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

maximum likelihood erasure decoding of ldpc codes pivoting algorithms and code design this paper investigates efficient maximum likelihood ml decoding of low density parity check ldpc codes over erasure channels a set of algorithms referred to as pivoting algorithms is developed the aim is to limit the average number of pivots or reference variables from which all the other erased symbols are recovered iteratively the suggested algorithms exhibit different trade offs between complexity of the pivoting phase and average number of pivots moreover a systematic procedure to design ldpc code ensembles for efficient ml decoding is proposed numerical results illustrate that the designed ldpc codes achieve a near optimum performance very close to the singleton bound at least down to a codeword error rate level 10 8 with an affordable decoding complexity for one of the presented codes and algorithms a software implementation has been developed which is capable to provide data rates above 1 5 gbps on a commercial computing platform

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

permutation decoding and the stopping redundancy hierarchy of cyclic and extended cyclic codes we introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel we derive lower and upper bounds for the stopping redundancy hierarchy via lovasz s local lemma lll and bonferroni type inequalities and specialize them for codes with cyclic parity check matrices based on the observed properties of parity check matrices with good stopping redundancy characteristics we develop a novel decoding technique termed automorphism group decoding that combines iterative message passing and permutation decoding we also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size simulation results demonstrate that for a large number of algebraic codes the performance of the new decoding method is close to that of maximum likelihood ml decoding

ldpc codes over the bec bounds and decoding algorithms the performance of maximum likelihood ml decoding on the binary erasure channel for finite length low density parity check ldpc codes from two random ensembles is studied a tightened union type upper bound on the ml decoding error probability based on the precise coefficients of the average weight spectrum is presented for ldpc codes from the gallager ensemble and the richardson urbanke ensemble new upper bounds on the ml decoding performance based on computing the rank of submatrices of the code parity check matrix are derived a new lower bound on the ml decoding threshold followed from the latter error probability bound is obtained an improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well a new low complexity near ml decoding algorithm for quasi cyclic ldpc codes is proposed and simulated its performance is compared to the simulated belief propagation and ml decoding performance and simulated performance of the best known improved iterative decoding techniques as well as with the derived upper bounds on the ml decoding performance and with decoding thresholds obtained by the density evolution technique

exploiting coding theory for classification we propose a multiclass to binary decomposition strategy founded on coding theory we use low density parity check codes a very effective family of binary block codes exploiting the algebraic properties of the code we handle both coding and decoding two decoding rules are proposed that provide many advantages over known strategies several experiments are performed that show significant performance improvements a powerful strategy for the classification of multiple classes is to create a classifier ensemble that decomposes the polychotomy into several dichotomies the central issue when designing a multiclass to binary decomposition scheme is the definition of both the coding matrix and the decoding algorithm in this study we propose a new classification system based on low density parity check codes which is a very effective class of binary block codes the main idea is to exploit the algebraic properties of the codes to generate the codewords for the coding matrix and to define two decoding approaches which allow us to detect and recover possible errors or rejects produced by the dichotomizers experiments based on benchmark datasets demonstrated that the proposed approach provides a statistically significant improvement in terms of the classification performance compared with state of the art decomposition strategies

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

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

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

symbolic extensions and smooth dynamical systems let f x x be a homeomorphism of the compact metric space x a symbolic extension of f x is a subshift on a finite alphabet g y which has f as a topological factor we show that a generic c1 non hyperbolic i e non anosov area preserving diffeomorphism of a compact surface has no symbolic extensions for r 1 we exhibit a residual subset mathcal r of an open set mathcal u of cr diffeomorphisms of a compact surface such that if f in mathcal r then any possible symbolic extension has topological entropy strictly larger than that of f these results complement the known fact that any c diffeomorphism has symbolic extensions with the same entropy we also show that cr generically on surfaces homoclinic closures which contain tangencies of stable and unstable manifolds have hausdorff dimension two

turbo decoding on the binary erasure channel finite length analysis and turbo stopping sets this paper is devoted to the finite length analysis of turbo decoding over the binary erasure channel bec the performance of iterative belief propagation decoding of low density parity check ldpc codes over the bec can be characterized in terms of stopping sets we describe turbo decoding on the bec which is simpler than turbo decoding on other channels we then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure apply turbo decoding until the transmitted codeword has been recovered or the decoder fails to progress further then the set of erased positions that will remain when the decoder stops is equal to the unique maximum size turbo stopping set which is also a subset of the set of erased positions furthermore we present some improvements of the basic turbo decoding algorithm on the bec the proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results finally we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption and an efficient enumeration algorithm of small size turbo stopping sets for a particular interleaver the solution is based on the algorithm proposed by garello et al in 2001 to compute an exhaustive list of all low weight codewords in a turbo code

on ensembles of low density parity check codes asymptotic distance distributions we derive expressions for the average distance distributions in several ensembles of regular low density parity check codes ldpc among these ensembles are the standard one defined by matrices having given column and row sums ensembles defined by matrices with given column sums or given row sums and an ensemble defined by bipartite graphs

on achievable rates and complexity of ldpc codes over parallel channels bounds and applications a variety of communication scenarios can be modeled by a set of parallel channels upper bounds on the achievable rates under maximum likelihood ml decoding and lower bounds on the decoding complexity per iteration of ensembles of low density parity check ldpc codes are presented the communication of these codes is assumed to take place over statistically independent parallel channels where the component channels are memoryless binary input and output symmetric the bounds are applied to ensembles of punctured ldpc codes where the puncturing patterns are either random or possess some structure our discussion is concluded by a diagram showing interconnections between the new theorems and some previously reported results

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

distance distributions in ensembles of irregular low density parity check codes we derive asymptotic expressions for the average distance distributions in ensembles of irregular low density parity check ldpc codes the ensembles are defined by matrices with given profiles of column and row sums

results on the improved decoding algorithm for low density parity check codes over the binary erasure channel in this correspondence we first investigate some analytical aspects of the recently proposed improved decoding algorithm for low density parity check ldpc codes over the binary erasure channel bec we derive a necessary and sufficient condition for the improved decoding algorithm to successfully complete decoding when the decoder is initialized to guess a predetermined number of guesses after the standard message passing terminates at a stopping set furthermore we present improved bounds on the number of bits to be guessed for successful completion of the decoding process when a stopping set is encountered under suitable conditions we derive a lower bound on the number of iterations to be performed for complete decoding of the stopping set we then present a superior novel improved decoding algorithm for ldpc codes over the binary erasure channel bec the proposed algorithm combines the observation that a considerable fraction of unsatisfied check nodes in the neighborhood of a stopping set are of degree two and the concept of guessing bits to perform simple and intuitive graph theoretic manipulations on the tanner graph the proposed decoding algorithm has a complexity similar to previous improved decoding algorithms finally we present simulation results of short length codes over bec that demonstrate the superiority of our algorithm over previous improved decoding algorithms for a wide range of bit error rates

tree expectation propagation for ml decoding of ldpc codes over the bec we propose a decoding algorithm for ldpc codes that achieves the maximum likelihood ml solution over the binary erasure channel bec in this channel the tree structured expectation propagation tep decoder improves the peeling decoder pd by processing check nodes of degree one and two however it does not achieve the ml solution as the tree structure of the tep allows only for approximate inference in this paper we provide the procedure to construct the structure needed for exact inference this algorithm denoted as generalized tree structured expectation propagation gtep modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph the gtep decoder upon completion either provides the unique ml solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ml solution we also explain the algorithm as a gaussian elimination method relating the gtep to other ml solutions compared to previous approaches it presents an equivalent complexity it exhibits a simpler graphical message passing procedure and most interesting the algorithm can be generalized to other channels

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

guessing facets polytope structure and improved lp decoder we investigate the structure of the polytope underlying the linear programming lp decoder introduced by feldman karger and wainwright we first show that for expander codes every fractional pseudocodeword always has at least a constant fraction of nonintegral bits we then prove that for expander codes the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword we further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as lp decoding that provably performs better the method is very simple it first applies ordinary lp decoding and when it fails it proceeds by guessing facets of the polytope and then resolving the linear program on these facets while the lp decoder succeeds only if the ml codeword has the highest likelihood over all pseudocodewords we prove that the proposed algorithm when applied to suitable expander codes succeeds unless there exists a certain number of pseudocodewords all adjacent to the ml codeword on the lp decoding polytope and with higher likelihood than the ml codeword we then describe an extended algorithm still with polynomial complexity that succeeds as long as there are at most polynomially many pseudocodewords above the ml codeword

parity check density versus performance of binary linear block codes over memoryless symmetric channels we derive lower bounds on the density of parity check matrices of binary linear codes which are used over memoryless binary input output symmetric mbios channels the bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit error probability vanishes for every mbios channel we construct a sequence of ensembles of regular low density parity check ldpc codes so that an upper bound on the asymptotic density of their parity check matrices scales similarly to the lower bound the tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right regular ldpc codes which was introduced by shokrollahi and which is known to achieve the capacity of this channel under iterative message passing decoding we show that this sequence of ensembles is asymptotically optimal in a sense to be defined in this paper strengthening a result of shokrollahi finally we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs and study their performance limitations over mbios channels the latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error correction codes

on decoding of low density parity check codes over the binary erasure channel this paper investigates decoding of low density parity check ldpc codes over the binary erasure channel bec we study the iterative and maximum likelihood ml decoding of ldpc codes on this channel we derive bounds on the ml decoding of ldpc codes on the bec we then present an improved decoding algorithm the proposed algorithm has almost the same complexity as the standard iterative decoding however it has better performance simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm we also provide some graph theoretic properties of different decoding algorithms of ldpc codes over the bec which we think are useful to better understand the ldpc decoding methods in particular for finite length codes

distance verification for classical and quantum ldpc codes the techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes then these techniques are considered for classical and quantum stabilizer low density parity check ldpc codes new complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of ldpc codes these bounds are expressed in terms of the erasure correcting capacity of the corresponding ensemble we also present a new irreducible cluster technique that can be applied to any ldpc code and takes advantage of parity checks sparsity for both the classical and quantum ldpc codes this technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances which also include all known families of the quantum stabilizer ldpc codes

computing the stopping distance of a tanner graph is np hard two decision problems related to the computation f stopping sets in tanner graphs are shown to be np complete it follows as a consequence that there exists no polynomial time algorithm for computing the stopping distance of a tanner graph unless p np

tree structure expectation propagation for ldpc decoding over the bec we present the tree structure expectation propagation tree ep algorithm to decode low density parity check ldpc codes over discrete memoryless channels dmcs expectation propagation generalizes belief propagation bp in two ways first it can be used with any exponential family distribution over the cliques in the graph second it can impose additional constraints on the marginal distributions we use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the ldpc code s tanner graph thanks to these additional constraints the tree ep marginal estimates for each variable in the graph are more accurate than those provided by bp we also reformulate the tree ep algorithm for the binary erasure channel bec as a peeling type algorithm tep and we show that the algorithm has the same computational complexity as bp and it decodes a higher fraction of errors we describe the tep decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters the solution of these equations is used to predict the tep decoder performance in both the asymptotic regime and the finite length regimes over the bec while the asymptotic threshold of the tep decoder is the same as the bp decoder for regular and optimized codes we propose a scaling law for finite length ldpc codes which accurately approximates the tep improved performance and facilitates its optimization

results on parity check matrices with optimal stopping and or dead end set enumerators the performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying tanner graph or equivalently the parity check matrix representing the code in this correspondence we introduce the notion of dead end sets to explicitly demonstrate this dependency the choice of the parity check matrix entails a tradeoff between performance and complexity we give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity check matrices further we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator

capacity achieving codes with bounded graphical complexity and maximum likelihood decoding in this paper the existence of capacity achieving codes for memoryless binary input output symmetric mbios channels under maximum likelihood ml decoding with bounded graphical complexity is investigated graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding irregular repeat accumulate ira codes are studied first utilizing the asymptotic average weight distribution aawd of these codes and invoking divsalar s bound on the binary input additive white gaussian noise biawgn channel it is shown that simple nonsystematic ira ensembles outperform systematic ira and regular low density parity check ldpc ensembles with the same graphical complexity and are at most 0 124 db away from the shannon limit however a conclusive result as to whether these nonsystematic ira codes can really achieve capacity cannot be reached motivated by this inconclusive result a new family of codes is proposed called low density parity check and generator matrix ldpc gm codes which are serially concatenated codes with an outer ldpc code and an inner low density generator matrix ldgm code it is shown that these codes can achieve capacity on any mbios channel using ml decoding and also achieve capacity on any bec using belief propagation bp decoding both with bounded graphical complexity moreover it is shown that under certain conditions these capacity achieving codes have linearly increasing minimum distances and achieve the asymptotic gilbert varshamov bound for all rates

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

asymptotic analysis and spatial coupling of counter braids a counter braid cb is a novel counter architecture introduced by lu et al in 2007 for per flow measurements on high speed links which can be decoded with low complexity using message passing mp cbs achieve an asymptotic compression rate under optimal decoding that matches the entropy lower bound of the flow size distribution in this paper we apply the concept of spatial coupling to cbs to improve the performance of the original cbs and analyze the performance of the resulting spatially coupled cbs sc cbs we introduce an equivalent bipartite graph representation of cbs with identical iteration by iteration finite length and asymptotic performance based on this equivalent representation we then analyze the asymptotic performance of single layer cbs and sc cbs under the mp decoding algorithm proposed by lu et al in particular we derive the potential threshold of the uncoupled system and show that it is equal to the area threshold we also derive the maxwell decoder for cbs and prove that the potential threshold is an upper bound on the maxwell decoding threshold which in turn is a lower bound on the maximum a posteriori map decoding threshold we then show that the area under the extended mp extrinsic information transfer curve defined for the equivalent graph computed for the expected residual cb graph when a peeling decoder equivalent to the mp decoder stops is equal to zero precisely at the area threshold this combined with the analysis of the maxwell decoder and simulation results leads us to the conjecture that the potential threshold is in fact equal to the maxwell decoding threshold and hence a lower bound on the map decoding threshold interestingly sc cbs do not show the well known phenomenon of threshold saturation of the mp decoding threshold to the potential threshold characteristic of spatially coupled low density parity check codes and other coupled systems however sc cbs yield better mp decoding thresholds than their uncoupled counterparts finally we also consider sc cbs as a compressed sensing scheme and show that low undersampling factors can be achieved

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

tight bounds for ldpc and ldgm codes under map decoding a new method for analyzing low density parity check ldpc codes and low density generator matrix ldgm codes under bit maximum a posteriori probability map decoding is introduced the method is based on a rigorous approach to spin glasses developed by francesco guerra it allows one to construct lower bounds on the entropy of the transmitted message conditional to the received one based on heuristic statistical mechanics calculations we conjecture such bounds to be tight the result holds for standard irregular ensembles when used over binary input output symmetric bios channels the method is first developed for tanner graph ensembles with poisson left degree distribution it is then generalized to multi poisson graphs and by a completion procedure to arbitrary degree distribution

non binary ldpc erasure codes with separated low degree variable nodes the code design of non binary low density parity check codes for the erasure channel under maximum a posteriori decoding is addressed in particular a partially structured ensemble of codes characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees is proposed the identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade off between waterfall performance error floor and decoding complexity as an example the performance curve of a short 256 128 code on the memoryless 16 ary erasure channel tightly approaches the singleton bound at least down to a codeword error rate of 10 9 at low decoding complexity

performance of low density parity check codes with linear minimum distance this correspondence studies the performance of the iterative decoding of low density parity check ldpc code ensembles that have linear typical minimum distance and stopping set size we first obtain a lower bound on the achievable rates of these ensembles over memoryless binary input output symmetric channels we improve this bound for the binary erasure channel we also introduce a method to construct the codes meeting the lower bound for the binary erasure channel then we give upper bounds on the rate of ldpc codes with linear minimum distance when their right degree distribution is fixed we compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble

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

maxwell construction the hidden bridge between iterative and maximum a posteriori decoding there is a fundamental relationship between belief propagation and maximum a posteriori decoding a decoding algorithm which is called the maxwell decoder is introduced and provides a constructive description of this relationship both the algorithm itself and the analysis of the new decoder are reminiscent of the maxwell construction in thermodynamics this paper investigates in detail the case of transmission over the binary erasure channel while the extension to general binary memoryless channels is discussed in a companion paper

fast and slow convergence to equilibrium for maxwellian molecules via wild sums we consider the spatially homogeneous boltzmann equation for maxwellian molecules and general finite energy initial data positive borel measures with finite moments up to order 2 we show that the coefficients in the wild sum converge strongly to the equilibrium and quantitatively estimate the rate we show that this depends on the initial data f essentially only through on the behavior near r 0 of the function j f r v 1 r v 2 df v these estimates on the terms in the wild sum yield a quantitative estimate in the strongest physical norm on the rate at which the solution converges to equilibrium as well as a global stability estimate we show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds these are the first examples of solutions of the boltzmann equation that converge to equilibrium more slowly than exponentially

bounds on the error probability of block codes over the q ary erasure channel in this paper tight bounds on the block error probability of linear block codes over order q finite fields for the q ary erasure channel under maximum likelihood ml decoding are developed upper bounds are obtained for uniform parity check ensembles sparse parity check ensembles general parity check ensembles e g gallager regular nonbinary low density parity check ensembles and for any given linear code with known distance spectrum the tightness of the upper bounds is confirmed both by the comparison with simple lower bounds and for gallager low density parity check ensembles by extensive monte carlo simulations exploiting the derived bounds it is shown how already for short blocks and small q 2 sparse ensembles attain block error probabilities close to those of idealized maximum distance separable mds codes down to low error probabilities whereas in the same regime binary codes show visible losses with respect to the singleton bound thanks to the accurate performance estimates the developed bounds can support the design of near optimum erasure correcting codes with short and moderate lengths

results on punctured low density parity check codes and improved iterative decoding techniques this paper first introduces an improved decoding algorithm for low density parity check ldpc codes over binary input output symmetric memoryless channels then some fundamental properties of punctured ldpc codes are presented it is proved that for any ensemble of ldpc codes there exists a puncturing threshold it is then proved that for any rates r1 and r2 satisfying 0 r1 r 2 1 there exists an ensemble of ldpc codes with the following property the ensemble can be punctured from rate r1 to r2 resulting in asymptotically good codes for all rates r1lesrlesr2 specifically this implies that rates arbitrarily close to one are achievable via puncturing bounds on the performance of punctured ldpc codes are also presented it is also shown that punctured ldpc codes are as good as ordinary ldpc codes for bec and arbitrary positive numbers r1 r2 1 the existence of the sequences of punctured ldpc codes that are capacity achieving for all rates r1 lesrlesr2 is shown based on the above observations a method is proposed to design good punctured ldpc codes over a broad range of rates finally it is shown that the results of this paper may be used for the proof of the existence of the capacity achieving ldpc codes over binary input output symmetric memoryless channels

abundance of c 1 robust homoclinic tangencies a diffeomorphism f has a c 1 robust homoclinic tangency if there is a c 1 neighbourhood u of f such that every diffeomorphism in g u has a hyperbolic setg depending contin uously on g such that the stable and unstable manifolds ofg have some non transverse intersection for every manifold of dimension greater than or equal to three we exhibit a lo cal mechanism blender horseshoes generating diffeomorphisms with c 1 robust homoclinic tangencies using blender horseshoes we prove that homoclinic classes of c 1 generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings of appropriate dimensions display c 1 robust homoclinic tangencies