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

an efficient algorithm to find all small size stopping sets of low density parity check matrices in this work we introduce an efficient algorithm to find all stopping sets of size less than some threshold of a fixed low density parity check ldpc matrix the solution is inspired by the algorithm proposed by rosnes and ytrehus in 2005 to find an exhaustive list of all small size turbo stopping sets in a turbo code the efficiency of the proposed algorithm is demonstrated by several numerical examples for instance we have applied the algorithm to the well known 3 5 regular 155 64 tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer also we have verified that the minimum stopping set size of the 4896 2474 ramanujan margulis code is indeed 24 and that the corresponding multiplicity is exactly 204 furthermore we have applied the algorithm to the ieee 802 16e ldpc codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes finally as an application we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity check matrix in order to remove all stopping sets in the corresponding tanner graph of size less than the minimum distance an extensive case study of the 155 64 tanner code illustrates the usefulness of the algorithm and we present a 110 times 155 redundant parity check matrix for this code with no stopping sets of size less than the minimum distance simulation results of iterative decoding on the binary erasure channel show performance improvements for low to medium erasure probabilities when this redundant parity check matrix is used for decoding

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

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

reconstruction guarantee analysis of basis pursuit for binary measurement matrices in compressed sensing recently binary 0 1 measurement matrices especially those from coding theory were introduced to compressed sensing dimakis et al found that the linear programming lp decoding of ldpc codes is very similar to the lp reconstruction of compressed sensing and they further showed that the sparse binary parity check matrices of good ldpc codes can be used as provably good measurement matrices for compressed sensing under basis pursuit bp moreover khajehnejad et al made use of girth to certify the good performances of sparse binary measurement matrices in this paper we examine the performance of binary measurement matrices with uniform column weight and arbitrary girth under bp for a fixed measurement matrix we first introduce a performance indicator w bp min called minimum bp weight and show that any k sparse signals could be exactly recovered by bp if and only if k le w bp min 1 2 then lower bounds of w bp min are studied borrowing ideas from the tree bound for the ldpc codes we obtain several explicit lower bounds of w bp min which improve on the previous results in some cases these lower bounds also imply explicit ell 1 ell 1 ell 2 ell 1 and ell infty ell 1 sparse approximation guarantees and further confirm that large girth has positive impacts on the performance of binary measurement matrices under bp

new constructions of quasi cyclic ldpc codes based on special classes of bibd s for the awgn and binary erasure channels this paper presents new methods for efficiently constructing encodable quasi cyclic low density parity check ldpc codes based on special balanced incomplete block designs bibds codes constructed perform well over both the additive white gaussian noise awgn and binary erasure channels with iterative decoding

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

performance analysis of ldpc coded diversity combining on rayleigh fading channels with impulsive noise spatial diversity is an effective method to mitigate the effects of fading and when used in conjunction with low density parity check ldpc codes it can achieve excellent error correcting performance noise added at each branch of the diversity combiner is generally assumed to be additive white gaussian noise but there are many applications where the received signal is impaired by noise with a non gaussian distribution in this paper we derive the exact bit error probability of different linear combining techniques on rayleigh fading channels with impulsive noise which is modeled using symmetric alpha stable distributions the relationship for the signal to noise ratios of these linear combiners is derived and then different non linear detectors are presented a detector based on the bi parameter cauchy gaussian mixture model is used and shows near optimal performance with a significant reduction in complexity when compared with the optimal detector furthermore the threshold signal to noise ratio of ldpc codes for different combining techniques on these channels is derived using density evolution and an estimation of the waterfall performance of ldpc codes is derived that reduces the gap between simulated and asymptotic performance

construction of near optimum burst erasure correcting low density parity check codes in this paper a simple and effective tool for the design of low density parity check ldpc codes for iterative correction of bursts of erasures is presented the design method consists of starting from the parity check matrix of an ldpc code and developing an optimized parity check matrix with the same performance over the memoryless erasure channel and suitable also for the iterative correction of single erasure bursts the parity check matrix optimization is performed by an algorithm called pivot searching and swapping pss algorithm it executes permutations of carefully chosen columns of the parity check matrix after a local analysis of particular variable nodes called stopping set pivots this algorithm can be in principle applied to any ldpc code if the input parity check matrix is designed to achieve a good performance over the memoryless erasure channel then the code obtained after the application of the algorithm provides a good joint correction of independent erasures and single erasure bursts numerical results are provided in order to show the algorithm effectiveness when applied to different categories of ldpc codes

high rate quantum low density parity check codes assisted by reliable qubits quantum error correction is an important building block for reliable quantum information processing a challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation a typical obstacle to constructing a variety of strong quantum error correcting codes is the complicated restrictions imposed on the structure of a code recently promising solutions to this problem have been proposed in quantum information science where in principle any binary linear code can be turned into a quantum error correcting code by assuming a small number of reliable quantum bits this paper studies how best to take advantage of these latest ideas to construct desirable quantum error correcting codes of very high information rate our methods exploit structured high rate low density parity check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths our approach to designing high rate quantum error correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes making it possible to deal with a situation where promising quantum analogues of low density parity check codes are difficult to find

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

quasi cyclic ldpc codes an algebraic construction rank analysis and codes on latin squares quasi cyclic ldpc codes are the most promising class of structured ldpc codes due to their ease of implementation and excellent performance over noisy channels when decoded with message passing algorithms as extensive simulation studies have shown in this paper an approach for constructing quasi cyclic ldpc codes based on latin squares over finite fields is presented by analyzing the parity check matrices of these codes combinatorial expressions for their ranks and dimensions are derived experimental results show that with iterative decoding algorithms the constructed codes perform very well over the awgn and the binary erasure channels

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

finite alphabet iterative decoders part i decoding beyond belief propagation on the binary symmetric channel we introduce a new paradigm for finite precision iterative decoding on low density parity check codes over the binary symmetric channel the messages take values from a finite alphabet and unlike traditional quantized decoders which are quantized versions of the belief propagation bp decoder the proposed finite alphabet iterative decoders faids do not propagate quantized probabilities or log likelihoods and the variable node update functions do not mimic the bp decoder rather the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code thereby rendering these decoders capable of outperforming the bp in the error floor region on certain column weight three codes of practical interest we show that there exist faids that surpass the floating point bp decoder in the error floor region while requiring only three bits of precision for the representation of the messages hence faids are able to achieve a superior performance at much lower complexity we also provide a methodology for the selection of faids that is not code specific but gives a set of candidate faids containing potentially good decoders in the error floor region for any column weight three code we validate the code generality of our methodology by providing particularly good three bit precision faids for a variety of codes with different rates and lengths

group theoretic analysis of cayley graph based cycle gf 2 p codes using group theory we analyze cycle gf 2p codes that use cayley graphs as their associated graphs first we show that through row and column permutations the parity check matrix h can be put in a concatenation form of row permuted block diagonal matrices encoding utilizing this form can be performed in linear time and in parallel second we derive a rule to determine the nonzero entries of h and present determinate and semi determinate codes our simulations show that the determinate and semi determinate codes have better performance than codes with randomly generated nonzero entries for gf 16 and gf 64 and have similar performance for gf 256 the constructed determinate and semi determinate codes over gf 64 and gf 256 can outperform the binary irregular counterparts of the same block lengths one distinct advantage for determinate and semi determinate codes is that they greatly reduce the storage cost of h for decoding the results in this correspondence are appealing for the implementation of efficient encoders and decoders for this class of promising ldpc codes especially when the block length is large

message passing algorithms for counting short cycles in a graph a message passing algorithm for counting short cycles in a graph is presented for bipartite graphs which are of particular interest in coding the algorithm is capable of counting cycles of length g g 2 2g 2 where g is the girth of the graph for a general non bipartite graph cycles of length g g 1 2g 1 can be counted the algorithm is based on performing integer additions and subtractions in the nodes of the graph and passing extrinsic messages to adjacent nodes the complexity of the proposed algorithm grows as o g e 2 where e is the number of edges in the graph for sparse graphs the proposed algorithm significantly outperforms the existing algorithms tailored for counting em short cycles in terms of computational complexity and memory requirements we also discuss a more generic and basic approach of counting short cycles which is based on matrix multiplication and provide a message passing interpretation for such an approach we then demonstrate that an efficient implementation of the matrix multiplication approach has essentially the same complexity as the proposed message passing algorithm

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

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

the design of rate compatible protograph ldpc codes this paper presents a simple yet effective method for designing nested families of ldpc codes rate compatible codes are essential for many communication applications e g hybrid automatic repeat request harq systems and their design is nontrivial due to the difficulty of simultaneously guaranteeing the quality of several related codes puncturing can be used to generate rate compatible ldpc codes but it produces a gap to capacity that in practice often significantly exceeds the gap of the mother code we propose an alternative method based on successively extending a high rate protograph the resulting codes not only inherit the advantages of protograph codes namely low encoding complexity and efficient decoding algorithms but also cover a wide range of rates and have very good performance with iterative decoding thresholds that are within 0 2 db of their capacity limits

error rate estimation of low density parity check codes decoded by quantized soft decision iterative algorithms this paper describes a combinatorial approach to estimate the error rate performance of low density parity check ldpc codes decoded by quantized soft decision iterative decoding algorithms the method is based on efficient enumeration of input vectors with small distances to a reference vector whose elements are selected to be the most reliable values from the input alphabet several techniques including modified cycle enumeration and the efficient derivation of problematic inputs for finer quantizers from those of coarser ones are employed to reduce the complexity of the enumeration the error rate estimate is derived by testing the input vectors of small distances followed by estimating the contribution of larger distance vectors we demonstrate by a number of examples that the proposed method provides accurate estimates of error rate with computational complexity much lower than that of monte carlo simulations especially at the error floor region

the rost nilpotence theorem for geometrically rational surfaces we show that the rost nilpotence theorem is true for geometrically rational surfaces over fields of characteristic zero

design of ldpc codes based on multipath emd strategies for progressive edge growth low density parity check ldpc codes are capable of achieving excellent performance and provide a useful alternative for the high performance applications however at medium to high signal to noise ratios an observable error floor arises from the loss of independence of messages passed under iterative graph based decoding in this paper the error floor performance of the short block length codes is improved by the use of a novel candidate selection metric in code graph construction the proposed multipath extrinsic message degree emd approach avoids harmful structures in the graph by evaluating certain properties of the cycles introduced in each edge placement we present multipath emd based designs for several structured ldpc codes including quasi cyclic and the irregular repeat accumulate codes in addition an extended class of the diversity achieving codes on the challenging block fading channel is proposed and considered with the multipath emd design this combined approach is demonstrated to provide the gains in decoder convergence and error rate performance a simulation study evaluates the performance of the proposed and existing state of the art methods

rost nilpotence and higher unramified cohomology abstract we develop an approach to proving the rost nilpotence principle involving higher unramified cohomology along with a decomposition of the diagonal technique we use this to prove the principle for certain threefolds over a perfect field

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

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

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

constructing short length irregular ldpc codes with low error floor trapping sets tss are known to cause error floors in regular and irregular low density parity check ldpc codes by avoiding major error contributing tss during the code construction process codes with low error floors can effectively be built in it has been shown that tss labeled as w u are considered as being equivalent under the automorphism of the graph and are therefore contributing equally to the error floor however tss with the same label w u are not identical in general particularly for the case of irregular ldpc codes in this paper we introduce a parameter e that can identify the number of distinguishable cycles in the connected subgraph induced by an elementary trapping set further we propose a code construction algorithm namely the progressive edge growth approximate minimum cycle set extrinsic message degree peg acse method that aims to avoid small elementary trapping sets etss particularly detrimental etss we also develop theorems evaluating the minimum possible etss formed by peg construction algorithms in general we compare the characteristics of the codes built using the proposed method and those built using peg only or peg approximate minimum cycle extrinsic message degree peg ace methods results from simulations show that the codes constructed using the proposed peg acse method produce lower error rates particularly at the high signal to noise snr region compared with codes constructed using other peg based algorithms

computing linear transformations with unreliable components we consider the problem of computing a binary linear transformation when all circuit components are unreliable two models of unreliable components are considered probabilistic errors and permanent errors we introduce the encoded technique that ensures that the error probability of the computation of the linear transformation is kept bounded below a small constant independent of the size of the linear transformation even when all logic gates in the computation are noisy by deriving a lower bound we show that in some cases the computational complexity of the encoded technique achieves the optimal scaling in error probability further we examine the gain in energy efficiency from the use of a voltage scaling scheme where gate energy is reduced by lowering the supply voltage we use a gate energy reliability model to show that tuning gate energy appropriately at different stages of the computation dynamic voltage scaling in conjunction with encoded can lead to orders of magnitude energy savings over the classical uncoded approach finally we also examine the problem of computing a linear transformation when noiseless decoders can be used providing upper and lower bounds to the problem

on finding bipartite graphs with a small number of short cycles and large girth the problem of finding bipartite tanner graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low density parity check ldpc codes in this paper we prove that for given integers alpha beta and gamma and degree sequences pi and pi the problem of determining whether there exists a simple bipartite graph with degree sequences pi pi that has at most alpha beta and gamma cycles of length four six and eight respectively is np complete this is proved by a two step polynomial time reduction from the 3 partition problem on the other hand using connections to linear hypergraphs we prove that given the degree sequence pi a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is pi and has a girth of at least six in addition to these complexity results we devise a quasi polynomial time algorithm that can construct a bipartite graph with given irregular degree sequences and a girth that increases logarithmically with the size of the graph compared to the well known progressive edge growth peg algorithm the proposed method in some cases results in tanner graphs with larger girth particularly for scenarios where the degree sequences of the graph are strictly enforced

counting short cycles in bipartite graphs a fast technique algorithm and a hardness result in this paper we propose a new technique based on the so called breadth first search algorithm to count the short cycles of a bipartite graph for a general bipartite graph with v nodes and girth g our technique has a time complexity of mathcal o v 2 delta to count g cycles and g 2 cycles and a time complexity of mathcal o v 2 delta 2 to count g 4 cycles where delta is the maximum node degree in the graph moreover for bi regular bipartite graphs the latter complexity is further reduced to mathcal o v 2 delta compared to the fastest known algorithm which has a complexity mathcal o g v 2 delta 2 the proposed method always has a lower complexity for counting g cycles and g 2 cycles it also has a lower complexity for counting g 4 cycles in bi regular graphs and in scenarios where g is increased with the size of the graph related to the problem of counting short cycles we also demonstrate using a long standing conjecture that there is no algorithm with time complexity less than mathcal o left v 2 frac 2 1 i right that can determine whether a given sparse bipartite graph has a cycle of length 4i an important application of the results presented here is to count the short cycles of tanner graphs of low density parity check ldpc codes

bounds on the size of parity check matrices for quasi cyclic low density parity check codes in this paper we investigate the cycle properties of quasi cyclic low density parity check qc ldpc codes using the sequence representation of a parity check matrix for a qc ldpc code we analyze a necessary and sufficient condition for a cycle of a given length to exist we then derive bounds which are necessary conditions for a qc ldpc code to have a given girth in terms of its parameters we also give a bound which is a sufficient condition for a qc ldpc code of a given girth to be constructed by a greedy algorithm the bounds derived here are applicable to any regular or irregular qc ldpc codes as well as they improve the existing bounds in many classes of regular ldpc codes

progressive transmission of images over fading channels using rate compatible ldpc codes in this paper we propose a combined source channel coding scheme for transmission of images over fading channels the proposed scheme employs rate compatible low density parity check codes along with embedded image coders such as jpeg2000 and set partitioning in hierarchical trees spiht the assignment of channel coding rates to source packets is performed by a fast trellis based algorithm we examine the performance of the proposed scheme over correlated and uncorrelated rayleigh flat fading channels with and without side information simulation results for the expected peak signal to noise ratio of reconstructed images which are within 1 db of the capacity upper bound over a wide range of channel signal to noise ratios show considerable improvement compared to existing results under similar conditions we also study the sensitivity of the proposed scheme in the presence of channel estimation error at the transmitter and demonstrate that under most conditions our scheme is more robust compared to existing schemes

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

rost nilpotence and etale motivic cohomology abstract a smooth projective scheme x over a field k is said to satisfy the rost nilpotence principle if any endomorphism of x in the category of chow motives that vanishes on an extension of the base field k is nilpotent we show that an etale motivic analogue of the rost nilpotence principle holds for all smooth projective schemes over a perfect field this provides a new approach to the question of rost nilpotence and allows us to obtain an elegant proof of rost nilpotence for surfaces as well as for birationally ruled threefolds over a field of characteristic 0

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

improved low density parity check accumulate ldpca codes we present improved constructions for low density parity check accumulate ldpca codes which are rate adaptive codes commonly used for distributed source coding dsc applications our proposed constructions mirror the traditional ldpca approach higher rate codes are obtained by splitting the check nodes in the decoding graph of lower rate codes beginning with a lowest rate mother code in a departure from the uniform splitting strategy adopted by prior ldpca codes however the proposed constructions introduce non uniform splitting of the check nodes at higher rates codes are designed by a global minimization of the average rate gap between the code operating rates and the corresponding theoretical lower bounds evaluated by density evolution in the process of formulating the design framework the paper also contributes a formal definition of ldpca codes performance improvements provided by the proposed non uniform splitting strategy over the conventional uniform splitting approach used in prior work are substantiated via density evolution based analysis and dsc codec simulations optimized designs for our proposed constructions yield codes with a lower average rate gap than conventional designs and alleviate the trade off between the performance at different rates inherent in conventional designs a software implementation is provided for the codec developed

optimal erasure selection of m ary pam signaling for errors and erasures decoding algorithms erasure selection for errors and erasures decoding algorithms is investigated and analyzed where declaring erasures provides additional gain for hard decision decoding to derive an analytical criterion for optimal erasure selection we first investigate erasure effects on decoding performance by which a metric function for erasure selection is derived from the derived metric a suboptimal threshold is defined which can be used for errors and erasures decoding of large code distance channel codes moreover application to m ary m 2 pam constellation is also described for verification of the proposed erasure threshold simulation results with popular channel codes are included

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

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

low complexity coded modulation for isi constrained channels we propose a low complexity pam based transmission scheme that is well suited for channels constrained by inter symbol interference isi and colored gaussian noise the scheme consists of coset codes constructed on multi dimensional lattices with a combination of low density parity check ldpc codes and classical reed solomon rs codes to approach the capacity of an isi constrained channel the code is easy to employ in conjunction with spectral shaping tomlinson harashima precoding and decision feedback equalization dfe the scheme performs within 2 2 5 db of un shaped channel capacity the sphere bound at very low ber s even with regular ldpc codes of modest block lengths we investigate dense multidimensional lattices such as the schlafli gosset barnes wall and leech lattices besides simple one dimensional lattices via the density evolution technique we show that the lattices reduce the noise threshold of belief propagation decoders we investigate the practical application of the proposed schemes to 10g base t an emerging ethernet standard over twisted pairs at 10 gbit sec a simple 1 dimensional scheme improves upon recent proposals by 0 5 1 db at ber s approaching 10 11 with half the ldpc coding complexity multi dimensional schemes are seen to reduce the complexity further

searching for voltage graph based ldpc tailbiting codes with large girth the relation between parity check matrices of quasi cyclic qc low density parity check ldpc codes and biadjacency matrices of bipartite graphs supports searching for powerful ldpc block codes using the principle of tailbiting compact representations of bipartite graphs based on convolutional codes can be found bounds on the girth and the minimum distance of ldpc block codes constructed in such a way are discussed algorithms for searching iteratively for ldpc block codes with large girth and for determining their minimum distance are presented constructions based on all one matrices steiner triple systems and qc block codes are introduced finally new qc regular ldpc block codes with girth up to 24 are given

minimum distance bounds by graph analysis the parity check matrix of a linear code is used to define a bipartite code constraint tanner graph in which bit nodes are connected to parity check nodes the connectivity properties of this graph are analyzed using both local connectivity and the eigenvalues of the associated adjacency matrix a simple lower bound on the minimum distance of the code is expressed in terms of the two largest eigenvalues for a more powerful bound local properties of the subgraph corresponding to a minimum weight word in the code are used to create an optimization problem whose solution is a lower bound on the code s minimum distance linear programming gives one bound the technique is illustrated by applying it to sparse block codes with parameters 7 3 4 and 42 23 6

construction of regular and irregular ldpc codes geometry decomposition and masking two algebraic methods for systematic construction of structured regular and irregular low density parity check ldpc codes with girth of at least six and good minimum distances are presented these two methods are based on geometry decomposition and a masking technique numerical results show that the codes constructed by these methods perform close to the shannon limit and as well as random like ldpc codes furthermore they have low error floors and their iterative decoding converges very fast the masking technique greatly simplifies the random like construction of irregular ldpc codes designed on the basis of the degree distributions of their code graphs

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

ldpc codes from generalized polygons we use the theory of finite classical generalized polygons to derive and study low density parity check ldpc codes the tanner graph of a generalized polygon ldpc code is highly symmetric inherits the diameter size of the parent generalized polygon and has minimum one half diameter to girth ratio we show formally that when the diameter is four or six or eight all codewords have even hamming weight when the generalized polygon has in addition an equal number of points and lines we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length a new minimum distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines finally we prove that all codes derived from finite classical generalized quadrangles are quasi cyclic and we give the explicit size of the circulant blocks in the parity check matrix our simulation studies of several generalized polygon ldpc codes demonstrate powerful bit error rate ber performance when decoding is carried out via low complexity variants of belief propagation

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

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

bounds on the minimum distance of punctured quasi cyclic ldpc codes recent work by divsalar has shown that properly designed protograph based low density parity check codes typically have minimum hamming distance linearly increasing with block length this fact rests on ensemble arguments over all possible expansions of the base protograph however when implementation complexity is considered the expansions are frequently selected from a smaller class of structured expansions for example protograph expansion by cyclically shifting connections generates a quasi cyclic qc code other recent work by smarandache and vontobel has provided upper bounds on the minimum distance of qc codes in this paper we generalize these bounds to punctured qc codes and then show how to tighten these for certain classes of codes we then evaluate these upper bounds for the family of protograph codes known as ar4ja codes that have been recommended for use in deep space communications in a standard established by the consultative committee for space data systems at block lengths larger than 4400 bits these upper bounds fall well below the ensemble lower bounds

decomposition methods for large scale lp decoding when binary linear error correcting codes are used over symmetric channels a relaxed version of the maximum likelihood decoding problem can be stated as a linear program lp this lp decoder can be used to decode error correcting codes at bit error rates comparable to state of the art belief propagation bp decoders but with significantly stronger theoretical guarantees however lp decoding when implemented with standard lp solvers does not easily scale to the block lengths of modern error correcting codes in this paper we draw on decomposition methods from optimization theory specifically the alternating direction method of multipliers admm to develop efficient distributed algorithms for lp decoding the key enabling technical result is a two slice characterization of the parity polytope the polytope formed by taking the convex hull of all codewords of the single parity check code this new characterization simplifies the representation of points in the polytope using this simplification we develop an efficient algorithm for euclidean norm projection onto the parity polytope this projection is required by the admm decoder and its solution allows us to use lp decoding with all its theoretical guarantees to decode large scale error correcting codes efficiently we present numerical results for ldpc codes of lengths more than 1000 the waterfall region of lp decoding is seen to initiate at a slightly higher snr than for sum product bp however an error floor is not observed for lp decoding which is not the case for bp our implementation of lp decoding using the admm executes as fast as our baseline sum product bp decoder is fully parallelizable and can be seen to implement a type of message passing with a particularly simple schedule

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

error rate estimation of low density parity check codes on binary symmetric channels using cycle enumeration the performance of low density parity check ldpc codes decoded by hard decision iterative decoding algorithms can be accurately estimated if the weight j and the number ej of the smallest error patterns that cannot be corrected by the decoder are known to obtain j and ej one would need to perform the direct enumeration of error patterns with weight i les j the complexity of enumeration increases exponentially with j essentially as nj where n is the code block length this limits the application of direct enumeration to codes with small n and j in this letter we approximate j and ej by enumerating and testing the error patterns that are subsets of short cycles in the code s tanner graph this reduces the computational complexity by several orders of magnitude compared to direct enumeration making it possible to estimate the error rates for almost any practical ldpc code to obtain the error rate estimates we propose an algorithm that progressively improves the estimates as larger cycles are enumerated through a number of examples we demonstrate that the proposed method can accurately estimate both the bit error rate ber and the frame error rate fer of regular and irregular ldpc codes decoded by a variety of hard decision iterative decoding algorithms

design of versatile eira codes for parallel decoders in this paper we propose a semi random technique for the generation of a class of eira codes a popular class of irregular ldpc codes that can be encoded in linear time suited to partially parallel decoder implementations the suggested technique tries to avoid both low weight codewords and nearcodewords which limit the performance of the belief propagation decoder its effectiveness is verified by comparison with literature results the obtained codes are versatile in terms of code rate and block length and they are characterized by a low error floor

on the uep capabilities of several ldpc construction algorithms this paper analyzes construction algorithms for low density parity check ldpc codes with respect to their unequal error protection uep capabilities we show that the choice of code construction algorithm highly affects the performance and uep properties of ldpc codes with identical degree distributions our results provide an explanation to disagreements in earlier research

design of protograph ldpc codes for partial response channels we investigate the performance of the protograph low density parity check ldpc codes which have been shown to possess simple structures and outstanding error performance over additive white gaussian noise awgn channels over partial response pr channels using the finite length extrinsic information transfer exit algorithm due to the intersymbol interference isi caused by the pr channels we observe that the conventional protograph ldpc codes do not perform well in terms of error rates we further propose a new design scheme and construct three new types of protograph ldpc codes unlike conventional protograph ldpc codes in which the highest degree variable nodes are punctured the new protograph ldpc codes have their lowest degree variable nodes punctured moreover some edge re connections are made in one of the proposed codes the exit chart analysis the convergence analysis and the bit error rate simulation have shown that all three new codes outperform the conventional protograph ldpc codes moreover two of the proposed codes are superior to the regular column weight 3 ldpc code and thus they are good alternatives compared to other error correction codes for use in data storage systems

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

expectation propagation as turbo equalizer in isi channels in probabilistic equalization of channels with inter symbol interference the bcjr algorithm and its approximations become intractable for high order modulations even for moderate channel dispersions in this paper we introduce a novel soft equalizer to approximate the symbol a posteriori probabilities app where the expectation propagation ep algorithm is used to provide an accurate estimation this new soft equalizer is presented as a block solution denoted as block ep bep where the structure of the matrices involved is exploited to reduce the complexity order to mathcal o l n 2 i e linear in the length of the channel l and quadratic in the frame length n the solution is presented in complex valued formulation within a turbo equalization scheme this algorithm can be cast as a linear minimum mean squared error lmmse turbo equalization with double feedback architecture where constellations being discrete is a restriction exploited by the ep that provides a first refinement of the app in the experiments included the bep exhibits a robust performance regardless of the channel response with gains in the range 1 5 5 db compared with the lmmse equalization

correctness of local probability propagation in graphical models with loops graphical models such as bayesian networks and markov networks represent joint distributions over a set of variables by means of a graph when the graph is singly connected local propagation rules of the sort proposed by pearl 1988 are guaranteed to converge to the correct posterior probabilities recently a number of researchers have empirically demonstrated good performance of these same local propagation schemes on graphs with loops but a theoretical understanding of this performance has yet to be achieved for graphical models with a single loop we derive an analytical relationship between the probabilities computed using local propagation and the correct marginals using this relationship we show a category of graphical models with loops for which local propagation gives rise to provably optimal maximum a posteriori assignments although the computed marginals will be incorrect we also show how nodes can use local information in the messages they receive in order to correct their computed marginals we discuss how these results can be extended to graphical models with multiple loops and show simulation results suggesting that some properties of propagation on single loop graphs may hold for a larger class of graphs specifically we discuss the implication of our results for understanding a class of recently proposed error correcting codes known as turbo codes

ldpc decoding scheduling for faster convergence and lower error floor this paper presents a maximum mutual information increase m 2 i 2 based algorithm that can be used to arrange low density parity check ldpc decoding schedules for faster convergence where the increase is used to guide the arrangement of the fixed decoding schedule the predicted mutual information for the messages to be updated is used in the calculation of the increase by looking ahead for several decoding stages a high order prediction can be realized which can then be used to devise a schedule with an even faster convergence for a single received frame different decoding results can be obtained using different schedules and hence schedule diversity that lowers the error floor resultant from the dominant trapping sets is proposed by adopting the m 2 i 2 based schedule together with the schedule diversity both the convergence speed in the waterfall region and the error rate in the floor region can be improved

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

permutation modules and chow motives of geometrically rational surfaces abstract we prove that the chow motive with integral coefficient of a geometrically rational surfaces s over a perfect field k is zero dimensional if and only if the picard group of k k s where k is an algebraic closure of k is a direct summand of a gal k k permutation module and s possesses a zero cycle of degree one as shown by colliot thelene in a letter to the author which we have reproduced in the appendix this is in turn equivalent to s having a zero cycle of degree 1 and ch 0 k s k s being torsion free

motivic decomposition of isotropic projective homogeneous varieties we give a decomposition of the chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties as an application we prove a generalization of rost s nilpotence theorem

on girth conditioning for low density parity check codes low density parity check ldpc codes are gaining interest for high data rate applications in both terrestrial and spatial communications they can be designed and studied through a bipartite graph whose characteristics affect the performance this paper proposes a low complexity method to improve the performance of ldpc codes by selectively removing some cycles from the associated bipartite graph the method is based on a modified version of the breadth first search bfs algorithm that we call modified bfs mbfs which is applied to find cycles and a greedy procedure to eliminate them throughout the paper we will give a detailed description of the algorithm proposed and analytically study its complexity simulation results show that this girth conditioning method applied to some classes of codes whose structure allows further optimization can lead to a significative complexity reduction and a performance improvements with respect to other methods

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

an algorithm for counting short cycles in bipartite graphs let g u spl cup w e be a bipartite graph with disjoint vertex sets u and w edge set e and girth g this correspondence presents an algorithm for counting the number of cycles of length g g 2 and g 4 incident upon every vertex in u spl cup w the proposed cycle counting algorithm consists of integer matrix operations and its complexity grows as o gn sup 3 where n max u w

divide and concur and difference map bp decoders for ldpc codes the divide and concur dc algorithm introduced by gravel and elser can be considered a competitor to the belief propagation bp algorithm in that both algorithms can be applied to a wide variety of constraint satisfaction optimization and inference problems we show that dc can be interpreted as a message passing algorithm on a normal factor graph the difference map dynamics of the dc algorithm enables it to avoid traps which may be related to the trapping sets or pseudo codewords that plague bp decoders of low density parity check ldpc codes in the error floor regime we investigate two decoders for ldpc codes based on these ideas the first decoder is based directly on dc while the second decoder borrows the important difference map concept from the dc algorithm and translates it into a bp like decoder we show that this difference map belief propagation dmbp decoder has dramatically improved error floor performance compared to standard bp decoders while maintaining a similar computational complexity we present simulation results for ldpc codes comparing dc and dmbp decoders with other decoders based on sum product bp linear programming and mixed integer linear programming we also describe the close relation of the dmbp decoder to reweighted min sum algorithms including those recently proposed by ruozzi and tatikonda

on the construction of structured ldpc codes free of small trapping sets we present a method to construct low density parity check ldpc codes with low error floors on the binary symmetric channel codes are constructed so that their tanner graphs are free of certain small trapping sets these trapping sets are selected from the trapping set ontology for the gallager a b decoder they are selected based on their relative harmfulness for a given decoding algorithm we evaluate the relative harmfulness of different trapping sets for the sum product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets we apply this method to construct structured ldpc codes to facilitate the discussion we give a new description of structured ldpc codes whose parity check matrices are arrays of permutation matrices this description uses latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the tanner graph of a code to be free of four cycles

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

finding the exhaustive list of small fully absorbing sets and designing the corresponding low error floor decoder this work provides an efficient exhaustive search algorithm for finding all small fully absorbing sets fass of any arbitrary low density parity check ldpc code the proposed algorithm is based on the branch bound principle for solving np complete problems in particular given any ldpc code the problem of finding all fass of size less than t is formulated as an integer programming problem for which a new branch bound algorithm is devised with new node selection and tree trimming mechanisms the resulting algorithm is capable of finding all fass of size 7 for ldpc codes of length 1000 when limiting the fass of interest to those with the number of violated parity check nodes 3 the proposed algorithm is capable of finding all such fass of size 14 for ldpc codes of lengths 1000 the resulting exhaustive list of small fass is then used to devise a new efficient post processing low error floor ldpc decoder the numerical results show that by exploiting the exhaustive list of small fass the proposed post processing decoder can significantly lower the error floor performance of a given ldpc code for various example codes of length 3000 the proposed post processing decoder lowers the error floor by a couple of orders of magnitude when compared to the standard belief propagation decoder and by an order of magnitude when compared to other existing low error floor decoders

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

integral motives relative krull schmidt principle and maranda type theorems in the present article we investigate properties of the category of integral grothendieck chow motives over a field we discuss the krull schmidt principle for integral motives provide a complete list of generalized severi brauer varieties with indecomposable integral motive and exploit a relation between the category of motives of twisted flag varieties and integral p adic representations of finite groups

low density parity check codes from transversal designs with improved stopping set distributions this paper examines the construction of low density parity check ldpc codes from transversal designs based on sets of mutually orthogonal latin squares mols by transferring the concept of configurations in combinatorial designs to the level of latin squares we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes based on large sets of simple structured mols we derive powerful constraints for the choice of suitable subsets leading to improved stopping set distributions for the corresponding codes we focus on ldpc codes with column weight 4 but the results are also applicable for the construction of codes with higher column weights finally we show that a subclass of the presented codes has quasi cyclic structure which allows low complexity encoding

efficient algorithm for finding dominant trapping sets of ldpc codes this paper presents an efficient algorithm for finding the dominant trapping sets of a low density parity check ldpc code the algorithm can be used to estimate the error floor of ldpc codes or as a tool to design ldpc codes with low error floors for regular codes the algorithm is initiated with a set of short cycles as the input for irregular codes in addition to short cycles variable nodes with low degree and cycles with low approximate cycle extrinsic message degree ace are also used as the initial inputs the initial inputs are then expanded recursively to dominant trapping sets of increasing size at the core of the algorithm lies the analysis of the graphical structure of dominant trapping sets and the relationship of such structures to short cycles low degree variable nodes and cycles with low ace the algorithm is universal in the sense that it can be used for an arbitrary graph and that it can be tailored to find a variety of graphical objects such as absorbing sets and zyablov pinsker trapping sets known to dominate the performance of ldpc codes in the error floor region over different channels and for different iterative decoding algorithms simulation results on several ldpc codes demonstrate the accuracy and efficiency of the proposed algorithm in particular the algorithm is significantly faster than the existing search algorithms for dominant trapping sets

on motivic decompositions arising from the method of bialynicki birula recently v chernousov s gille and a merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the bruhat decomposition using the method of a bialynicki birula and a corollary which is essentially due to s del bano i generalize this decomposition to the case of a possibly anisotropic smooth projective variety homogeneous under the action of an isotropic reductive group this answers a question of n karpenko

an algorithm for coprime matrix fraction description using sylvester matrices abstract several algorithms for the computation of coprime matrix fraction descriptions have been proposed in the past here we explore some of the properties of sylvester matrices and develop a different approach to the problem which is based on singular value decompositions and therefore avoids problems of numerical ill conditioning