially coupled low density parity check sc ldpc code ensembles defined over the general linear group on the binary erasure channel in particular we prove the threshold saturation of belief propagation decoding to the so called potential threshold using the proof technique based on potential functions introduced by yedla et al assuming that the potential function exists we rewrite the density evolution of nonbinary sc ldpc codes in an equivalent vector recursion form which is suited for the use of the potential function we then discuss the existence of the potential function for the general case of vector recursions defined by multivariate polynomials and give a method to construct it we define a potential function in a slightly more general form than the one by yedla et al in order to make the technique based on potential functions applicable to the case of nonbinary ldpc codes we show that the potential function exists if a solution to a carefully designed system of linear equations exists furthermore we numerically show the existence of a solution to the system of linear equations for a large number of nonbinary ldpc code ensembles which allows us to define their potential function and thus prove threshold saturation

a multilevel algorithm for large eddy simulation of turbulent compressible flows abstract a multilevel method for large eddy simulation of turbulent compressible flows is proposed the method relies on the splitting of the turbulent flowfield into several frequency bands in space and time each band being associated to a specific computational grid in physical space this allows to take into account in a deterministic way the information contained on finer grids a subgrid model adapted to such a decomposition based on a generalization of the germano s identity to multilevel decomposition is also introduced the approach is validated by several multilevel simulations in a subsonic plane channel flow configuration for a low and a high value of the reynolds number while reductions of the cpu times up to 80 are obtained

a lattice boltzmann method for nonlinear disturbances around an arbitrary base flow in this paper we address the problem of the time evolution of a perturbation around a steady base flow with the use of the lattice boltzmann method lbm this approach named base flow lattice boltzmann method is of great interest in particular for aeroacoustic fields where the acoustic perturbation on the one hand is almost exclusively influenced by the large scale average structures of the underlying flow and on the other hand has a low effect on the large structures the method is implemented for weakly compressible flows and the results of the base flow lattice boltzmann are compared with the standard single relaxation time lbm the boundary conditions for the base flow lattice boltzmann method are discussed as well as the implementation of outflow conditions for acoustic waves

trellis based extended min sum algorithm for non binary ldpc codes and its hardware structure in this paper we present an improvement and a new implementation of a simplified decoding algorithm for non binary low density parity check codes nb ldpc in galois fields gf q the base algorithm that we use is the extended min sum ems algorithm which has been widely studied in the recent literature and has been shown to approach the performance of the belief propagation bp algorithm with limited complexity in our work we propose a new way to compute modified configuration sets using a trellis representation of incoming messages to check nodes we call our modification of the ems algorithm trellis ems t ems in the t ems the algorithm operates directly on the deviation space by considering a trellis built from differential messages which serves as a new reliability measure to sort the configurations we show that this new trellis representation reduces the computational complexity without any performance degradation in addition we show that our modifications of the algorithm allows to greatly reduce the decoding latency by using a larger degree of hardware parallelization

design of spatially coupled ldpc codes over gf q for windowed decoding in this paper we study spatially coupled low density parity check sc ldpc codes over finite fields gf q q geq 2 and develop design rules for q ary sc ldpc code ensembles based on their iterative belief propagation decoding thresholds with particular emphasis on low latency windowed decoding wd we consider transmission over both the binary erasure channel bec and the binary input additive white gaussian noise channel biawgnc and present results for a variety of j k regular sc ldpc code ensembles constructed over gf q using protographs thresholds are calculated using the protograph versions of q ary density evolution for the bec and the q ary extrinsic information transfer analysis for the biawgnc we show that the wd of q ary sc ldpc codes provides significant threshold gains compared with corresponding uncoupled q ary ldpc block code ldpc bc ensembles when the window size w is large enough and that these gains increase as the finite field size q 2 m increases moreover we demonstrate that the new design rules provide wd thresholds that are close to capacity even when both m and w are relatively small thereby reducing decoding complexity and latency the analysis further shows that compared with standard flooding schedule decoding the wd of q ary sc ldpc code ensembles results in significant reductions in both the decoding complexity and the decoding latency and that these reductions increase as m increases for the applications with a near threshold performance requirement and a constraint on decoding latency we show that using q ary sc ldpc code ensembles with moderate q 2 instead of their binary counterparts results in reduced decoding complexity

optimized design of finite length separable circulant based spatially coupled codes an absorbing set based analysis in this paper we characterize the finite length performance of separable circulant based spatially coupled scb sc ldpc codes for transmission over the additive white gaussian noise channel for a general class of finite length graph based codes it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime we first present the mathematical conditions for the existence of absorbing sets in binary scb sc codes this analysis enables us to find the exact number of absorbing sets as a function of the design parameters in particular our results show that the choice of the cutting vector affects the number of absorbing sets and therefore the error floor performance of the code for a fixed column weight we find provably optimal cutting vectors that result in the least number of absorbing sets furthermore we extend our analysis to nonbinary scb sc codes where we show that the choice of the cutting vector is not as critical as in the binary case we provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary scb sc codes by informed selection of edge labels our simulation results show the superior error floor performance of our designed binary and nonbinary scb sc codes compared with binary unstructured and nonbinary quasi cyclic sc codes available in the open literature

a variational scheme for hyperbolic obstacle problems abstract we consider an obstacle problem for possibly non local wave equations and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme we provide the corresponding numerical implementation and raise some open questions

quasi cyclic ldpc codes an algebraic construction this paper presents two new large classes of qc ldpc codes one binary and one non binary codes in these two classes are constructed by array dispersions of row distance constrained matrices formed based on additive subgroups of finite fields experimental results show that codes constructed perform very well over the awgn channel with iterative decoding based on belief propagation codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry ldpc codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low complexity reliability based iterative decoding algorithms such as binary message passing decoding algorithms non binary codes decoded with a fast fourier transform based sum product algorithm achieve significantly large coding gains over reed solomon codes of the same lengths and rates decoded with either the hard decision berlekamp massey algorithm or the algebraic soft decision kotter vardy algorithm they have potential to replace reed solomon codes in some communication or storage systems where combinations of random and bursts of errors or erasures occur

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

distance bounds for an ensemble of ldpc convolutional codes an ensemble of j k regular low density parity check ldpc convolutional codes is introduced and existence type lower bounds on the minimum distance dl of code segments of finite length l and on the free distance dfree are derived for sufficiently large constraint lengths v the distances are shown to grow linearly with v and the ratio dl v approaches the ratio dfee v for large l moreover the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for gallager s ensemble of j k regular ldpc block codes

an analysis of the block error probability performance of iterative decoding asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes n and the number of iterations i go to infinity three classes of codes are considered these are gallager s regular low density parity check ldpc codes tanner s generalized ldpc gldpc codes and the turbo codes due to berrou et al it is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability p sub b but also the block frame error probability p sub b goes to zero as n and i go to infinity

weight distributions of regular low density parity check codes over finite fields the average weight distribution of a regular low density parity check ldpc code ensemble over a finite field is thoroughly analyzed in particular a precise asymptotic approximation of the average weight distribution is derived for the small weight case and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved based on this analysis a general result including all previous results as special cases is established for the minimum distance of individual codes in a regular ldpc code ensemble

on the use of a high order overlapping grid method for coupling in cfd caa this paper presents a theoretical analysis and two applications of a high order overlapping grid method for coupling cartesian and curvilinear grids developed in order to simulate aerodynamic noise first the overlapping grid method based on lagrange interpolating polynomials is described and a theoretical analysis of the interpolation operator is then carried out it shows that the interpolation generates spurious modes that depend on the wavenumbers of the signal besides it also gives the optimal conditions in which interpolation can be applied then an application to the simulation of the aeroacoustic noise generated by the vortex shedding behind a cylinder is presented during this simulation it appears that interpolation can create some spurious acoustic modes in regions where hydrodynamic fluctuations are significant as predicted by the theoretical analysis it is shown that these spurious modes disappear when a refined cartesian grid is used 26 points per wavelength of the vortex shedding were found to be adequate in this study at last the simulation of the aerodynamic noise of a three element high lift wing profile has then been carried out for this application the main acoustic source at the slat trailing edge is represented analytically the propagation of the generated acoustic wave is simulated with a mean flow at rest and with a steady turbulent mean flow computed by rans the first application allows us to assess the method by comparing the results to a reference solution the second one shows that the influence of a non uniform mean flow on the directivity of an acoustic source can be observed in complex geometries this application therefore shows that the proposed coupling method is well adapted to complex geometries that are usually met in industrial applications

two enhanced reliability based decoding algorithms for nonbinary ldpc codes the weighted bit reliability based wbrb algorithm for nonbinary ldpc codes suffers certain loss of symbol reliability thus this paper enhances its soft decision version by passing multiple symbol reliability instead of bit reliability furthermore it demonstrates that plurality robustly indicates symbol reliability of extrinsic information sums thus this paper enhances the hard decision version by introducing symbol reliability from plurality analysis results show that these two enhanced decoding algorithms significantly outperform the wbrb algorithm with reasonable overhead

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

reliability based joint detection decoding algorithm for nonbinary ldpc coded modulation systems this paper studies an extension and improvement of the joint detection decoding algorithm for nonbinary ldpc coded modulation systems the iterative joint detection decoding ijdd algorithm in 1 combines nonbinary ldpc decoding with signal detection based on the hard message passing strategy resulting in significantly reduced decoding complexity however it applies only to majority logic decodable nonbinary ldpc codes with high column weight for nonbinary ldpc codes with low column weight a noticeable performance loss will be incurred to handle this problem we propose a reliability based iterative joint detection decoding also termed improved ijdd algorithm which combines the accumulated reliability of symbols based on the one step majority logic decoding mlgd algorithm and a chase like local list decoding algorithm simulation results show that the improved ijdd algorithm outperforms the ijdd algorithm by about 0 3 db using nonbinary ldpc codes with high column weight and by about 3 db using nonbinary ldpc codes with low column weight d v 4 while maintaining the low complexity of decoding compared to the fft qspa the proposed algorithm has a performance degradation of 0 5 db in the high column weight regime and about 1 db in the low column weight regime

minimum distance and trapping set analysis of protograph based ldpc convolutional codes low density parity check ldpc convolutional codes have been shown to be capable of achieving capacity approaching performance with iterative message passing decoding in the first part of this paper using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio we show that several ensembles of regular and irregular ldpc convolutional codes derived from protograph based ldpc block codes have the property that the free distance grows linearly with respect to the constraint length i e the ensembles are asymptotically good in particular we show that the free distance to constraint length ratio of the ldpc convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding ldpc block code ensembles a large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal to noise ratios under maximum likelihood decoding when suboptimal decoding methods are employed there are many factors that affect the performance of a code recently it has been shown that so called trapping sets are a significant factor affecting decoding failures of ldpc codes over the additive white gaussian noise channel with iterative message passing decoding in the second part of this paper we study the trapping sets of the asymptotically good protograph based ldpc convolutional codes considered earlier by extending the theory presented in part one and using similar bounding techniques we show that the size of the smallest non empty trapping set grows linearly with the constraint length for these ensembles

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

the compressible hybrid rans les formulation using an additive operator in this work the compressible governing equations for the hybrid reynolds averaged large eddy simulations are formally derived by applying a hybrid filter to the navier stokes equations this filter is constructed by linearly combining the reynolds average rans and large eddy simulation les operators with a continuous blending function the derived hybrid equations include additional terms that represent the interactions between rans and les formulations the relevance of these terms is investigated in flat plate turbulent boundary layer simulations and indicate that these additional terms play a fundamental role in compensating for the turbulence that is neither modeled nor resolved in the transition region between rans and les results also show that when the additional terms are included the calculations are not very sensitive to the blending function implemented in the hybrid filter in the contrary when these terms are neglected and a step function is implemented in the hybrid filter nonphysical discontinuities are predicted in the flow statistics

efficient encoding of quasi cyclic low density parity check codes efficient encoding of quasi cyclic low density parity check codes quasi cyclic qc low density parity check ldpc codes form an important subclass of ldpc codes these codes have encoding advantage over other types of ldpc codes this paper addresses the issue of efficient encoding of qc ldpc codes two methods are presented to find the generator matrices of qc ldpc codes in systematic circulant form from their parity check matrices given in circulant form based on the systematic circulation form of the generator matrix of a qc ldpc code various types of encoding circuits using simple shift registers are devised it is shown that the encoding complexity of a qc ldpc code is linearly proportional to the number of parity bits of the code for serial encoding and to the length of the code for high speed parallel encoding

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

transactions papers constructions of nonbinary quasi cyclic ldpc codes a finite field approach this paper is concerned with construction of efficiently encodable nonbinary quasi cyclic ldpc codes based on finite fields four classes of nonbinary quasi cyclic ldpc codes are constructed experimental results show that codes constructed perform well with iterative decoding using a fast fourier transform based q ary sum product algorithm and they achieve significant coding gains over reed solomon codes of the same lengths and rates decoded with either algebraic hard decision berlekamp massey algorithm or algebraic soft decision kotter vardy algorithm

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

accumulate repeat accumulate codes in this paper we propose an innovative channel coding scheme called accumulate repeat accumulate ara codes this class of codes can be viewed as serial turbo like codes or as a subclass of low density parity check ldpc codes and they have a projected graph or protograph representation this allows for high speed iterative decoding implementation using belief propagation an ara code can be viewed as precoded repeat accumulate ra code with puncturing or as precoded irregular repeat accumulate ira code where simply an accumulator is chosen as the precoder the amount of performance improvement due to the precoder will be called precoding gain using density evolution on their associated protographs we find some rate 1 2 ara codes with a maximum variable node degree of 5 for which a minimum bit snr as low as 0 08 db from channel capacity threshold is achieved as the block size goes to infinity such a low threshold cannot be achieved by ra ira or unstructured irregular ldpc codes with the same constraint on the maximum variable node degree furthermore by puncturing the inner accumulator we can construct families of higher rate ara codes with thresholds that stay close to their respective channel capacity thresholds uniformly iterative decoding simulation results are provided and compared with turbo codes in addition to iterative decoding analysis we analyzed the performance of ara codes with maximum likelihood ml decoding by obtaining the weight distribution of these codes and through existing tightest bounds we have shown that the ml snr threshold of ara codes also approaches very closely to that of random codes these codes have better interleaving gain than turbo codes

computations of flow past a circular cylinder using a continuous turbulence model modeling is calculated with reynolds averaged navier stokes equations using a function that is uniquely determined by the ratio of the resolved small scale length to the integral length scale the unified treatment of reynolds averaged navier stokes large eddy simulation could be achieved with this continuous model by controlling the coarseness of the mesh the continuous reynolds averaged navier stokes large eddy simulation model is initially tested with flow past a circular cylinder at the reynolds number re 3900 both constant integral length scales and varied integral length scales based on the mixing length are used in the simulations the mesh dependence is also studied with both coarse and fine meshes the comparisons are made and analyzed between the results with different integral length scales and different mesh sizes these results are also compared with the experimental results and reynolds averaged navier stokes results the results demonstrate that with the continuous modeling the large eddy simulation like simulation can be achieved by solving reynolds averaged navier stokes equations alone the spatially varied integral length scale is necessary to capture the more accurate turbulence quantities for anisotropic wall bounded turbulent flow

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

variational approach to evolutionary free boundary problems abstract the analysis of evolutionary nonlocal problems represented by volume constrained motion is discussed in the physical sense these problems can describe the motion of droplets on a solid plane the variational approach called the discrete morse flow combined with a fluid dynamics model appears to be a suitable tool

a consistent dual mesh framework for hybrid les rans modeling in this work we propose a consistent turbulence modeling framework for hybrid les rans modeling in this framework the filtered and reynolds averaged navier stokes rans equations are solved simultaneously in the whole domain on their respective meshes consistency between the two solutions is achieved in terms of velocity pressure and turbulent quantities through additional drift terms in the corresponding equations this approach leads to clean conditions at the les rans interfaces note that this general framework does not depend on the specific choice of les and rans models a hybrid les rans solver is developed within this framework and used to simulate the flow in a plane channel and that in a channel with periodic hills the results demonstrate that the hybrid solver leads to significantly improved results with moderate computational overhead compared to traditional les making it a promising candidate for industrial flow simulations

on the asymptotic weight and stopping set distribution of regular ldpc ensembles in this correspondence we estimate the variance of weight and stopping set distribution of regular low density parity check ldpc ensembles using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular ldpc ensemble has its weight distribution and stopping set distribution close to respective ensemble averages we are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average

a general class of commutative filters for les in complex geometries a class of filters for large eddy simulations of turbulent inhomogeneous flows is presented a general set of rules for constructing discrete filters in complex geometry is given and examples of such filters are presented with these filters the commutation error between numerical differentiation and filtering can be made arbitrarily small allowing for derivation of a consistent set of equations for the large scale field the application of such filters for explicit filtering in large eddy simulations and the issue of boundary conditions for the filtered field are also discussed

symbol flipping decoding algorithms based on prediction for non binary ldpc codes this paper constructs an objective function for symbol flipping decoding algorithms considering not only soft reliability but also hard reliability the maximization of this objective function indicates that the flipping metric should involve both the information before and after symbol flipping while the existing algorithms consider the information before symbol flipping theoretical analysis shows that such prediction mechanism together with hard reliability can significantly improve the error performance of symbol flipping algorithms simulation results show that the proposed algorithms provide effective tradeoff between error performance and complexity for decoding non binary ldpc codes

time invariant quasi cyclic spatially coupled ldpc codes based on packings this paper presents two packings derived from balanced incomplete block designs to construct quasi cyclic spatially coupled ldpc convolutional codes sc ldpc ccs the construction gives time invariant codes since the sub blocks corresponding to each time instant of the parity check matrix are identical moreover it provides flexible design rates and constraint lengths simulation results show that the proposed packing based sc ldpc ccs outperform the existing time invariant codes and perform closely to the time varying protograph based codes

quasi cyclic ldpc codes for fast encoding in this correspondence we present a special class of quasi cyclic low density parity check qc ldpc codes called block type ldpc b ldpc codes which have an efficient encoding algorithm due to the simple structure of their parity check matrices since the parity check matrix of a qc ldpc code consists of circulant permutation matrices or the zero matrix the required memory for storing it can be significantly reduced as compared with randomly constructed ldpc codes we show that the girth of a qc ldpc code is upper bounded by a certain number which is determined by the positions of circulant permutation matrices the b ldpc codes are constructed as irregular qc ldpc codes with parity check matrices of an almost lower triangular form so that they have an efficient encoding algorithm good noise threshold and low error floor their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

spatially coupled ensembles universally achieve capacity under belief propagation we investigate spatially coupled code ensembles for transmission over the binary erasure channel it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble this explains why convolutional ldpc ensembles originally introduced by felstrom and zigangirov perform so well over this channel we show that the equivalent result holds true for transmission over general binary input memoryless output symmetric channels more precisely given a desired error probability and a gap to capacity we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding in fact most codes in this ensemble have this property the quantifier universal refers to the single ensemble code that is good for all channels but we assume that the channel is known at the receiver the key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble we conclude by discussing some interesting open problems

spatially coupled ldpc codes constructed from protographs in this paper we construct protograph based spatially coupled low density parity check ldpc codes by coupling together a series of l disjoint or uncoupled ldpc code tanner graphs into a single coupled chain by varying l we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary l we demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design capacity approaching iterative belief propagation bp decoding thresholds and linear growth of minimum distance with block length in particular we show that for sufficiently large l the bp thresholds on both the binary erasure channel and the binary input additive white gaussian noise channel saturate to a particular value significantly better than the bp decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled ldpc code when all variable nodes in the coupled chain have degree greater than two asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good ldpc codes with fast convergence rates and bp thresholds close to the shannon limit further the gap to capacity decreases as the density of the graph increases opening up a new way to construct capacity achieving codes on memoryless binary input symmetric output channels with low complexity bp decoding

multi block large eddy simulations of turbulent boundary layers time developing turbulent boundary layers over an isothermal flat plate at free stream mach numbers of 0 3 and 0 7 are computed using an explicit finite difference method on structured multi block grids the size of each block is adjusted depending on the dimension of the largest structures present locally in the flow this alleviates the cost of calculations in which the wall layer is resolved and may result in substantial savings of memory and cpu time if several layers are used in the calculations presented the near wall region is computed using a domain with a spanwise length l o 820 which is sufficient to contain several streaks this grid block is repeated periodically in the spanwise direction the outer layer which contains larger structures is computed using a domain that is twice as wide l o 1640 although the flow at the interface between the blocks has a periodicity length determined by the inner layer block within a few grid points longer wavelengths are generated the velocity statistics and rms intensities compare well with single block calculations that use substantially more grid points

zonal detached eddy simulation of the flow around a high lift configuration a zonal hybrid reynolds averaged navier stokes large eddy simulation rans les approach called zonal des used to handle a two dimensional high lift configuration with deployed slat and flap is presented this method allows to reduce significantly the cost of an accurate numerical prediction of the unsteady flow around wings compared to a complete les some issues concerning grid generation as well as the use of zonal detached eddy simulation for a multi element airfoil are discussed the basic planar grid has 250 000 points and the finest spanwise grid has 31 points with az c 0 002 the effort is geared toward detailed comparison of the numerical results with the europiv2 experimental particle image velocimetry data including both mean and fluctuating properties of the velocity field arnott and al

large girth nonbinary qc ldpc codes of various lengths in this paper we construct nonbinary quasi cyclic low density parity check qc ldpc codes whose parity check matrices consist of an array of square sub matrices which are either zero matrices or circulant permutation matrices we propose a novel method to design the shift offset values of the circulant permutation sub matrices so that the code length can vary while maintaining a large girth extensive monte carlo simulations demonstrate that the obtained codes of a wide range of rates from 1 2 to 8 9 with length from 1000 to 10000 bits have very good performance over both awgn and rayleigh fading channels furthermore the proposed method is extended to design multiple nonbinary qc ldpc codes simultaneously where each individual code can achieve large girth with variable lengths the proposed codes are appealing to practical adaptive systems where the block length and code rate need to be adaptively adjusted depending on traffic characteristics and channel conditions

low complexity decoding for non binary ldpc codes in high order fields in this paper we propose a new implementation of the extended min sum ems decoder for non binary ldpc codes a particularity of the new algorithm is that it takes into accounts the memory problem of the non binary ldpc decoders together with a significant complexity reduction per decoding iteration the key feature of our decoder is to truncate the vector messages of the decoder to a limited number nm of values in order to reduce the memory requirements using the truncated messages we propose an efficient implementation of the ems decoder which reduces the order of complexity to nm log2 nm this complexity starts to be reasonable enough to compete with binary decoders the performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction which is shown both with a simulated density evolution approach and actual simulations

compact disturbance equations for aeroacoustic simulations a set of compact disturbance equations is developed for high accuracy and efficient aeroacoustic simulations the compact disturbance equations in their complete form are an exact rearrangement of the navier stokes equations but incorporate various linear and nonlinear disturbance equations such as the linearized euler equations and the linearized navier stokes equations their attractive mathematical properties facilitate the implementations of the full compact disturbance equations and the reduced equations in essentially the same form as in existing computational fluid dynamics solvers with minor modifications a high resolution unsteady simulation in a reduced domain can be embedded inside a less expensive reynolds averaged navier stokes solution of flow in a larger possibly very complex configuration this results in a hybrid reynolds averaged navier stokes large eddy simulation method with reduced meshing difficulties and computational costs but improved grid qualities and more accurate boundary

bit reliability based low complexity decoding algorithms for non binary ldpc codes this paper presents bit reliability based majority logic decoding mlgd algorithms for non binary ldpc codes the proposed algorithms pass only one galois field element and its reliability along each edge of the tanner graph of a non binary ldpc code since their reliability updates are in terms of bits rather than symbols they are more efficient than traditional mlgd based decoding algorithms by weighting the soft reliability of the extrinsic information sums based on their hard reliability the proposed algorithms can achieve good error performance for non binary ldpc codes with various column weights moreover their computational complexity and memory consumption are remarkably reduced compared with existing mlgd based decoding algorithms as a result they provide effective tradeoffs between error performance and complexity for decoding of non binary ldpc codes

meshless approach for wall treatment in large eddy simulation abstract an extended meshless approach for wall modelling in large eddy simulation of incompressible flows that may take into account several terms of navier stokes equations the pressure gradient the unsteady term buoyancy effects is presented the method is based on an analytical integration of the momentum equations along the normal direction to the wall after a detailed description of the algorithm the present approach is applied to a two dimensional periodic hill flow at a bulk reynolds number of 7100 it is shown that taking into account the instantaneous pressure gradient has no effect on the mean values of the velocity however a substantial improvement is obtained concerning the rms values

multiple access demodulation in the lifted signal graph with spatial coupling demodulation in a random multiple access channel is considered where the signals are chosen uniformly randomly with unit energy it is shown that by lifting replicating the graph of this system and randomizing the graph connections a simple iterative cancellation demodulator achieves the same performance as an optimal symbol by symbol detector of the original system the iterative detector has a complexity that is linear in the number of users while the direct optimal approach is known to be np hard however the maximal system load of this lifted graph is limited to alpha even for large signal to noise ratios snrs the system is interference limited spatial coupling between subsequent lifted graphs is introduced and anchoring the initial graphs the achievable system load alpha can go to infinity as the snr goes to infinity our results apply to several well documented system proposals such as interleave division multiple access partitioned spreading and certain forms of multiple input multiple output communications

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