high rate girth eight low density parity check codes on rectangular integer lattices this letter introduces a combinatorial construction of girth eight high rate low density parity check codes based on integer lattices the parity check matrix of a code is defined as a point line incidence matrix of a 1 configuration based on a rectangular integer lattice and the girth eight property is achieved by a judicious selection of sets of parallel lines included in a configuration a class of codes with a wide range of lengths and column weights is obtained the resulting matrix of parity checks is an array of circulant matrices

error correction capability of column weight three ldpc codes under the gallager a algorithm part ii the relation between the girth and the error correction capability of column weight three ldpc codes under the gallager a algorithm is investigated it is shown that a column weight three ldpc code with tanner graph of girth g 10 can correct all error patterns with up to g 2 1 errors in at most g 2 iterations of the gallager a algorithm for codes with tanner graphs of girth g 8 it is shown that girth alone cannot guarantee correction of all error patterns with up to g 2 1 errors under the gallager a algorithm sufficient conditions to correct g 2 1 errors are then established by studying trapping sets

construction of quasi cyclic ldpc codes for awgn and binary erasure channels a finite field approach in the late 1950s and early 1960s finite fields were successfully used to construct linear block codes especially cyclic codes with large minimum distances for hard decision algebraic decoding such as bose chaudhuri hocquenghem bch and reed solomon rs codes this paper shows that finite fields can also be successfully used to construct algebraic low density parity check ldpc codes for iterative soft decision decoding methods of construction are presented ldpc codes constructed by these methods are quasi cyclic qc and they perform very well over the additive white gaussian noise awgn binary random and burst erasure channels with iterative decoding in terms of bit error probability block error probability error floor and rate of decoding convergence collectively particularly they have low error floors since the codes are qc they can be encoded using simple shift registers with linear complexity

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

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

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

pseudocodewords of tanner graphs this paper presents a detailed analysis of pseudocodewords of tanner graphs pseudocodewords arising on the iterative decoder s computation tree are distinguished from pseudocodewords arising on finite degree lifts lower bounds on the minimum pseudocodeword weight are presented for the bec bsc and awgn channel some structural properties of pseudocodewords are examined and pseudocodewords and graph properties that are potentially problematic with min sum iterative decoding are identified an upper bound on the minimum degree lift needed to realize a particular irreducible lift realizable pseudocodeword is given in terms of its maximal component and it is shown that all irreducible lift realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure examples and different tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed the results obtained provide some insights in relating the structure of the tanner graph to the pseudocodeword distribution and suggest ways of designing tanner graphs with good minimum pseudocodeword weight

near shannon limit quasi cyclic low density parity check codes this letter presents two classes of quasi cyclic low density parity check codes that perform close to the shannon limit

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

on trapping sets and guaranteed error correction capability of ldpc codes and gldpc codes the relation between the girth and the guaranteed error correction capability of left regular low density parity check ldpc codes when decoded using the bit flipping serial and parallel algorithms is investigated a lower bound on the size of variable node sets which expand by a factor of at least 3 4 is found based on the moore bound this bound combined with the well known expander based arguments leads to a lower bound on the guaranteed error correction capability the decoding failures of the bit flipping algorithms are characterized using the notions of trapping sets and fixed sets the relation between fixed sets and a class of graphs known as cage graphs is studied upper bounds on the guaranteed error correction capability are then established based on the order of cage graphs the results are extended to left regular and right uniform generalized ldpc codes it is shown that this class of generalized ldpc codes can correct a linear number of worst case errors in the code length under the parallel bit flipping algorithm when the underlying tanner graph is a good expander a lower bound on the size of variable node sets which have the required expansion is established

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

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

control of singular problem via differentiation of a min max abstract for a smooth domain in r n with boundary 0 1 we are concerned with the wave equation y y s in q t 0 t with t at source term satisfying s s l 1 0 t l 2 a dirichlet condition is imposed on 0 and we consider an absorbing condition y n uy 0 in 0 t gl 1 where u is the control parameter we introduce the cost function j u 1 2 t 0 1 uy 2 d d t and using the min max formulation of j we by pasas the sensitivity analysis of u y and obtain the gradient of j with a usual adjoint problem we first present an abstract frame for this kind of problems using the differentiability results of a min max 1 2 which we very shortly deduce here we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

efficient encoding of quasi cyclic low density parity check codes efficient encoding of quasi cyclic low density parity check codes quasi cyclic qc low density parity check ldpc codes form an important subclass of ldpc codes these codes have encoding advantage over other types of ldpc codes this paper addresses the issue of efficient encoding of qc ldpc codes two methods are presented to find the generator matrices of qc ldpc codes in systematic circulant form from their parity check matrices given in circulant form based on the systematic circulation form of the generator matrix of a qc ldpc code various types of encoding circuits using simple shift registers are devised it is shown that the encoding complexity of a qc ldpc code is linearly proportional to the number of parity bits of the code for serial encoding and to the length of the code for high speed parallel encoding

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

lowering the error floor of ldpc codes using cyclic liftings cyclic liftings are proposed to lower the error floor of low density parity check ldpc codes the liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets we derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle of length l in the base graph such that the inverse image of in the lifted graph consists of only cycles of length strictly larger than l the proposed method is universal in the sense that it can be applied to any ldpc code over any channel and for any iterative decoding algorithm it also preserves important properties of the base code such as degree distributions and in some cases the code rate the constructed codes are quasi cyclic and thus attractive from a practical point of view the proposed method is applied to both structured and random codes over the binary symmetric channel bsc the error floor improves consistently by increasing the lifting degree and the results show significant improvements in the error floor compared to the base code a random code of the same degree distribution and block length and a random lifting of the same degree similar improvements are also observed when the codes designed for the bsc are applied to the additive white gaussian noise awgn channel

asymptotic spectra of trapping sets in regular and irregular ldpc code ensembles we evaluate the asymptotic normalized average distributions of a class of combinatorial configurations in random regular and irregular binary low density parity check ldpc code ensembles among the configurations considered are trapping and stopping sets these sets represent subsets of variable nodes in the tanner graph of a code that play an important role in determining the height and point of onset of the error floor in its performance curve the techniques used for deriving the spectra include large deviations theory and statistical methods for enumerating binary matrices with prescribed row and column sums these techniques can also be applied in a setting that involves more general structural entities such as subcodes and or minimal codewords that are known to characterize other important properties of soft decision decoders of linear block codes

estimation of bit and frame error rates of finite length low density parity check codes on binary symmetric channels a method for estimating the performance of low density parity check ldpc codes decoded by hard decision iterative decoding algorithms on binary symmetric channels bscs is proposed based on the enumeration of the smallest weight error patterns that cannot be all corrected by the decoder this method estimates both the frame error rate fer and the bit error rate ber of a given ldpc code with very good precision for all crossover probabilities of practical interest through a number of examples we show that the proposed method can be effectively applied to both regular and irregular ldpc codes and to a variety of hard decision iterative decoding algorithms compared with the conventional monte carlo simulation the proposed method has a much smaller computational complexity particularly for lower error rates

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

quasi cyclic low density parity check codes with girth larger than 12 a quasi cyclic qc low density parity check ldpc code can be viewed as the protograph code with circulant permutation matrices or circulants in this correspondence we find all the subgraph patterns of protographs of qc ldpc codes having inevitable cycles of length 2i i 6 7 8 9 10 i e the cycles that always exist regardless of the shift values of circulants it is also derived that if the girth of the protograph is 2g g 2 its protograph code cannot have the inevitable cycles of length smaller than 6g based on these subgraph patterns we propose new combinatorial construction methods of the protographs whose protograph codes can have girth larger than or equal to 14 or 18 we also propose a couple of shift value assigning rules for circulants of a qc ldpc code guaranteeing the girth 14

minimum pseudoweight and minimum pseudocodewords of ldpc codes in this correspondence we study the minimum pseudoweight and minimum pseudocodewords of low density parity check ldpc codes under linear programming lp decoding first we show that the lower bound of kelley sridhara xu and rosenthal on the pseudoweight of a nonzero pseudocodeword of an ldpc code whose tanner graph has girth greater than is tight if and only if this pseudocodeword is a real multiple of a codeword then the lower bound of kashyap and vardy on the stopping distance of an ldpc code is proved to be also a lower bound on the pseudoweight of a nonzero pseudocodeword of an ldpc code whose tanner graph has girth and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword using these results we further obtain that for some ldpc codes there are no other minimum pseudocodewords except the real multiples of minimum weight codewords this means that the lp decoding for these ldpc codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of lp decoding and maximum likelihood decoding approaches as the signal to noise ratio snr tends to infinity finally some ldpc codes are listed to illustrate these results

error correction capability of column weight three ldpc codes in this paper the error correction capability of column weight three low density parity check ldpc codes when decoded using the gallager a algorithm is investigated it is proved that a necessary condition for a code to correct all error patterns with up to k ges 5 errors is to avoid cycles of length up to 2k in its tanner graph as a consequence of this result it is shown that given any alpha 0 exist n such that forall n n no code in the ensemble of column weight three codes can correct all alphan or fewer errors the results are extended to the bit flipping algorithms

on the hardness of approximating stopping and trapping sets we prove that approximating the size of stopping and trapping sets in tanner graphs of linear block codes and more restrictively the class of low density parity check ldpc codes is np hard the ramifications of our findings are that methods used for estimating the height of the error floor of moderate and long length ldpc codes based on stopping and trapping set enumeration cannot provide accurate worst case performance predictions for most codes

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

finding all small error prone substructures in ldpc codes it is proven in this work that it is np complete to exhaustively enumerate small error prone substructures in arbitrary finite length low density parity check ldpc codes two error prone patterns of interest include stopping sets for binary erasure channels becs and trapping sets for general memoryless symmetric channels despite the provable hardness of the problem this work provides an exhaustive enumeration algorithm that is computationally affordable when applied to codes of practical short lengths n ap 500 by exploiting the sparse connectivity of ldpc codes the stopping sets of size les 13 and the trapping sets of size les11 can be exhaustively enumerated the central theorem behind the proposed algorithm is a new provably tight upper bound on the error rates of iterative decoding over becs based on a tree pruning technique this upper bound can be iteratively sharpened until its asymptotic order equals that of the error floor this feature distinguishes the proposed algorithm from existing non exhaustive ones that correspond to finding lower bounds of the error floor the upper bound also provides a worst case performance guarantee that is crucial to optimizing ldpc codes when the target error rate is beyond the reach of monte carlo simulation numerical experiments on both randomly and algebraically constructed ldpc codes demonstrate the efficiency of the search algorithm and its significant value for finite length code optimization

singular minimizers for regular one dimensional problems in the calculus of variations i i f x u x u x dx ja where a b is a finite interval denotes d dx f f x u p is c 0 and fpp 0 regularity we consider the problem of minimizing in the set a of absolutely continuous functions u a b r satisfying u a a u b 3 where a and 5 are given constants as is well known if a minimizer u of i in a is lipschitz continuous then u is smooth and satisfies the euler lagrange equation

error correction on a tree an instanton approach we introduce a method that allows analytical or semianalytical estimating of the post error correction bit error rate ber when a forward error correction is utilized for transmitting information through a noisy channel the generic method that applies to a variety of error correction schemes in the regimes where the ber is low is illustrated using the example of a finite size code approximated by a treelike structure exploring the statistical physics formulation of the problem we find that the ber decreases with the signal to noise ratio nonuniformly i e crossing over through a sequence of phases the higher the signal to noise ratio the lower the symmetry of the phase dominating ber the last 50 years have witnessed a tremendous increase in the amount of data transmitted through various communication systems this also leads to tightening of the conditions for error free data transfer in the presence of noise and other transmission related impairments the problem of dealing with errors in information flow has a fundamental importance and has been studied extensively in information and coding theory in 1948 shannon 1 proved that applying an error corrected code can result in an error free communication in the thermodynamic limit of an infinitely long message as long as the rate of transmitted information is kept below a certain value known as the channel capacity constructing well performing capacity approaching yet practical codes has been a challenge until the discovery that some classes of codes based on random construction 2 can achieve near optimum performance when used for transmission over white additive gaussian noise channels 3 in the past few years several codes have been designed with performances very close to this limit 4 generally these codes are referred to as codes on graphs and their prime examples are low density parity check ldpc codes and turbo codes a linear block code for which ldpc codes are an example can be represented as the set

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

shortened array codes of large girth one approach to designing structured low density parity check ldpc codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity check matrix contain all the variables involved in short cycles this approach is especially effective if the parity check matrix of a code is a matrix composed of blocks of circulant permutation matrices as is the case for the class of codes known as array codes we show how to shorten array codes by deleting certain columns of their parity check matrices so as to increase their girth the shortening approach is based on the observation that for array codes and in fact for a slightly more general class of ldpc codes the cycles in the corresponding tanner graph are governed by certain homogeneous linear equations with integer coefficients consequently we can selectively eliminate cycles from an array code by only retaining those columns from the parity check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles we provide ramsey theoretic estimates for the maximum number of columns that can be retained from the original parity check matrix with the property that the sequence of their indices avoid solutions to various types of cycle governing equations this translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles simulation results show that for the codes considered shortening them to increase the girth can lead to significant gains in signal to noise ratio snr in the case of communication over an additive white gaussian noise awgn channel

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

eliminating trapping sets in low density parity check codes by using tanner graph covers we discuss error floor asympotics and present a method for improving the performance of low density parity check ldpc codes in the high snr error floor region the method is based on tanner graph covers that do not have trapping sets from the original code the advantages of the method are that it is universal as it can be applied to any ldpc code channel decoding algorithm and it improves performance at the expense of increasing the code length without losing the code regularity without changing the decoding algorithm and under certain conditions without lowering the code rate the proposed method can be modified to construct convolutional ldpc codes also the method is illustrated by modifying tanner mackay and margulis codes to improve performance on the binary symmetric channel bsc under the gallager b decoding algorithm decoding results on awgn channel are also presented to illustrate that optimizing codes for one channel decoding algorithm can lead to performance improvement on other channels