a class of quasi cyclic ldpc codes over gf 2 m low density parity check ldpc codes over gf 2m are an extension of binary ldpc codes performances of gf 2m ldpc codes have been shown to be higher than binary ldpc codes but the complexity of the encoders decoders increases hence there is a substantial lack of implementations for ldpc codes over gf 2m codes this paper presents a class of quasi cyclic ldpc codes over gf 2m these codes can alleviate the encoding decoding complexity without excessive loss of performances it is shown how from a performance point of view such codes are better than m times bigger binary codes and as good as 2m longer binary codes

multiplicity estimates on group varieties tions the field k will be the field q of algebraic numbers we remark at this point that the results remain true if we take instead of the field of complex numbers c its p adic analogue c for some fixed prime p and for k a corresponding subfield kp of cp we restrict ourselves to the complex case in order to avoid some minor complications appearing in the p adic domain these complications always arise when analytic functions come in our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p adic domain these difficulties can be avoided by a purely algebraic approach if we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero but we prefer to avoid such an approach in order to keep the text understandable also for those who are mainly interested in the applications in transcendence

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

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

breaking the trapping sets in ldpc codes check node removal and collaborative decoding trapping sets strongly degrade performance of low density parity check ldpc codes in the low error rate region this creates significant difficulties for the deployment of ldpc codes to low error rate applications such as storage and wireless systems with no or limited retransmission options we propose a novel technique for breaking trapping sets based on collaborative decoding that utilizes two different decoding modes while the main decoding mode executes message passing based on the original parity check matrix of the corresponding ldpc code the sub decoding mode operates on a modified parity check matrix formed by removing a portion of check nodes in the factor graph representation of the given code the modified parity check matrix is designed to promote a passing of correct information into erroneous variable nodes in the trapping set theoretical properties of the proposed trapping set breaking technique have been established based on the notion of the improved separation for the trapped variable nodes simulation results show that the proposed collaborative ldpc decoding scheme switching between the two decoding modes back and forth effectively breaks dominant trapping sets of various known types of regular and irregular ldpc codes

diagnosability of stochastic discrete event systems we investigate diagnosability of stochastic discrete event systems we define the notions of a and aa diagnosability for stochastic automata these notions are weaker than the corresponding notion of diagnosability for logical automata introduced by sampath et al through the construction of a stochastic diagnoser we determine offline conditions necessary and sufficient to guarantee a diagnosability and sufficient to guarantee aa diagnosability we also show how the stochastic diagnoser can be used for on line diagnosis of failure events we illustrate the results through two examples from hvac systems

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

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

iterative decoding of one step majority logic deductible codes based on belief propagation previously the belief propagation bp algorithm has received a lot of attention in the coding community mostly due to its near optimum decoding for low density parity check ldpc codes and its connection to turbo decoding in this paper we investigate the performance achieved by the bp algorithm for decoding one step majority logic decodable osmld codes the bp algorithm is expressed in terms of likelihood ratios rather than probabilities as conventionally presented the proposed algorithm fits better the decoding of osmld codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding ldpc codes although it has been believed that osmld codes are far inferior to ldpc codes we show that for medium code lengths say between 200 1000 bits the bp decoding of osmld codes can significantly outperform bp decoding of their equivalent ldpc codes the reasons for this behavior are elaborated

sequential decoding of low density parity check codes by adaptive reordering of parity checks decoding algorithms in which unpruned codeword trees are generated from an ordered list of parity checks are investigated the order is computed from the received message and low density parity check codes are used to help control the growth of the tree simulation results are given for the binary erasure channel they suggest that for the small erasure probability the method is computationally feasible at rates above the computational cutoff rate

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

an improved decoding algorithm for finite geometry ldpc codes in this letter an improved bit flipping decoding algorithm for high rate finite geometry low density parity check fg ldpc codes is proposed both improvement in performance and reduction in decoding delay are observed by flipping multiple bits in each iteration our studies show that the proposed algorithm achieves an appealing tradeoff between performance and complexity for fg ldpc codes

solving sparse linear systems of equations over finite fields using bit flipping algorithm let fq be the finite field with q elements we give an algorithm for solving sparse linear systems of equations over fq when the coefficient matrix of the system has a specific structure here called relatively connected this algorithm is based on a well known decoding algorithm for low density parity check codes called bit flipping algorithm we modify and extend this hard decision decoding algorithm the complexity of this algorithm is linear in terms of the number of columns n and the number of nonzero coefficients of the matrix per iteration the maximum number of iterations is bounded above by m the number of equations

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

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

construction of high performance block and convolutional multi edge type qc ldpc codes we present code sieving method for construction of long length high performance block and convolutional multi edge type qc ldpc codes met qc ldpcc under the fixed point quantized reduced complexity box plus qrcbp decoder proposed method use iterative decoding threshold protograph sieving and simulated annealing method for qc lifting with cycles emd maximization using proposed method we construct base matrix of met ldpc code with size 92 20 maximal column weight 17 with weight one circulants code rate 0 8 2 variable nodes punctured which have iterative decoding threshold eb no 2 333 db under 200 iterations of qrcbp over an awgn channel with qpsk modulation on bit error rate below 10 0 289 db from shannon limit simulation results of block with code length n 261924 and met qc ldpcc codes equivalent block length n 785772 constructed using simulated annealing lifting of base matrix under 32 iterations of qrcbp show eb no 2 56db eb no 2 478db below ber10 15 respectively cole s importance sampling under qrcbp show that proposed codes error floor ber 10 16

reduced complexity iterative decoding of low density parity check codes based on belief propagation two simplified versions of the belief propagation algorithm for fast iterative decoding of low density parity check codes on the additive white gaussian noise channel are proposed both versions are implemented with real additions only which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed also these two algorithms do not require any knowledge about the channel characteristics both algorithms yield a good performance complexity trade off and can be efficiently implemented in software as well as in hardware with possibly quantized received values

a walsh hadamard coded spectral efficient full frequency diversity ofdm system a full frequency diversity cofdm fd cofdm scheme based on sub channel multiplexing is proposed to improve the performance of cofdm without sacrificing spectral efficiency since the channel suffers the frequency selective fading the received signal contains strong inter symbol interference isi to obtain the diversity and code gains an iterative receiving algorithm is presented by jointly using diversity combining interference cancellation and soft decoding the proposed iterative receiving algorithm properly decodes the information bits in the strong isi environment consequently all the subcarriers of fd cofdm participate in diversity so that the system can achieve full frequency diversity in most practical channels the new receiving algorithm is relatively simple compared to the maximum likelihood estimation algorithm and is therefore a better alternative to balance the algorithm complexity and system performance

design of ldpc decoders for improved low error rate performance quantization and algorithm choices many classes of high performance low density parity check ldpc codes are based on parity check matrices composed of permutation submatrices we describe the design of a parallel serial decoder architecture that can be used to map any ldpc code with such a structure to a hardware emulation platform high throughput emulation allows for the exploration of the low bit error rate ber region and provides statistics of the error traces which illuminate the causes of the error floors of the 2048 1723 reed solomon based ldpc rs ldpc code and the 2209 1978 array based ldpc code two classes of error events are observed oscillatory behavior and convergence to a class of non codewords termed absorbing sets the influence of absorbing sets can be exacerbated by message quantization and decoder implementation in particular quantization and the log tanh function approximation in sum product decoders strongly affect which absorbing sets dominate in the errorfloor region we show that conventional sum product decoder implementations of the 2209 1978 array based ldpc code allow low weight absorbing sets to have a strong effect and as a result elevate the error floor dually quantized sum product decoders and approximate sum product decoders alleviate the effects of low weight absorbing sets thereby lowering the error floor

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

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

near optimum universal belief propagation based decoding of low density parity check codes in this paper we propose a belief propagation bp based decoding algorithm which utilizes normalization to improve the accuracy of the soft values delivered by a previously proposed simplified bp based algorithm the normalization factors can be obtained not only by simulation but also importantly theoretically this new bp based algorithm is much simpler to implement than bp decoding as it requires only additions of the normalized received values and is universal i e the decoding is independent of the channel characteristics some simulation results are given which show this new decoding approach can achieve an error performance very close to that of bp on the additive white gaussian noise channel especially for low density parity check ldpc codes whose check sums have large weights the principle of normalization can also be used to improve the performance of the max log map algorithm in turbo decoding and some coding gain can be achieved if the code length is long enough

random edge local complementation with applications to iterative decoding of high density parity check codes we describe the application of edge local complementation elc to a tanner graph associated with a binary linear code c various properties of elc are described mainly the special case of isomorphic elc operations and the relationship to the automorphism group of the code aut c and the generalization of elc to weight bounding elc wb elc operations under which the number of edges remains upper bounded elc generates all systematic parity check matrices the orbit of the code so wb elc facilitates a restriction to low weight matrices of this orbit we propose using elc and wb elc as a source of diversity to improve iterative soft input soft output decoding of high density parity check hdpc codes with the sum product algorithm spa a motivation of elc enhanced spa decoding is locality that diversity is achieved by local graph action and is well suited to the local actions that constitute the spa and allows for parallel software implementation simulation data on the error rate performance of the proposed spa elc and spa wbelc iterative decoding algorithms is shown for several hdpc codes a gain is reported over spa decoding and over a recently proposed algorithm to decode hdpc codes using permutations from aut c elc enhanced decoding extends the scope of iterative decoding to codes with trivial aut c

iterative optimum symbol by symbol decoding and fast correlation attacks the relation between fast correlation attacks on stream ciphers based on linear feedback shift registers lfsrs and the hartmann rudolph 1976 algorithm for optimal symbol by symbol decoding of linear block codes is theoretically investigated by iterating approximate versions of the hr algorithm novel iterative probabilistic decoding algorithms for fast correlation attacks are developed in particular these algorithms can be applied when a large number of nonorthogonal parity checks are used a number of properties of the proposed iterated versions of the hr algorithm including convergence conditions and critical bit error rates are derived

low complexity affine mmse detector for iterative detection decoding mimo ofdm systems iterative turbo processing between detection and decoding shows near capacity performance on a multiple antenna system combining iterative processing with optimum front end detection is particularly challenging because the front end maximum a posteriori map algorithm has a computational complexity that is exponential sub optimum detector such as the soft interference cancellation linear minimum mean square error sic lmmse detector with near front end map performance has been proposed in the literature the asymptotic computational complexity of sic lmmse is o nt 2nr ntnr 3 ntmc2mc per detection decoding cycle where nt is number of transmit antenna nr is number of receive antenna and mc is modulation size a lower complexity detector is the hard interference cancellation lmmse hic lmmse detector hic lmmse has asymptotic complexity of o nt 2nr ntmc2mc but suffers extra performance degradation in this paper two front end detection algorithms are introduced that not only achieve asymptotic computational complexity of o nt 2nr ntnr 2 gamma beta 1 ntmc2mc where gamma beta is a function with discrete output 1 2 3 nt and o ntmc2mc respectively simulation results demonstrate that the proposed low complexity detection algorithms offer exactly same performance as their full complexity counterpart in an iterative receiver while being computational more efficient

reduced complexity decoding of ldpc codes various log likelihood ratio based belief propagation llr bp decoding algorithms and their reduced complexity derivatives for low density parity check ldpc codes are presented numerically accurate representations of the check node update computation used in llr bp decoding are described furthermore approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check node update or symbol node update or both in particular two main approaches for simplified check node updates are presented that are based on the so called min sum approximation coupled with either a normalization term or an additive offset term density evolution is used to analyze the performance of these decoding algorithms to determine the optimum values of the key parameters and to evaluate finite quantization effects simulation results show that these reduced complexity decoding algorithms for ldpc codes achieve a performance very close to that of the bp algorithm the unified treatment of decoding techniques for ldpc codes presented here provides flexibility in selecting the appropriate scheme from performance latency computational complexity and memory requirement perspectives

two bit flipping decoding algorithms for low density parity check codes in this letter a low complexity decoding algorithm for binary linear block codes is applied to low density paritycheck ldpc codes and improvements are described namely an extension to soft decision decoding and a loop detection mechanism for soft decoding only one real valued addition per code symbol is needed while the remaining operations are only binary as in the hard decision case the decoding performance is considerably increased by the loop detection simulation results are used to compare the performance with other known decoding strategies for ldpc codes with the result that the presented algorithms offer excellent performances at smaller complexity

applicability of n vicinity method for calculation of free energy of ising model abstract here we apply the n vicinity method of approximate calculation of the partition function to an ising model with the nearest neighbor interaction on d dimensional hypercube lattice we solve the equation of state for an arbitrary dimension d and analyze the behavior of the free energy as expected for large dimensions d 3 the system demonstrates a phase transition of the second kind in this case we obtain a simple analytical expression for the critical value of the inverse temperature when 3 d 7 this expression is in a very good agreement with the results of computer simulations in the case of small dimensions d 1 2 there is a noticeable discrepancy with the known exact results

a decoding algorithm for finite geometry ldpc codes in this paper we develop a new low complexity algorithm to decode low density parity check ldpc codes the developments are oriented specifically toward low cost yet effective decoding of high rate finite geometry fg ldpc codes the decoding procedure updates iteratively the hard decision received vector in search of a valid codeword in the vector space only one bit is changed in each iteration and the bit selection criterion combines the number of failed checks and the reliability of the received bits prior knowledge of the signal amplitude and noise power is not required an optional mechanism to avoid infinite loops in the search is also proposed our studies show that the algorithm achieves an appealing tradeoff between performance and complexity for fg ldpc codes

turbo decoding of product codes using adaptive belief propagation the adaptive belief propagation abp algorithm was recently proposed by jiang and narayanan for the soft decoding of reed solomon rs codes in this paper simplified versions of this algorithm are investigated for the turbo decoding of product codes the complexity of the turbo oriented adaptive belief propagation tab algorithm is significantly reduced by moving the matrix adaptation step outside of the belief propagation iteration loop a reduced complexity version of the tab algorithm that offers a trade off between performance and complexity is also proposed simulation results for the turbo decoding of product codes show that belief propagation based on adaptive parity check matrices is a practical alternative to the currently very popular chase pyndiah algorithm

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

reduced lists of error patterns for maximum likelihood soft decoding a method whereby a substantially reduced family of error patterns called survivors may be created for maximum likelihood soft decoding is introduced the family of survivor depends on the received word a decoder based on this approach first forms the survivors then scores them rather than obtaining the survivors by online elimination of error patterns the use of predetermined lists that represent all families of survivors is proposed

fault diagnosis in discrete event systems framework and model reduction a state based approach for online passive fault diagnosis in systems modeled as finite state automata is presented in this framework the system and the diagnoser the fault detection system do not have to be initialized at the same time furthermore no information about the state or even the condition failure status of the system before the initiation of diagnosis is required the design of the fault detection system in the worst case has exponential complexity a model reduction scheme with polynomial time complexity is introduced to reduce the computational complexity of the design diagnosability of failures is studied and necessary and sufficient conditions for failure diagnosability are derived

improved random redundant iterative hdpc decoding an iterative algorithm for soft input soft output siso decoding of classical algebraic cyclic block codes is presented below inspired by other approaches for high performance belief propagation bp decoding this algorithm requires up to 10 times less computational complexity than other methods that achieve similar performance by utilizing multiple bp decoders and using random permutation taken from the permutation group of the code this algorithm reaches near maximum likelihood performance a computational complexity comparison of the proposed algorithm versus other methods is presented as well this includes complexity versus performance analysis allowing one to trade between the former and the latter according to ones needs