a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

design of finite length irregular protograph codes with low error floors over the binary input awgn channel using cyclic liftings we propose a technique to design finite length irregular low density parity check ldpc codes over the binary input additive white gaussian noise awgn channel with good performance in both the waterfall and the error floor region the design process starts from a protograph which embodies a desirable degree distribution this protograph is then lifted cyclically to a certain block length of interest the lift is designed carefully to maximize the components of the approximate cycle extrinsic message degree ace spectrum of the code s tanner graph in a greedy fashion as a consequence the designed code would perform well in the error floor region moreover the proposed construction results in quasi cyclic codes which are attractive in practice due to simple encoder and decoder implementation simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions

response to nagle s criticism of my proposed definition of the entropy in a recent paper nagle criticized the new definition of entropy that i had proposed in an earlier work in the examples for which nagle claims my definition fails he took a formula that i had derived for one set of experiments and used it to represent my definition for other experiments however the formulas obtained from my definition depend on the specific experimental observables if my definition is correctly applied to nagle s experiments no contradictions remain

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

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

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

structured ira codes performance analysis and construction in this letter we present design techniques for structured irregular repeat accumulate s ira codes with low error rate floors these s ira codes need not be quasi cyclic permitting flexibility in code dimension length and rate we present a simple ensemble estimate of the level of the error rate floor of finite length ira codes on the additive white gaussian noise channel this performance estimate provides guidance on the choice of ira code column weights which yield low floors we also present two design algorithms for s ira codes accompanied by software and hardware based performance results which demonstrate their low floors lastly we present two design algorithms for multirate s ira code families implementable by a single encoder decoder

topological features of large fluctuations to the interior of a limit cycle received 29 october 1996 we investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves with overwhelming probability when it fluctuates far away from a stable state our emphasis is on systems that perform self sustained periodic vibrations and have an unstable focus inside a stable limit cycle we show that in the vicinity of the unstable focus the flow field of optimal paths generically displays a pattern of singularities in particular it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types the switching line spirals into the focus and has a self similar structure depending on the behavior of the system near the focus it may be smooth or have finite length branches our results are based on an analysis of the topology of the lagrangian manifold for an auxiliary purely dynamical problem that determines the optimal paths we illustrate our theory by studying both theoretically and numerically a van der pol oscillator driven by weak white noise s1063 651x 97 02703 7 many nonlinear dissipative systems display stable periodic oscillations i e their state spaces contain stable limit cycles 1 examples include lasers 2 radiofrequency generators 1 a chemical 3 a and biological 3 b systems in many cases the vibrations are nonsinusoidal as in a multimode laser with strong mode coupling 4 passive optically bistable elements 5 and various sorts of engineering structure 6 fluctuations in periodically oscillating systems play an extremely important role they eventually destroy periodicity of the self sustained oscillations so that the oscillation phase becomes random and a stationary probability distribution in the state space of the system is formed 7 8 this distribution reaches a maximum at the cycle does not vary much along the cycle and falls off rapidly in directions transverse to the cycle the shape of the maximum depends on the character of small transverse fluctuations away from the cycle if the fluctuations are due to the system being perturbed by weak noise of intensity d the transverse falloff will usually be approximately gaussian 7 9 with standard deviation proportional to d 1 2 as d 0

cyclic and quasi cyclic ldpc codes on constrained parity check matrices and their trapping sets this paper is concerned with construction and structural analysis of both cyclic and quasi cyclic codes particularly low density parity check ldpc codes it consists of three parts the first part shows that a cyclic code given by a parity check matrix in circulant form can be decomposed into descendant cyclic and quasi cyclic codes of various lengths and rates some fundamental structural properties of these descendant codes are developed including the characterization of the roots of the generator polynomial of a cyclic descendant code the second part of the paper shows that cyclic and quasi cyclic descendant ldpc codes can be derived from cyclic finite geometry ldpc codes using the results developed in the first part of the paper this enlarges the repertoire of cyclic ldpc codes the third part of the paper analyzes the trapping set structure of regular ldpc codes whose parity check matrices satisfy a certain constraint on their rows and columns several classes of finite geometry and finite field cyclic and quasi cyclic ldpc codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances consequently their error floor performances are dominated by their minimum distances

quantized iterative message passing decoders with low error floor for ldpc codes the error floor phenomenon observed with ldpc codes and their graph based iterative message passing mp decoders is commonly attributed to the existence of error prone substructures variously referred to as near codewords trapping sets absorbing sets or pseudocodewords in a tanner graph representation of the code many approaches have been proposed to lower the error floor by designing new ldpc codes with fewer such substructures or by modifying the decoding algorithm using a theoretical analysis of iterative mp decoding in an idealized trapping set scenario we show that a contributor to the error floors observed in the literature may be the imprecise implementation of decoding algorithms and in particular the message quantization rules used we then propose a new quantization method q 1 bit quasi uniform quantization that efficiently increases the dynamic range of messages thereby overcoming a limitation of conventional quantization schemes finally we use the quasi uniform quantizer to decode several ldpc codes that suffer from high error floors with traditional fixed point decoder implementations the performance simulation results provide evidence that the proposed quantization scheme can for a wide variety of codes significantly lower error floors with minimal increase in decoder complexity

error floor approximation for ldpc codes in the awgn channel this paper addresses the prediction of error floors of low density parity check codes transmitted over the additive white gaussian noise channel using a linear state space model to estimate the behavior of the sum product algorithm spa decoder in the vicinity of trapping sets tss we study the performance of the spa decoder in the log likelihood ratio llr domain as a function of the llr saturation level when applied to several widely studied codes the model accurately predicts a significant decrease in the error floor as the saturation level is allowed to increase for nonsaturating decoders however we find that the state space model breaks down after a small number of iterations due to the strong correlation of llr messages we then revisit richardson s importance sampling methodology for estimating error floors due to tss when those floors are too low for monte carlo simulation we propose modifications that account for the behavior of a nonsaturating decoder and present the resulting error floor estimates for the margulis code these estimates are much lower significantly steeper and more sensitive to iteration count than those previously reported

ldpc codes on partial geometries construction trapping set structure and puncturing many known constructions of ldpc codes can be placed in a general framework using the notion of partial geometries based on this notion the structure of such ldpc codes can be analyzed using a geometric approach that illuminates important properties of their parity check matrices in this approach trapping sets are represented by subgeometries of the geometry used to construct the code based on the incidence relations between lines and points in this geometry the structure of trapping sets is investigated on the other hand it is shown that removing a subgeometry corresponding to a trapping set gives a punctured matrix which can be used as a parity check matrix of an ldpc code this relates trapping sets represented by subgeometries and punctured matrices represented by the residual geometries the null spaces of these punctured matrices are ldpc codes which inherit many of the good structural properties of the original code hence new ldpc codes with various lengths and rates can be obtained by puncturing an ldpc code constructed based on a partial geometry furthermore these punctured matrices and codes can be used in a two phase decoding scheme to correct combinations of errors and erasures

selection of parity check equations for the iterative message passing detection of m sequences we consider the joint detection and decoding of m sequences the receiver has to decide whether an m sequence is received and possibly to decode its initial state to do so it implements an iterative message passing decoding algorithm that operates on a parity check matrix built upon a number of reference parity check equations satisfied by the m sequence this matrix concatenates several elementary parity check matrices which are derived from reference equations unlike the conventional decoding case the detection problem imposes to consider false alarms that may occur when the decoder is only fed with noise while absorbing sets are known to be responsible for the error floor phenomenon of iterative message passing decoders we show that they may have a beneficial effect on the detection performance in that they may prevent the decoder to produce false alarms we further compute the number of hybrid cycles of length six and eight in the tanner graph of the decoder and use the minimization of this number as criterion to derive an algorithm for selecting the reference parity check equations this algorithm was found to be efficient for minimizing the probability of false alarm and decreases also the probability of wrong detection in the very small snr region this has been achieved at the cost of a reduction of the probability of correct detection

use of hamiltonian mechanics in systems driven by colored noise the evaluation of the path integral representation for stochastic processes in the weak noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics we show that even when the noise is colored these may be put into a hamiltonian form which leads to better insights and improved numerical treatments we concentrate on solving hamilton s equations over an infinite time interval in order to determine the leading order contribution to the mean escape time for a bistable potential the paths may be oscillatory and inherently unstable in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy we look at two systems in some detail the underdamped langevin equation driven by external exponentially correlated noise and the overdamped langevin equation driven by external quasi monochromatic noise we deduce that caustics focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes

low floor decoders for ldpc codes one of the most significant impediments to the use of ldpc codes in many communication and storage systems is the error rate floor phenomenon associated with their iterative decoders the error floor has been attributed to certain subgraphs of an ldpc code s tanner graph induced by so called trapping sets we show in this paper that once we identify the trapping sets of an ldpc code of interest a sum product algorithm spa decoder can be custom designed to yield floors that are orders of magnitude lower than floors of the the conventional spa decoder we present three classes of such decoders 1 a bi mode decoder 2 a bit pinning decoder which utilizes one or more outer algebraic codes and 3 three generalized ldpc decoders we demonstrate the effectiveness of these decoders for two codes the rate 1 2 2640 1320 margulis code which is notorious for its floors and a rate 0 3 640 192 quasi cyclic code which has been devised for this study although the paper focuses on these two codes the decoder design techniques presented are fully generalizable to any ldpc code

stationary probability distribution near stable limit cycles far from hopf bifurcation points we obtain analytic results for the stationary probability distribution in the vicinity of a stable limit cycle for markov systems described by a fokker planck equation or a birth death master equation the results apply best for ranges of parameters removed from hopf bifurcation points as a by product we demonstrate that there holds a liouville like theorem for the stationary probability distribution the product of the velocity along the limit cycle times the area of the cross section of the probability distribution transverse to the cycle is a constant a numerical simulation of a chemical model system with a limit cycle shows good agreement with the analytic results

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

statistics of distinguishable particles and resolution of the gibbs paradox of the first kind in physics there are two distinct paradoxes which are both known as the gibbs paradox this article is concerned with only one of them the false increase in entropy which is calculated from the process of combining two gases of the same kind consisting of distinguishable particles in the following this paradox will be referred to as the gibbs paradox of the first kind gp1 two particles are said to be distinguishable if they are either non identical that is if they have different properties or if they are identical and there are microstates which change under transposition of the two particles the gp1 is demonstrated and subsequently analyzed the analysis shows that for quantum or classical systems of distinguishable particles it is generally uncertain of which particles they consist the neglect of this uncertainty is the root of the gp1 for the statistical description of a system of distinguishable particles an underlying set of particles containing all particles that in principle qualify for being part of the system is assumed to be known of which elements of this underlying particle set the system is composed differs from microstate to microstate thus the system is described by an ensemble of possible particle compositions the uncertainty about the particle composition contributes to the entropy of the system systems for which all possible particle compositions are equiprobable will be called harmonic classical systems of distinguishable identical particles are harmonic as a matter of principle quantum or classical systems of non identical particles are not necessarily harmonic since for them the composition probabilities depend individually on the preparation of the system harmonic systems with the same underlying particle set are always correlated hence for harmonic systems the entropy is no longer additive and loses its thermodynamic meaning a quantity derived from entropy is introduced the reduced entropy which for harmonic systems replaces the entropy as thermodynamic potential for identical classical particles the equivalence in particular with respect to the second law of thermodynamics between distinguishability and indistinguishability is proved the resolution of the gp1 is demonstrated applying the previously found results

on the bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

stochastic analysis of a hopf bifurcation master equation approach the effect of inhomogeneous fluctuations in a reaction diffusion system exhibiting a hopf bifurcation is analyzed using the master equation approach a taylor expansion of the logarithm of the stationary probability known as the stochastic potential is calculated this procedure displays marked analogies with the theory of normal forms the critical potential reduced to its local expansion around an arbitrary point of the limit cycle brings out the essential role played by the phase of the oscillating variables a comparison with the langevin analysis of walgraefet al j chem phys 78 6 3043 1983 is performed

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

escape problem for irreversible systems the problem of noise induced escape from a metastable state arises in physics chemistry biology systems engineering and other areas the problem is well understood when the underlying dynamics of the system obey detailed balance when this assumption fails many of the results of classical transition rate theory no longer apply and no general method exists for computing the weak noise asymptotic behavior of fundamental quantities such as the mean escape time in this paper we present a general technique for analyzing the weak noise limit of a wide range of stochastically perturbed continuous time nonlinear dynamical systems we simplify the original problem which involves solving a partial differential equation into one in which only ordinary differential equations need be solved this allows us to resolve some old issues for the case when detailed balance holds when it does not hold we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path we also present results on short time behavior and discuss the possibility of ital focusing along the escape path

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

from cages to trapping sets and codewords a technique to derive tight upper bounds on the minimum size of trapping sets and minimum distance of ldpc codes cages defined as regular graphs with minimum number of nodes for a given girth are well studied in graph theory trapping sets are graphical structures responsible for error floor of low density parity check ldpc codes and are well investigated in coding theory in this paper we make connections between cages and trapping sets in particular starting from a cage or a modified cage we construct a trapping set in multiple steps based on the connection between cages and trapping sets we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable regular ldpc codes with a given variable degree and girth the derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets considering that non zero codewords are a special case of trapping sets we also derive tight upper bounds on the minimum weight of such codewords i e the minimum distance of variable regular ldpc codes as a function of variable degree and girth

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

quasi equilibrium closure hierarchies for the boltzmann equation abstract after boltzmann and gibbs the notion of disorder in statistical physics relates to ensembles not to individual states this disorder is measured by the logarithm of ensemble volume the entropy but recent results about measure concentration effects in analysis and geometry allow us to return from the ensemble based point of view to a state based one at least partially in this paper the order disorder problem is represented as a problem of relation between distance and measure the effect of strong order disorder separation for multiparticle systems is described the phase space could be divided into two subsets one of them set of disordered states has almost zero diameter the second one has almost zero measure the symmetry with respect to permutations of particles is responsible for this type of concentration dynamics of systems with strong order disorder separation has high average acceleration squared which can be interpreted as evolution through a series of collisions acceleration dominated dynamics the time arrow direction from order to disorder follows from the strong order disorder separation but inverse for systems in space of symmetric configurations with sticky boundaries the way back from disorder to order is typical natural selection recommendations for mining of molecular dynamics results are also presented

graph based message passing schedules for decoding ldpc codes we study a wide range of graph based message passing schedules for iterative decoding of low density parity check ldpc codes using the tanner graph tg of the code and for different nodes and edges of the graph we relate the first iteration in which the corresponding messages deviate from their optimal value corresponding to a cycle free graph to the girths and the lengths of the shortest closed walks in the graph using this result we propose schedules which are designed based on the distribution of girths and closed walks in the tg of the code and categorize them as node based versus edge based unidirectional versus bidirectional and deterministic versus probabilistic these schedules in some cases outperform the previously known schedules and in other cases provide less complex alternatives with more or less the same performance the performance complexity tradeoff and the best choice of schedule appear to depend not only on the girth and closed walk distributions of the tg but also on the iterative decoding algorithm and channel characteristics we examine the application of schedules to belief propagation sum product over additive white gaussian noise awgn and rayleigh fading channels min sum max sum over an awgn channel and gallager s algorithm a over a binary symmetric channel

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

regarding the entropy of distinguishable particles the conventional way of explaining gibbs paradox as due to the distinguishability of particles has been challenged recently and a new fundamental definition for the entropy has been proposed that gives the same entropy for distinguishable particles as for indistinguishable particles in this paper arguments are presented that strongly favor the conventional definition of entropy and its resolution of gibbs paradox

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

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

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

construction of low density parity check codes by superposition this paper presents a superposition method for constructing low density parity check ldpc codes several classes of structured ldpc codes are constructed codes in these classes perform well with iterative decoding and their tanner graphs have girth at least six