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

approximate analytical solutions of systems of pdes by homotopy analysis method in this paper the homotopy analysis method ham is applied to obtain series solutions to linear and nonlinear systems of first and second order partial differential equations pdes the ham solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions it is shown in particular that the solutions obtained by the variational iteration method vim are only special cases of the ham solutions

stable trace formula cuspidal tempered terms consider a connected reductive group g over a number field f for technical reasons we assume that the derived group of g is simply connected see l1 in l3 langlands partially stabilizes the trace formula for g after making certain assumptions he writes the elliptic regular part of the trace formula for g as a linear combination of the elliptic g regular parts of the stable trace formulas for the elliptic endoscopic groups h of g the function f used in the stable trace formula forh is obtained from the function f used in the trace formula for g by transferring orbital integrals

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 matrix theoretic approach to the construction of non binary quasi cyclic ldpc codes this paper presents two simple and very flexible methods for constructing non binary nb quasi cyclic qc ldpc codes the proposed construction methods have several known ingredients including base array masking binary to non binary replacement and matrix dispersion by proper choice and combination of these ingredients nb qc ldpc codes with excellent performance can be constructed the constructed codes can be decoded with a reduced complexity iterative decoding scheme which significantly reduces the hardware implementation complexity

non binary decoder diversity for dense or locally dense parity check codes in this paper a new and promising framework called non binary decoder diversity is presented based on the observation that different non binary tanner graphs of the same binary code decoded with a non binary belief propagation decoder can have distinct convergence behaviors and fixed points the goal of this work is to propose a decoder with linear complexity in the blocklength and with performance close to maximum likelihood decoding this framework is especially interesting for binary codes which are dense or locally dense and for which the usual binary iterative decoders perform far from the optimum curves by using the diversity brought by decoding distinct tanner graphs of the same code the proposed technique has very good decoding performance for three very different test cases which are known to be complex decoding problems i near maximum likelihood decoding mld of bch codes on the bpsk awgn channel ii performance results which outperform bounded distance decoding of bch codes over a binary symmetric channel bsc and finally iii decoding performance better than the bcjr based turbo decoder for parallel duo binary turbo codes

counting integral points in certain homogeneous spaces the leading term of asymptotic formula of the number of integral points in non compact symmetric homogeneous spaces of semi simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the brauer manin obstruction the similar result is also true for homogeneous spaces of reductive groups with some restriction as application we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and prove the leading term of asymptotic formula of the number of integral matrices with a fixed irreducible characteristic polynomial over z studied by eskin mozes shah is equal to the product of the number of local integral solutions over all primes although the density function defined by borovoi and rudnick is not trivial in general we also answer a question raised by borovoi and rudnick for comparing the number of integral symmetric matrices with the given determinant with the product of local densities c 2015 elsevier all rights reserved

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

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

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

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

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

maximum girth cylinder type block circulant ldpc codes in this paper a particular class of block circulant low density parity check bc ldpc codes referred to as cylinder type bc ldpc ctbc ldpc codes is studied we represent a cylinder type parity check matrix h by a graph called the block structure graph of h and denoted by bsg h using the properties of bsg h we show that ctbc matrices with column weight two and girth an arbitrary multiple of 8 can be constructed while for a ctbc matrix h with column weight l 3 this girth cannot exceed 12 an algorithm generating ctbc ldpc codes of arbitrary possible girth is given the algorithm produces a large group of codes with flexible rate length and girth from performance perspective over awgn channel the maximum girth 12 ctbc ldpc codes with column weight at least three generated by the given algorithm are at least as good as their random like counterpart codes and outperform the codes of the same size constructed by successive level growth progressive edge growth and partition and shift methods

progressive algebraic soft decision decoding of reed solomon codes the algebraic soft decision decoding asd algorithm is a polynomial time soft decoding algorithm for reed solomon rs codes it outperforms both the algebraic hard decision decoding ahd and the conventional unique decoding algorithms but with a high computational cost this paper proposes a progressive asd pasd algorithm that enables the conventional asd algorithm to perform decoding with an adjustable designed factorization output list size ols the ols is enlarged progressively leading to an incremental computation for the interpolation and an enhanced error correction capability multiple factorizations are performed in order to find out the intended message polynomial which will be validated by a cyclic redundant check crc code the incremental interpolation constraints are introduced to characterize the progressive decoding the validity analysis of the algorithm shows the pasd algorithm is a natural and computationally saving generalization of the asd algorithm delivering the same interpolation solution the average decoding complexity of the algorithm is further theoretically characterized revealing its dependence on the channel condition the simulation results further validate the analysis by showing that the average decoding complexity can be converged to the minimal level in a good channel condition finally performance evaluation shows the pasd algorithm preserves the error correction capability of the asd algorithm

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

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

multivalued behavior for a two level system using homotopy analysis method abstract we use the homotopy analysis method ham to solve the system of equations modeling the two level system and extract results which will pinpoint to turbulent behavior we look at multivalued solutions as indicative of turbulence or turbulent like behavior we take different specific cases which result in multivalued velocities the solutions are in series form and application of ham ensures convergence in some region

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

two low complexity reliability based message passing algorithms for decoding non binary ldpc codes this paper presents two low complexity reliability based message passing algorithms for decoding ldpc codes over non binary finite fields these two decoding algorithms require only finite field and integer operations and they provide effective trade off between error performance and decoding complexity compared to the non binary sum product algorithm they are particularly effective for decoding ldpc codes constructed based on finite geometries and finite fields

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

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

short non binary low density parity check codes for phase noise channels this paper considers the design of short non binary low density parity check ldpc codes over finite fields of order m for channels with phase noise in particular m ary differential phase shift keying dpsk modulated code symbols are transmitted over an additive white gaussian noise awgn channel with the wiener phase noise at the receiver side non coherent detection takes place with the help of a multi symbol detection algorithm followed by a non binary decoding step both the detector and the decoder operate on a joint factor graph as a benchmark finite length bounds and information rate expressions are computed and compared with the codeword error rate cer performance as well as the iterative threshold of the obtained codes as a result performance within 1 2 db from finite length bounds is obtained down to a cer of 10 3

noisy gradient descent bit flip decoding for ldpc codes a modified gradient descent bit flipping gdbf algorithm is proposed for decoding low density parity check ldpc codes on the binary input additive white gaussian noise channel the new algorithm called noisy gdbf ngdbf introduces a random perturbation into each symbol metric at each iteration the noise perturbation allows the algorithm to escape from undesirable local maxima resulting in improved performance a combination of heuristic improvements to the algorithm are proposed and evaluated when the proposed heuristics are applied ngdbf performs better than any previously reported gdbf variant and comes within 0 5 db of the belief propagation algorithm for several tested codes unlike other previous gdbf algorithms that provide an escape from local maxima the proposed algorithm uses only local fully parallelizable operations and does not require computing a global objective function or a sort over symbol metrics making it highly efficient in comparison the proposed ngdbf algorithm requires channel state information which must be obtained from a signal to noise ratio snr estimator architectural details are presented for implementing the ngdbf algorithm complexity analysis and optimizations are also discussed

integral points of bounded height on compactifications of semi simple groups we study the asymptotic distribution of integral points of bounded height on partial bi equivariant compactifications of semi simple groups of adjoint type

a numerical solution of the coupled sine gordon equation using the modified decomposition method the modified decomposition method has been implemented for solving a coupled sine gordon equation we consider a system of coupled sine gordon equation which models one dimensional nonlinear wave processes in two component media by using an initial value system the numerical solutions of coupled sine gordon equation have been represented graphically

nonbinary hybrid ldpc codes in this paper a new class of low density parity check ldpc codes named hybrid ldpc codes is introduced hybrid ldpc codes are characterized by an irregular connectivity profile and heterogeneous orders of the symbols in the codeword it is shown in particular that the class of hybrid ldpc codes can be asymptotically characterized and optimized using density evolution de framework and a technique to maximize the minimum distance of the code is presented numerical assessment of hybrid ldpc code performances is provided by comparing them to protograph based and multiedge type met ldpc codes hybrid ldpc codes are shown to allow to achieve an interesting tradeoff between good error floor performance and good waterfall region with nonbinary coding techniques

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

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

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

the variational iteration method for solving linear and nonlinear systems of pdes in this work the variational iteration method vim is used for analytic treatment of the linear and nonlinear systems of partial differential equations the method reduces the calculation size and overcomes the difficulty of handling nonlinear terms numerical examples are examined to highlight the significant features of the vim method the method shows improvements over existing numerical techniques

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

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

non binary ldpc code design for the poisson ppm channel this paper investigates the design of non binary protograph low density parity check codes for the poisson channel with m ary pulse position modulation the field order over which the code is constructed is matched to the pulse position modulation order yielding a coded modulation scheme the optimization of the low density parity check code structure is performed via protograph density evolution on a surrogate m ary erasure channel the surrogate design is illustrated to be not only accurate but also robust for a range of practical values of channel background noise and various modulation orders as a result the proposed codes show excellent performance over the poisson channel with pulse position modulation outperforming competing schemes as a side product of this paper finite length benchmarks on the block error probability are provided together with a union bound to characterize the code performance in the error floor region

selective avoidance of cycles in irregular ldpc code construction this letter explains the effect of graph connectivity on error floor performance of low density parity check ldpc codes under message passing decoding a new metric called extrinsic message degree emd measures cycle connectivity in bipartite graphs of ldpc codes using an easily computed estimate of emd we propose a viterbi like algorithm that selectively avoids small cycle clusters that are isolated from the rest of the graph this algorithm is different from conventional girth conditioning by emphasizing the connectivity as well as the length of cycles the algorithm yields codes with error floors that are orders of magnitude below those of random codes with very small degradation in capacity approaching capability

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

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

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

le defaut d approximation forte dans les groupes lineaires connexes let g be a connected linear algebraic group over a number field k we establish an exact sequence describing the closure of the group g k of rational points of g in the group of adelic points of g this exact sequence describes the defect of strong approximation on g in terms of the algebraic brauer group of g in particular we deduce from those results that the integral brauer manin obstruction on a torsor under the group g is the only obstruction to the existence of an integral point on this torsor we also obtain a non abelian poitou tate exact sequence for the galois cohomology of the linear group g the main ingredients in the proof of those results are the local and global duality theorems for complexes of k tori of length two and the abelianization maps in galois cohomology introduced by borovoi

integral points for groups of multiplicative type abstract we construct a finite subgroup of brauer manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type as an application the strong approximation theorem for linear algebraic groups of multiplicative type is established moreover the sum of two integral squares over some quadratic fields is discussed

integral points for multi norm tori we construct a finite subgroup of brauer manin obstruction for detecting the existence of integral points on integral models of principle homogeneous spaces of multi norm tori several explicit examples are provided

construction of non binary quasi cyclic ldpc codes by arrays and array dispersions transactions papers this paper presents two algebraic methods for constructing high performance and efficiently encodable nonbinary quasi cyclic ldpc codes based on arrays of special circulant permutation matrices and multi fold array dispersions codes constructed based on these methods perform well over the awgn and other types of channels with iterative decoding based on belief propagation experimental results show that over the awgn channel these non binary quasi cyclic ldpc codes significantly outperform 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 also presented in this paper is a class of asymptotically optimal ldpc codes for correcting bursts of erasures codes constructed also perform well over flat fading channels non binary quasi cyclic ldpc codes have a great potential to replace reed solomon codes in some applications in communication environments and storage systems for combating mixed types of noises and interferences

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

nonuniform error correction using low density parity check codes this correspondence introduces a framework to design and analyze low density parity check ldpc codes over nonuniform channels we study ldpc codes for channels with nonuniform noise distributions rate adaptive coding and unequal error protection first we propose a technique to design ldpc codes for volume holographic memory vhm systems for which the noise distribution is nonuniform we show that the proposed coding scheme has an easy design procedure and results in efficient codes for holographic memories an important property of the proposed technique is the design of the codes that have a low error floor and low variable node degrees while maintaining performance close to the shannon limit we then show that punctured ldpc codes can be studied as a special case of our design methodology for nonuniform channels finally we propose a method to generate ldpc codes that can provide unequal error protection in addition to having a good overall performance moreover the highly protected bits can be decoded without requiring the entire word to be decoded

searching for binary and nonbinary block and convolutional ldpc codes a unified approach to search for and optimize codes determined by their sparse parity check matrices is presented replacing the nonzero elements of a binary parity check matrix the base or parent matrix either by circulants or by companion matrices of elements from a finite field gf 2 m we obtain quasi cyclic low density parity check ldpc block codes and binary images of nonbinary ldpc block codes respectively by substituting monomials of a formal variable d we obtain the polynomial description of an ldpc convolutional code a set of performance measures applicable to different classes of ldpc codes is considered and a greedy algorithm for code performance optimization is presented the heart of the new optimization algorithm is a fast procedure for searching for ldpc codes with large girth of their tanner graphs for a few classes of ldpc codes examples of codes combining good error correcting performance with compact representation are obtained in particular we present optimized convolutional ldpc codes and conclude that the ldpc block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered moreover a specific channel model can easily be embedded into the optimization loop thereby the code can be optimized for a specific channel the efficiency of such an optimization is demonstrated via an example of faster than nyquist ftn signaling using ldpc codes the ftn strategy combined with a rate r 1 2 ldpc code of length 64 800 optimized for effective data rate r 3 4 gains more than 0 5 db compared with the standard ldpc codes of the same rate and length the obtained gain corresponds to transmission at the capacity of the binary input additive white gaussian noise channel in most numerical examples we consider codes with bidiagonal structure of the parity check matrix this restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards

generalized binary representation for the nonbinary ldpc code with decoder design in this paper we consider the performance optimized nonbinary low density parity check code over general linear group i e c a new methodology for constructing the binary representation generalized binary representation gbr of c is proposed which can be optimized with regard to both degree distributions and girth as to the decoding of the gbr we develop a low complexity hybrid parallel decoding process it is shown that the decoding performance of the gbr under the proposed binary decoding process could closely approach the decoding performance of its mother code c under nonbinary belief propagation decoding a simple code optimization algorithm for the gbr is also provided simulations show the comparative results and justify the advantages of the proposed constructions

stochastic decoding of ldpc codes over gf q despite the outstanding performance of non binary low density parity check ldpc codes over many communication channels they are not in widespread use yet this is due to the high implementation complexity of their decoding algorithms even those that compromise performance for the sake of simplicity in this paper we present three algorithms based on stochastic computation to reduce the decoding complexity the first is a purely stochastic algorithm with error correcting performance matching that of the sum product algorithm spa for ldpc codes over galois fields with low order and a small variable node degree we also present a modified version which reduces the number of decoding iterations required while remaining purely stochastic and having a low per iteration complexity the second algorithm relaxed half stochastic rhs decoding combines elements of the spa and the stochastic decoder and uses successive relaxation to match the error correcting performance of the spa furthermore it uses fewer iterations than the purely stochastic algorithm and does not have limitations on the field order and variable node degree of the codes it can decode the third algorithm nox is a fully stochastic specialization of rhs for codes with a variable node degree 2 that offers similar performance but at a significantly lower computational complexity we study the performance and complexity of the algorithms noting that all have lower per iteration complexity than spa and that rhs can have comparable average per codeword computational complexity and nox a lower one

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

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

source coding with side information at the decoder and uncertain knowledge of the correlation this paper considers the problem of lossless source coding with side information at the decoder when the correlation model between the source and the side information is uncertain four parametrized models representing the correlation between the source and the side information are introduced the uncertainty on the correlation appears through the lack of knowledge on the value of the parameters for each model we propose a practical coding scheme based on non binary low density parity check codes and able to deal with the parameter uncertainty at the encoder the choice of the coding rate results from an information theoretical analysis then we propose decoding algorithms that jointly estimate the source vector and the parameters as the proposed decoder is based on the expectation maximization algorithm which is very sensitive to initialization we also propose a method to produce first a coarse estimate of the parameters

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

gradient descent bit flipping algorithms for decoding ldpc codes a novel class of bit flipping bf algorithm for decoding low density parity check ldpc codes is presented the proposed algorithms which are referred to as gradient descent bit flipping gdbf algorithms can be regarded as simplified gradient descent algorithms the proposed algorithms exhibit better decoding performance than known bf algorithms such as the weighted bf algorithm or the modified weighted bf algorithm for several ldpc codes

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