waveform relaxation for reaction diffusion equations abstract we report a new waveform relaxation wr algorithm for general semi linear reaction diffusion equations the superlinear rate of convergence of the new wr algorithm is proved and we also show the advantages of the new approach superior to the classical wr algorithms by the estimation on iteration errors the corresponding discrete wr algorithm for reaction diffusion equations is presented and further the parallelism of the discrete wr algorithm is analyzed moreover the new approach is extended to handle the coupled reaction diffusion equations numerical experiments are carried out to verify the effectiveness of the theoretic work

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

reversible low density parity check codes low density parity check ldpc codes may be decoded using a circuit implementation of the sum product algorithm which maps the factor graph of the code by reusing the decoder for encoding both tasks can be performed using the same circuit thus reducing area and verification requirements motivated by this iterative encoding techniques based upon the graphical representation of the code are proposed code design constraints which ensure encoder convergence are presented and then used to design iteratively encodable codes while also preventing 4 cycle creation we show how the jacobi method for iterative matrix inversion can be applied to finite field matrices viewed as message passing and employed as the core of an iterative encoder we present an algebraic construction of 4 cycle free iteratively encodable codes using circulant matrices analysis of these codes identifies a weakness in their structure due to a repetitive pattern in the factor graph the graph supports pseudo codewords of low pseudo weight in order to remove the repetitive pattern in the graph we propose a recursive technique for generating iteratively encodable codes the new codes offer flexibility in the choice of code length and rate and performance that compares well to randomly generated quasi cyclic and extended euclidean geometry codes

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

simplification of packet symbol decoding with errors deletions misordering of packets and no sequence numbers in this paper a new method is described which builds on mitzenmacher s idea of adding a different pseudorandom number to each packet to help decode packet symbol low density codes with deletions errors and out of order reception without sequence numbers the new method has lower decoding complexity than the original method the most basic form of the new method applies to any parity check code structure but is limited to a rather small number of packets in the code decoding success is slightly inferior to an ideal erasure channel which would require sequence numbering and error detection in each packet error detection is needed only for the whole code amounting to usually less than one bit per packet symbol moreover if error detection can resolve one of a small number of alternatives the ordered case performs almost as well as the ideal erasure channel ways are shown to modify the basic algorithm for use with long codes possibly approaching the erasure channel capacity limit

reed solomon and simplex codes for peak to average power ratio reduction in ofdm new schemes for peak to average power ratio reduction in orthogonal frequency division multiplexing ofdm systems are proposed reed solomon rs and simplex codes are employed to create a number of candidates from which the best are selected thereby in contrast to existing approaches the codes are arranged over a number of ofdm frames rather than over the carriers hence a combination of the principles of multiple signal representation with selection as done in selected mapping and the use of channel coding is present in particular in multiple antenna transmission the proposed schemes do not cause any additional delay but due to the utilization of the dimension space additional gains can be achieved moreover the schemes are very flexible due to the selection step any criterion of optimality can be taken into account besides multiple antenna transmission packet transmission is briefly considered which moreover covers the appealing similarities with incremental redundancy check schemes in automatic repeat request arq applications and with decoding of codes transmitted over the erasure channel the performance of the schemes is using some approximations derived analytically and is covered by numerical results that are in very good agreement with the theory significant gains can be achieved with these very flexible and versatile methods

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

impact of redundant checks on the lp decoding thresholds of ldpc codes feldman et al 11 asked whether the performance of the linear programming lp decoder can be improved by adding redundant parity checks to tighten the lp relaxation we prove in this paper that for low density parity check codes even if we include all redundant parity checks asymptotically there is no gain in the lp decoder threshold on the binary symmetric channel under certain conditions on the base tanner graph first we show that if the tanner graph has bounded check degree and satisfies a condition which we call asymptotic strength then including high degree redundant parity checks in the lp does not significantly improve the threshold of the lp decoder in the following sense for each constant delta 0 there is a constant k 0 such that the threshold of the lp decoder containing all redundant checks of degree at most k improves by at most delta upon adding to the lp all redundant checks of degree larger than k we conclude that if the graph satisfies an additional condition which we call rigidity then including all redundant checks does not improve the threshold of the base lp we call the graph asymptotically strong if the lp decoder corrects a constant fraction of errors even if the log likelihood ratios of the correct variables are arbitrarily small by building on a construction due feldman et al 9 and its recent improvement by viderman 24 we show that asymptotic strength follows from sufficiently large variable to check expansion we also give a geometric interpretation of asymptotic strength in terms pseudocodewords we call the graph rigid if the minimum weight of a sum of check nodes involving a cycle tends to infinity as the block length tends to infinity under the assumptions that the graph girth is logarithmic and the minimum check degree is at least 3 rigidity is equivalent to the nondegeneracy property that adding at least logarithmically many checks does not give a constant weight check we argue that nondegeneracy is a typical property of random check regular tanner graphs

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

reliable approximate solution of systems of volterra integro differential equations with time dependent delays volterra integro differential equations with time dependent delay arguments can provide us with realistic models of many real world phenomena delayed lokta volterra predator prey systems arise in ecology and are well known examples of delay volterra integro differential equations dvides first introduced by volterra in 1928 we investigate the numerical solution of systems of dvides using an adaptive stepsize selection strategy we will present a generic variable stepsize approach for solving systems of neutral dvides based on an explicit continuous runge kutta method using defect error control and study the convergence of the resulting numerical method for various kinds of delay arguments we will show that the global error of the numerical solution can be effectively and reliably controlled by monitoring the size of the defect of the approximate solution and adjusting the stepsize on each step of the integration numerical results will be presented to demonstrate the effectiveness of this approach

analysis of verification based decoding on the q ary symmetric channel for large q a new verification based message passing decoder for low density parity check ldpc codes is introduced and analyzed for the q ary symmetric channel q sc rather than passing messages consisting of symbol probabilities this decoder passes lists of possible symbols and marks some lists as verified the density evolution de equations for this decoder are derived and used to compute decoding thresholds if the maximum list size is unbounded then one finds that any capacity achieving ldpc code for the binary erasure channel can be used to achieve capacity on the q sc for large q the decoding thresholds are also computed via de for the case where each list is truncated to satisfy a maximum list size constraint simulation results are also presented to confirm the de results during the simulations we observed differences between two verification based decoding algorithms introduced by luby and mitzenmacher that were implicitly assumed to be identical in this paper the node based algorithms are evaluated via analysis and simulation the probability of false verification fv is also considered and techniques are discussed to mitigate the fv optimization of the degree distribution is also used to improve the threshold for a fixed maximum list size finally the proposed algorithm is compared with a variety of other algorithms using both density evolution thresholds and simulation results

supersymmetry algebra cohomology i definition and general structure this paper concerns standard supersymmetry algebras in diverse dimensions involving bosonic translational generators and fermionic supersymmetry generators a cohomology related to these supersymmetry algebras termed supersymmetry algebra cohomology and corresponding primitive elements are defined by means of a brst becchi rouet stora tyutin type coboundary operator a method to systematically compute this cohomology is outlined and illustrated by simple examples

finite length scaling for iteratively decoded ldpc ensembles we investigate the behavior of iteratively decoded low density parity check ldpc codes over the binary erasure channel in the so called ldquowaterfall region rdquo we show that the performance curves in this region follow a simple scaling law we conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture the scaling law together with the error floor expressions developed previously can be used for a fast finite length optimization

on the design of fast convergent ldpc codes for the bec an optimization approach the complexity performance trade off is a fundamental aspect of the design of low density parity check ldpc codes in this paper we consider ldpc codes for the binary erasure channel bec use code rate for performance metric and number of decoding iterations to achieve a certain residual erasure probability for complexity metric we first propose a quite accurate approximation of the number of iterations for the bec moreover a simple but efficient utility function corresponding to the number of iterations is developed using the aforementioned approximation and the utility function two optimization problems w r t complexity are formulated to find the code degree distributions we show that both optimization problems are convex in particular the problem with the proposed approximation belongs to the class of semi infinite problems which are computationally challenging to be solved however the problem with the proposed utility function falls into the class of semi definite programming sdp and thus the global solution can be found efficiently using available sdp solvers numerical results reveal the superiority of the proposed code design compared to existing code designs from literature

the extraction and complexity limits of graphical models for linear codes two broad classes of graphical modeling problems for codes can be identified in the literature constructive and extractive problems the former class of problems concern the construction of a graphical model in order to define a new code the latter class of problems concern the extraction of a graphical model for a fixed given code the design of a new low density parity check code for some given criteria e g target block length and code rate is an example of a constructive problem the determination of a graphical model for a classical linear block code that implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem this work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes the primary focus of this work is a study of the space of all graphical models for a fixed given code the tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound the forest inducing cut set bound fi csb the proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools e g the csb and can be viewed as a generalization of the square root bound for tail biting trellises to graphical models with arbitrary cyclic topologies searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations that are shown to span this space finally heuristics for extracting novel graphical models for linear block codes using these transformations are investigated

second order weight distributions a fundamental property of codes the second order weight distribution is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles a series of results parallel to those for weight distributions is established for second order weight distributions in particular an analogue of macwilliams identities is proved the second order weight distributions of regular ldpc code ensembles are then computed as easy consequences the second moments of weight distributions of regular ldpc code ensembles are obtained furthermore the application of second order weight distributions in random coding approach is discussed the second order weight distributions of the ensembles generated by a so called 2 good random generator or parity check matrix are computed where a 2 good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple wise properties of sequences it is shown that the 2 good property is reflected in the second order weight distribution which thus plays a fundamental role in some well known problems in coding theory and combinatorics an example of linear intersecting codes is finally provided to illustrate this fact

convergence analysis of an arnoldi order reduced runge kutta method for integro differential equations of pantograph type in this paper we consider a numerical method for linear functional integro differential equations with pantograph delays to deal with the pantograph delays we introduce a geometrically increasing mesh and propose a new kind of runge kutta methods based on the arnoldi order reduced technique the convergence analysis of the new method is presented and the method is proved to be at least first order the numerical experiments attached show that the new method can achieve almost the same accuracy as the corresponding runge kutta method for some properly chosen krylov subspaces

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

on correcting bursts and random errors in vector symbol n k cyclic codes in this communication simple methods are shown for correcting bursts of large size and bursts combined with random errors using vector symbols and primarily vector xor and feedback shift register operations one result is that any n k cyclic code with minimum distance 2 can correct all full vector symbol error bursts of length n k 1 or less if the error vectors are linearly independent if the bursts are not full but contain some error free components the capability of correcting bursts up to n k or less is code dependent also vector symbol decoding with reed solomon component codes can correct very simply with probability ges 1 n n k 2 r all cases of e les n k 1 r bit random errors in any cyclic span of length les n k the techniques often work when there is linear dependence in cases where most errors are in a burst but a small number of errors are outside the solution given error correcting capability can be broken down into a simple solution for the small number of outside errors followed by a simple subtraction to reveal all the error values in the burst part

design and performance of turbo gallager codes the most powerful channel coding schemes namely those based on turbo codes and low density parity check ldpc gallager codes have in common the principle of iterative decoding however the relative coding structures and decoding algorithms are substantially different this paper shows that recently proposed novel coding structures bridge the gap between these two schemes in fact with properly chosen component convolutional codes a turbo code can be successfully decoded by means of the decoding algorithm used for ldpc codes i e the belief propagation algorithm working on the code tanner graph these new turbo codes are here nicknamed turbo gallager codes besides being interesting from a conceptual viewpoint these schemes are important on the practical side because they can be decoded in a fully parallel manner in addition to the encoding complexity advantage of turbo codes the low decoding complexity allows the design of very efficient channel coding schemes

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

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

a note on the spectra and pseudospectra of waveform relaxation operators for linear differential algebraic equations in this note we derived the expressions of the spectra and pseudospectra of waveform relaxation operators for general linear differential algebraic equations of index one the expressions are generalizations of those of lumsdaine and wu siam j sci comput 18 1997 pp 286 304 and the derivation here is nontrivial numerical experiments are given to verify the theoretical results

a super attainable order in collocation methods for differential equations with proportional delay abstract we are concerned with the pantograph differential equation y t ay t by qt f t t 0 y 0 y 0 with proportional delay qt 0 q 1 in the literatures it is known that if we choose some proper m collocation points for m 2 then collocation leads to a superconvergence result of order p 2 m 1 at the first mesh point t h in this paper in such collocation solution to the above equation we show that there are cases for some 0 q 1 such that the attainable order at the first mesh point t h becomes a super attainable order just o h 2 m 2 numerical experiments of such 0 q 1 are also presented

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

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

parity check matrices separating erasures from errors most decoding algorithms of linear codes in general are designed to correct or detect errors however many channels cause erasures in addition to errors in principle decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code for any given linear code and any given maximum number of correctable erasures parity check matrices are introduced that yield parity check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures these matrices allow for the separation of erasures from errors to facilitate decoding several constructions of such separating parity check matrices are presented to reduce decoding complexity separating parity check matrices with small number of rows are preferred the minimum number of rows in a parity check matrix separating a given maximum number of erasures is called the separating redundancy upper and lower bounds on the separating redundancies are derived in particular it is shown that the separating redundancies tend to grow linearly with the number of rows in full rank parity check matrices of codes the separating redundancies of some classes of codes are determined for some maximum numbers of erasures

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

generalized low density parity check codes based on hadamard constraints in this paper we consider the design and analysis of generalized low density parity check gldpc codes in awgn channels the gldpc codes are specified by a bipartite tanner graph as with standard ldpc codes but with the single parity check constraints replaced by general coding constraints in particular we consider imposing hadamard code constraints at the check nodes for a low rate approach termed ldpc hadamard codes we introduce a low complexity message passing based iterative soft input soft output siso decoding algorithm which employs the a posteriori probability app fast hadamard transform fht for decoding the hadamard check codes at each decoding iteration the achievable capacity with the gldpc codes is then discussed a modified ldpc hadamard code graph is also proposed we then optimize the ldpc hadamard code ensemble using a low complexity optimization method based on approximating the density evolution by a one dimensional dynamic system represented by an extrinsic mutual information transfer exit chart simulation results show that the optimized ldpc hadamard codes offer better performance in the low rate region than low rate turbo hadamard codes but also enjoy a fast convergence rate a rate 0 003 ldpc hadamard code with large block length can achieve a bit error rate ber performance of 10 5 at 1 44 db which is only 0 15 db away from the ultimate shannon limit 1 592 db and 0 24 db better than the best performing low rate turbo hadamard codes

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

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

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

a fresh look at coding for q ary symmetric channels this paper studies coding schemes for the q ary symmetric channel based on binary low density parity check ldpc codes that work for any alphabet size q 2m m n thus complementing some recently proposed packet based schemes requiring large q first theoretical optimality of a simple layered scheme is shown then a practical coding scheme based on a simple modification of standard binary ldpc decoding is proposed the decoder is derived from first principles and using a factor graph representation of a front end that maps q ary symbols to groups of m bits connected to a binary code the front end can be processed with a complexity that is linear in m log2q an extrinsic information transfer chart analysis is carried out and used for code optimization finally it is shown how the same decoder structure can also be applied to a larger class of q ary channels

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

optimal rate delay tradeoffs and delay mitigating codes for multipath routed and network coded networks via multiterminal information theory a framework is presented for deriving fundamental rate delay tradeoffs that delay mitigating codes must have when utilized over multipath routed and random linear network coded networks the rate delay tradeoff is formulated as a calculus problem on a capacity region of a related abstracted broadcast channel given this general framework for studying such rate delay tradeoffs the extreme case of uniform networks in which each possible received packet arrival order is equally likely is considered for these networks the rate delay calculus problem is simplified to an integer programming problem which for small numbers of packets may be solved explicitly or for larger numbers of packets may be accurately approximated through the calculus of variations by appropriate relaxation of an integer constraint explicit expressions for the rate delay tradeoff in uniform networks are presented in the special cases of i constant packet inter arrival times and ii exponential independent and identically distributed i i d packet arrival times finally the delay mitigating codes achieving these rate delay tradeoffs are discussed

polynomial time low density parity check codes with rates very close to the capacity of the q ary random deletion channel for large q we demonstrate polynomial time deletion codes based on the verification based decoding paradigm that come arbitrarily close to capacity the random deletion channel takes n symbols from a q ary alphabet and each symbol is deleted independently with probability p taking advantage of recent improvements on the results of luby and mitzenmacher for verification based decoding by shokrollahi and wang we show how to design for any epsi 0 and sufficiently large n and q deletion codes with the following properties the rate is 1 p 1 epsi the failure probability is no 1 epsi2 q and the computational complexity for encoding and decoding is no 1 epsi2 log q we also extend these schemes to obtain the same results even if the undeleted symbols are also transposed arbitrarily and if a sufficiently small number of random symbols are inserted

approaching capacity at high rates with iterative hard decision decoding a variety of low density parity check ldpc ensembles have now been observed to approach capacity with message passing decoding however all of them use soft i e non binary messages and a posteriori probability decoding of their component codes in this paper we show that one can approach capacity at high rates using iterative hard decision decoding hdd of generalized product codes specifically a class of spatially coupled generalized ldpc codes with bose chaudhuri hocquengham component codes is considered and it is observed that in the high rate regime they can approach capacity under the proposed iterative hdd these codes can be seen as generalized product codes and are closely related to braided block codes an iterative hdd algorithm is proposed that enables one to analyze the performance of these codes via density evolution

hierarchical and high girth qc ldpc codes we present an approach to designing capacity approaching high girth low density parity check ldpc codes that are friendly to hardware implementation and compatible with some desired input code structure defined using a protograph the approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi cyclic hqc ldpc codes whereas the parity check matrices of standard quasi cyclic qc ldpc codes are composed of circulant submatrices those of hqc ldpc codes are composed of a hierarchy of circulant submatrices that are in turn constructed from circulant submatrices and so on through some number of levels next we present a girth maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high girth hqc ldpc code subject to bounds imposed by the fact that hqc codes are still quasi cyclic finally we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a squashing procedure that results in a high girth qc ldpc code although not a hierarchical one we illustrate our approach with three design examples of qc ldpc codes two girth 10 codes of rates 1 3 and 0 45 and one girth 8 code of rate 0 7 all of which are obtained from protographs of one sided spatially coupled codes

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

stability of runge kutta methods for delay integro differential equations we study stability of runge kutta rk methods for delay integro differential equations with a constant delay on the basis of the linear equation du dt lu t mu t k t t u d where l m k are constant complex matrices in particular we show that the same result as in the case k 0 koto bit 34 1994 262 267 holds for this test equation i e every a stable rk method preserves the delay independent stability of the exact solution whenever a step size of the form h m is used where m is a positive integer

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

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

supersymmetry algebra cohomology iii primitive elements in four and five dimensions the primitive elements of the supersymmetry algebra cohomology as defined in a previous paper are computed for standard supersymmetry algebras in four and five dimensions for all signatures of the metric and any number of supersymmetries

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

discretized stability and error growth of the nonautonomous pantograph equation this paper is concerned with the stability properties of runge kutta methods for the pantograph equation a functional differential equation with a proportional delay the focus is on nonautonomous equations both linear and nonlinear cases are considered sufficient and necessary conditions for the asymptotic stability of the numerical solution of general neutral pantograph equations are given an upper bound for the error growth is investigated for algebraically stable methods applied to nonneutral equations finally some stability results are extended to the case of a more general class of equations

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

b r s algebras analysis of the consistency equations in gauge theory we compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms by using the appropriate cohomological and algebraic methods we do it for all dimensions of spacetime and all structure groups with reductive lie algebras

optimal superconvergence orders of iterated collocation solutions for volterra integral equations with vanishing delays in this paper we analyze the optimal convergence properties of collocation approximations to solutions of volterra integral equations of the second kind with vanishing variable delays the focus of the analysis is on the superconvergence of the iterated collocation approximation corresponding to collocation in the space of discontinuous piecewise polynomials of degree m 1 geq 0 we show that on uniform meshes the iterated collocation solution possesses the global superconvergence order m 1 and that the solution s order of local superconvergence at the nodes of the underlying mesh cannot exceed m 2 in sharp contrast to problems with nonvanishing delays moreover the optimal order p ast m 2 can be attained only under suitable assumptions this result also resolves a conjecture of 1997 regarding the attainable order of local superconvergence for volterra integral equations containing a proportional delay qt with q in 0 1

a tight rate bound and matching construction for locally recoverable codes with sequential recovery from any number of multiple erasures this paper considers the natural extension of locally recoverable codes lrc to the case of t 1 erased symbols while several approaches have been proposed for the handling of multiple erasures in the approach considered here the t erased symbols are recovered in succession each time contacting at most r other symbols for assistance under the local recovery constraint this sequential approach is the most general and hence offers the maximum possible code rate we characterize the rate of an lrc with sequential recovery for any r geq 3 and any t by first deriving an upper bound on the code rate and then constructing a binary code achieving this optimal rate the upper bound derived here proves an earlier conjecture our approach permits us to deduce the structure of the parity check matrix of a rate optimal lrc with sequential recovery the derived structure of parity check matrix leads to a graphical description of the code used in code construction a subclass of binary codes that are both rate and block length optimal are shown to correspond to certain regular graphs known as moore graphs that have the smallest number of vertices for a given girth a connection with tornado codes is also made

algebraic construction of quasi cyclic ldpc codes for the awgn and erasure channels this paper is concerned with construction of quasi cyclic qc low density parity check ldpc codes for three different types of channels the additive white gaussian noise the binary random erasure and the binary burst erasure channels two algebraic methods for systematic construction of qc ldpc codes are presented codes constructed perform well over all three types of channels

design of repeat accumulate codes for iterative detection and decoding extrinsic information transfer exit charts are used to design systematic and nonsystematic repeat accumulate ra codes for iterative detection and decoding the convergence problems of nonsystematic ra codes are solved by introducing a biregular or doped layer of check nodes as examples such nonsystematic codes are designed for multi input multi output mimo fading channels and are shown to operate close to capacity

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

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

new insights into weighted bit flipping decoding a natural relationship between weighted bit flipping wbf decoding and belief propagation like bp like decoding is explored this understanding can help us develop wbf algorithms from bp like algorithms for min sum decoding one can find that its wbf algorithm is the algorithm proposed by jiang et al for bp decoding we propose a new wbf algorithm and show its performance advantage the proposed wbf algorithms are parallelized to achieve rapid convergence two efficient simulation based procedures are proposed for the optimization of the associated thresholds

ldpc codes for 2d arrays binary codes over 2d arrays are very useful in data storage where each array column represents a storage device or unit that may suffer failure in this paper we propose a new framework for probabilistic construction of codes on 2d arrays instead of a pure combinatorial erasure model used in traditional array codes we propose a mixed combinatorial probabilistic model of limiting the number of column failures and assuming a binary erasure channel in each failing column for this model we give code constructions and detailed analysis that allow sustaining a large number of column failures with graceful degradation in the fraction of erasures correctable in failing columns another advantage of the new framework is that it uses low complexity iterative decoding the key component in the analysis of the new codes is to analyze the decoding graphs induced by the failed columns and infer the decoding performance as a function of the code design parameters as well as the array size and failure parameters a particularly interesting class of codes called probabilistically maximum distance separable mds array codes gives fault tolerance that is equivalent to traditional mds array codes the results also include a proof that the 2d codes outperform standard 1d low density parity check codes

bounds on achievable rates of ldpc codes used over the binary erasure channel we derive upper bounds on the maximum achievable rate of low density parity check ldpc codes used over the binary erasure channel bec under gallager s decoding algorithm given their right degree distribution we demonstrate the bounds on the ensemble of right regular ldpc codes and compare them with an explicit left degree distribution constructed from the given right degree

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

supersymmetry algebra cohomology iv primitive elements in all dimensions from d 4 to d 11 the primitive elements of the supersymmetry algebra cohomology as defined in previous work are derived for standard supersymmetry algebras in dimensions d 5 11 for all signatures of the related clifford algebras of gamma matrices and all numbers of supersymmetries the results are presented in a uniform notation along with results of previous work for d 4 and derived by means of dimensional extension from d 4 up to d 11

local brst cohomology and covariance the paper provides a framework for a systematic analysis of the local brst cohomology in a large class of gauge theories the approach is based on the cohomology of s d in the jet space of fields and antifields s and d being the brst operator and exterior derivative respectively it relates the brst cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields the latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions noether currents and gauge anomalies as well as of the equations of motion

construction of low density parity check codes based on balanced incomplete block designs this correspondence presents a method for constructing structured regular low density parity check ldpc codes based on a special type of combinatoric designs known as balance incomplete block designs codes constructed by this method have girths at least 6 and they perform well with iterative decoding furthermore several classes of these codes are quasi cyclic and hence their encoding can be implemented with simple feedback shift registers

degree distribution design for ldpc codes a derivative matching approach a deterministic method to design degree distributions for low density parity check codes over the binary erasure channel is proposed this method consists of matching the first and high order derivatives of the extrinsic information transfer exit function of the variable node set to the corresponding derivatives of the inverse exit function of the check node set in order to reduce the gap between the two curves in the exit chart a sufficient condition for a check concentrated distribution to achieve derivative matching up to some order is first obtained and then a deterministic design algorithm enabled by the fourier budan theorem is developed exploiting this sufficient condition a comparison with other deterministic design techniques is also provided revealing the potential of the proposed algorithm

high rate quasi cyclic low density parity check codes derived from finite affine planes this paper shows that several attractive classes of quasi cyclic qc low density parity check ldpc codes can be obtained from affine planes over finite fields one class of these consists of duals of one generator qc codes presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum weight codewords further it is shown that the minimum hamming distance of a code in this class is equal to its minimum additive white gaussian noise awgn pseudoweight also discussed is a class consisting of codes from circulant permutation matrices and an explicit formula for the rank of the parity check matrix is presented for these codes additionally it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2 the construction is similar to the derivation of reed solomon rs based ldpc codes presented by chen and djurdjevic experimental results show that a number of high rate qc ldpc codes with excellent error performance are contained in these classes

mean field and mixed mean field iterative decoding of low density parity check codes in this paper the mean field mf and mixed mean field mmf algorithms for decoding low density parity check ldpc codes are considered the mf principle is well established in statistical physics and artificial intelligence instead of using a single completely factorized approximated distribution as in the mf approach the mixed mf algorithm forms a weighted average of several mf distributions as an approximation of the true posterior probability distribution the mf decoding algorithm for linear block codes is derived and shown to be an approximation of the a posteriori probability app decoding algorithm the mf approach is then developed in the context of iterative decoding and presented as an approximation of the popular belief propagation decoding method these results are extended to iterative decoding with the mmf algorithm simulation results show that the mf and mmf decoding algorithms yield a good performance complexity tradeoff especially when employed for decoding ldpc codes based on finite geometries

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

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

the universal operation of ldpc codes over scalar fading channels root and varaiya proved the existence of a code that can communicate reliably over any linear gaussian channel for which the channel mutual information level exceeds the transmitted rate this paper provides several examples of scalar single input single output fading channels and shows that on these channels the performance of low density parity check ldpc codes lies in close proximity to the performance limits identified by root and varaiya specifically we consider periodic fading channels and partial band jamming pbj channels a special case of periodic fading is the variation of signal to noise ratio across orthogonal frequency division modulation subchannels the robustness of ldpc codes to periodic fading and pbj across parameterizations of these different channels is demonstrated through the consistency of the required mutual information to provide a specified bit error rate for the periodic fading case the gaussian approximation to density evolution has been adapted such that asymptotic threshold measures can be compared to simulated code performance in various periodic fading scenarios

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

fft based bp decoding of general ldpc codes over abelian groups we introduce a wide class of low density parity check ldpc codes large enough to include ldpc codes over finite fields rings or groups as well as some nonlinear codes a belief propagation decoding procedure with the same complexity as for the decoding of ldpc codes over finite fields is also presented moreover an encoding procedure is developed

efficient maximum likelihood decoding of ldpc codes over the binary erasure channel we propose an efficient maximum likelihood ml decoding algorithm for decoding low density parity check ldpc codes over the binary erasure channel bec we also analyze the computational complexity of the proposed algorithm

on convergence speed of capacity achieving sequences for erasure channel oswald and shokrollahi carried out a systematic study of capacity achieving sequences of low density parity check ldpc codes for the binary erasure channel two quantities mu and delta were suggested to measure the convergence speed of these sequences they gave an upper bound of mu for a certain type of capacity achieving sequences in this correspondence a lower bound of mu for this type of sequences is provided it follows that under a certain condition the exact value of mu can be determined which answers an open question posed by oswald and shokrollahi in their paper if the condition is not satisfied the upper bound may not be tight a counterexample is given

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

bounds on the expansion properties of tanner graphs this work focuses on the expansion properties of a tanner graph because they are known to be related to the performance of associated iterative message passing algorithms over various channels by analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a tanner graph lower bounds on these expansion properties are derived specifically for the binary erasure channel these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference set codes type i 2 d projective geometry low density parity check ldpc codes

optimal superconvergence results for delay integro differential equations of pantograph type we analyze the optimal global and local orders of superconvergence of collocation solutions u h on uniform meshes i h for delay volterra integro differential equations with proportional delay functions given by theta t qt 0 q 1 t in 0 t in particular we show that if u h is a continuous piecewise polynomial of degree m geq 2 and if collocation is at the gauss legendre points then the optimal order of local superconvergence on i h is p ast m 2 it turns out that the same order p ast holds for nonlinear strictly increasing delay functions vanishing at t 0 however on judiciously chosen geometric meshes collocation at the gauss points yields the order 2m varepsilon n where varepsilon n rightarrow 0 as the number n of mesh points tends to infinity optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis

expander graph arguments for message passing algorithms we show how expander based arguments may be used to prove that message passing algorithms can correct a linear number of erroneous messages the implication of this result is that when the block length is sufficiently large once a message passing algorithm has corrected a sufficiently large fraction of the errors it will eventually correct all errors this result is then combined with known results on the ability of message passing algorithms to reduce the number of errors to an arbitrarily small fraction for relatively high transmission rates the results hold for various message passing algorithms including gallager s hard decision and soft decision with clipping decoding algorithms our results assume low density parity check ldpc codes based on an irregular bipartite graph

binary intersymbol interference channels gallager codes density evolution and code performance bounds we study the limits of performance of gallager codes low density parity check ldpc codes over binary linear intersymbol interference isi channels with additive white gaussian noise awgn using the graph representations of the channel the code and the sum product message passing detector decoder we prove two error concentration theorems our proofs expand on previous work by handling complications introduced by the channel memory we circumvent these problems by considering not just linear gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols we compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify by simulation that the thresholds represent accurate predictions of the performance of the iterative sum product algorithm for finite but large block lengths we also demonstrate that for high rates the thresholds are very close to the theoretical limit of performance for gallager codes over isi channels if c denotes the capacity of a binary isi channel and if c sub i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed i i d binary random variables c sub i i d spl les c we prove that the maximum information rate achievable by the sum product decoder of a gallager coset code is upper bounded by c sub i i d the last topic investigated is the performance limit of the decoder if the trellis portion of the sum product algorithm is executed only once this demonstrates the potential for trading off the computational requirements and the performance of the decoder

verification based decoding for packet based low density parity check codes we introduce and analyze verification based decoding for low density parity check ldpc codes an approach specifically designed to manipulate data in packet sized units verification based decoding requires only linear time for both encoding and decoding and succeeds with high probability under random errors we describe how to utilize code scrambling to extend our results to channels with errors controlled by an oblivious adversary