asymptotic performance of single parity check product codes this article investigates the asymptotic performance of single parity check spc product codes pcs from a decoding point of view specifically the probability of bit error is bounded before and after the decoding of each dimension similar to the analysis of iterated codes by elias 1954 it is shown that the asymptotic probability of bit error can be driven to zero as the number of dimensions and hence the block length increases at signal to noise ratios snrs within 2 db of capacity over the additive white gaussian noise awgn channel

interior point decoding for linear vector channels based on convex optimization in the present paper a novel decoding algorithm for low density parity check ldpc codes based on convex optimization is presented the decoding algorithm which is referred to hereinafter as interior point decoding is designed for linear vector channels the linear vector channels include several practically important channels such as inter symbol interference channels and partial response pr channels it is shown that the maximum likelihood decoding mld rule for a linear vector channel can be relaxed to a convex optimization problem which is called a relaxed mld problem the proposed decoding algorithm is based on a numerical optimization technique known as the interior point method with barrier functions approximate variations of an interior point method based on the gradient descent and newton methods are used to solve the relaxed mld problem compared with a conventional joint message passing decoder from computer simulations it is observed that the proposed decoding algorithm achieves better ber performance on pr channels with less decoding complexity in several cases furthermore an extension of the proposed algorithm for high order modulation formats such as pam and qam is presented

adaptive cut generation algorithm for improved linear programming decoding of binary linear codes linear programming lp decoding approximates maximum likelihood ml decoding of a linear block code by relaxing the equivalent ml integer programming ip problem into a more easily solved lp problem the lp problem is defined by a set of box constraints together with a set of linear inequalities called parity inequalities that are derived from the constraints represented by the rows of a parity check matrix of the code and can be added iteratively and adaptively in this paper we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint or cut at a point in the unit hypercube then we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check rpc constraints that can eliminate pseudocodewords produced by the lp decoder often significantly improving the decoder error rate performance the cut generating algorithm is based upon a specific transformation of an initial parity check matrix of the linear block code we also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut generating algorithm simulation results for several low density parity check ldpc codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between lp decoding and ml decoding

the effect of interference of coloured additive and multiplicative white noises on escape rate in this paper we have calculated the escape rate from a meta stable state for coloured and correlated noise driven open systems based on the fokker planck description of the stochastic process we consider the effect of two correlation times due to the additive coloured noise and the correlation between additive coloured and multiplicative white noises the effect of the noise correlation strength on the rate has also been investigated

efficient linear programming decoding of hdpc codes we propose several improvements for linear programming lp decoding algorithms for high density parity check hdpc codes first we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts from redundant parity checks second we propose an efficient mixed integer decoder utilizing the branch and bound method we further enhance the proposed decoders by removing inactive constraints and by adapting the parity check matrix prior to decoding according to the channel observations based on simulation results the proposed decoders achieve near ml performance with reasonable complexity

decoding binary linear block codes using local search this paper presents a novel iterative hard decision decoding algorithm for binary linear block codes over a binary symmetric channel bsc the problem is formulated as a 0 1 integer programming problem which is known to be np hard when the crossover probability c of the channel is known the solution space of the decoding problem can be decreased to a sphere whose radius is related to c using the penalty function method the problem is reformulated on this reduced solution space then an iterative multi flip local search algorithm is designed to find the global solution of this decoding problem for a code with minimum distance d when the radius of the sphere is not greater than d 1 2 this algorithm has the maximum likelihood ml certificate property i e if the decoder outputs a codeword it is guaranteed to be the ml codeword compared to the probabilistic suboptimal iterative belief propagation bp decoder this approach has lower complexity and better performance numerical results show that in terms of speed and performance the proposed decoding method outperforms bp decoding in the error floor region

maximum likelihood soft decoding of binary block codes and decoders for the golay codes maximum likelihood soft decision decoding of linear block codes is addressed a binary multiple check generalization of the wagner rule is presented and two methods for its implementation one of which resembles the suboptimal forney chase algorithms are described besides efficient soft decoding of small codes the generalized rule enables utilization of subspaces of a wide variety thereby yielding maximum likelihood decoders with substantially reduced computational complexity for some larger binary codes more sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the 24 12 golay code yielding a computational gain factor of about 2 with respect to previous methods a ternary single check version of the wagner rule is applied for efficient soft decoding of the 12 6 ternary golay code

bistable kinetic model driven by correlated noises steady state analysis a simple rule to obtain the fokker planck equation for a general one dimensional system driven by correlated gaussian white noises is proved by the functional method the fokker planck equation obtained in this paper is applied to the bistable kinetic model we find the following for the steady state 1 in the ensuremath alpha d parameter plane ensuremath alpha is the strength of the additive noise and d is the multiplicative noise strength the critical curve separating the unimodal and bimodal regions of the stationary probability distribution spd of the model is shown to be affected by ensuremath lambda the degree of correlation of the noises as ensuremath lambda is increased the area of the bimodal region in the ensuremath alpha d plane is contracted 2 when we take a point fixed in the ensuremath alpha d plane and increase ensuremath lambda the form of the spd changes from a bimodal to a unimodal structure 3 the positions of the extreme value of the spd of the model sensitively depend on the strength of the multiplicative noise and weakly depend on the additive noise strength 4 for ensuremath lambda 1 the case of perfectly correlated noises when the parameters ensuremath alpha and d take values in the neighborhood of the line ensuremath alpha d in the ensuremath alpha d plane the spd s corresponding to the points ensuremath alpha dg1 and ensuremath alpha d1 exhibit a very different shape of divergence therefore the ratio ensuremath alpha d 1 plays the role of a critical ratio

the approximate maximum likelihood certificate the confidence in the reliability of a codeword output by some not necessarily optimal decoding algorithm is discussed a new property which relies on the linear programming lp decoder the approximate maximum likelihood certificate amlc is introduced to address this issue as follows first the channel output vector is decoded by some symmetric decoder d e g belief propagation or min sum algorithm decoding second the channel output vector is decoded by lp decoding third if the decoding result of d is a codeword its lp value is compared to the lp value of the lp decoding result the latter need not be a codeword if these two values are close the amlc holds using upper bounding techniques we show that the conditional frame error probability given that the amlc holds is with some degree of confidence below a threshold in channels with low noise this threshold is orders of magnitude lower than the simulated frame error rate and our bound holds with a very high degree of confidence this is in stark contrast with standard monte carlo simulation which would require excessively long runs to demonstrate like performance when the amlc holds our approach thus provides the decoder with extra error detection capability which is especially important in applications requiring high data integrity

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

fluctuations impact on a pattern forming model of population dynamics with non local interactions a model of interacting random walkers is presented and shown to give rise to patterns consisting in periodic arrangements of fluctuating particle clusters the model represents biological individuals that die or reproduce at rates depending on the number of neighbors within a given distance we evaluate the importance of the discrete and fluctuating character of this particle model on the pattern forming process to this end a deterministic mean field description including a linear stability and a weakly nonlinear analysis is given and compared with the particle model the deterministic approach is shown to reproduce some of the features of the discrete description in particular the existence of a finite wavelength instability stochasticity in the particle dynamics however has strong effects in other important aspects such as the parameter values at which pattern formation occurs or the nature of the homogeneous phase

pseudo codeword analysis of tanner graphs from projective and euclidean planes we consider coded data transmission over a binary input output symmetric memoryless channel using a binary linear code in order to understand the performance of maximum likelihood ml decoding one studies the codewords in particular the minimal codewords and their hamming weights in the context of linear programming lp decoding one s attention needs to be shifted to the pseudo codewords in particular to the minimal pseudo codewords and their pseudo weights in this paper we investigate some families of codes that have good properties under lp decoding namely certain families of low density parity check ldpc codes that are derived from projective and euclidean planes we study the structure of their minimal pseudo codewords and give lower bounds on their pseudo weight besides this main focus we also present some results that hold for pseudo codewords and minimal pseudo codewords of any tanner graph and we highlight how the importance of minimal pseudo codewords under lp decoding varies depending on which binary input output symmetric memoryless channel is used

on decoding of the 89 45 17 quadratic residue code in this paper three decoding methods of the 89 45 17 binary quadratic residue qr code to be presented are hard soft and linear programming decoding algorithms firstly a new hybrid algebraic decoding algorithm for the 89 45 17 qr code is proposed it uses the laplace formula to obtain the primary unknown syndromes as done in lin et al s algorithm when the number of errors v is less than or equal to 5 whereas gaussian elimination is adopted to compute the unknown syndromes when v 6 secondly an appropriate modification to the algorithm developed by chase is also given in this paper therefore combining the proposed algebraic decoding algorithm with the modified chase ii algorithm called a new soft decision decoding algorithm becomes a complete soft decoding of qr codes thirdly in order to further improve the error correcting performance of the code linear programming lp is utilized to decode the 89 45 17 qr code simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with lin et al s hard decoding algorithm and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate ber performance moreover the lp based decoding improves the error rate performance almost without increasing the decoding complexity when compared with the new soft decision decoding algorithm it provides a coding gain of 0 2 db at ber 2 x 10 6

iterative approximate linear programming decoding of ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of low density parity check ldpc codes is considered an iterative algorithm similar to min sum and belief propagation for efficient approximate solution of this problem was proposed by vontobel and koetter in this paper the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose in particular we are interested in obtaining a feasible vector in the lp decoding problem that is close to optimal in the following sense the distance normalized by the block length between the minimum and the objective function value of this approximate solution can be made arbitrarily small it is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length combined with previous results that have shown that the lp decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity this is achieved by first applying the iterative lp decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits and then correcting the remaining errors using some standard method these conclusions are also extended to generalized ldpc codes

spatially chaotic configurations and functional equations with rescaling the functional equation is associated with the appearance of spatially chaotic structures in amorphous glassy materials continuous compactly supported solutions of the above equation are of special interest we shall show that there are no such solutions for whereas such a solution exists for almost all the words for almost all q in the previous sentence cannot be omitted there are exceptional values of q in the interval for which there are no integrable solutions for example which is the reciprocal of the golden ratio is such an exceptional value more generally if is any pisot vijayaraghavan number or any salem number then is an exceptional value

generalization of the escape rate from a metastable state driven by external cross correlated noise processes we propose generalization of the escape rate from a metastable state for externally driven correlated noise processes in one dimension in addition to the internal non markovian thermal fluctuations the external correlated noise processes we consider are gaussian stationary in nature and are of ornstein uhlenbeck type based on a fokker planck description of the effective noise processes with finite memory we derive the generalized escape rate from a metastable state in the moderate to large damping limit and investigate the effect of degree of correlation on the resulting rate comparison of the theoretical expression with numerical simulation gives a satisfactory agreement and shows that by increasing the degree of external noise correlation one can enhance the escape rate through the dressed effective noise strength

internal fluctuations period doubling and chemical chaos the effects of internal molecular fluctuations on period doubling bifurcations and chaotic band merging are studied in a well stirred reactive chemical system where spatial degrees of freedom play no role the calculations are carried out using a stochastic model based on reactive lattice gas cellular automata molecular fluctuations arising from reactive collisions are incorporated in the model and the mass action rate equations are recovered in the mean field limit large system sizes can be studied and permit the investigation of the internal noise scaling structure in the periodic and chaotic regimes for the willamowski r ossler system the noise scaling exponent is consistent with that found for one dimensional quadratic maps subject to external noise the model provides a means to study small non equilibrium reacting systems where fluctuations may not be neglected

a new linear programming approach to decoding linear block codes in this paper we propose a new linear programming formulation for the decoding of general linear block codes different from the original formulation given by feldman the number of total variables to characterize a parity check constraint in our formulation is less than twice the degree of the corresponding check node the equivalence between our new formulation and the original formulation is proven the new formulation facilitates to characterize the structure of linear block codes and leads to new decoding algorithms in particular we show that any fundamental polytope is simply the intersection of a group of the so called minimum polytopes and this simplified formulation allows us to formulate the problem of calculating the minimum hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables we then propose a branch and bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem in addition we prove that for the family of single parity check spc product codes the fractional distance and the pseudodistance are both equal to the minimum distance finally we propose an efficient algorithm for decoding spc product codes with low complexity and maximum likelihood ml decoding performance

on decoding of the 73 37 13 quadratic residue code in this paper a method to search the set of syndromes indices needed in computing the unknown syndromes for the 73 37 13 quadratic residue qr code is proposed according to the resulting index sets one computes the unknown syndromes and thus finds the corresponding error locator polynomial by using an inverse free berlekamp massey bm algorithm based on the modified chase ii algorithm the performance of soft decision decoding for the 73 37 13 qr code is given this result is new moreover the error rate performance of linear programming lp decoding for the 73 37 13 qr code is also investigated and lp based decoding is shown to be significantly superior in performance to the algebraic soft decision decoding while requiring almost the same computational complexity in fact the algebraic hard decision and soft decision decoding of the 89 45 17 qr code outperforms that of the 73 37 13 qr code because the former has a larger minimal distance however experimental results indicate that the 73 37 13 qr code outperforms the 89 45 17 qr code with much fewer arithmetic operations when using the lp based decoding algorithms the pseudocodewords analysis partially explains this seemingly strange phenomenon

low complexity lp decoding of nonbinary linear codes linear programming lp decoding of low density parity check ldpc codes has attracted much attention in the research community in the past few years lp decoding has been derived for binary and nonbinary linear codes however the most important problem with lp decoding for both binary and nonbinary linear codes is that the complexity of standard lp solvers such as the simplex algorithm remains prohibitively large for codes of moderate to large block length to address this problem two low complexity lp lclp decoding algorithms for binary linear codes have been proposed by vontobel and koetter henceforth called the basic lclp decoding algorithm and the subgradient lclp decoding algorithm in this paper we generalize these lclp decoding algorithms to nonbinary linear codes the computational complexity per iteration of the proposed nonbinary lclp decoding algorithms scales linearly with the block length of the code a modified bcjr algorithm for efficient check node calculations in the nonbinary basic lclp decoding algorithm is also proposed which has complexity linear in the check node degree several simulation results are presented for nonbinary ldpc codes defined over z4 gf 4 and gf 8 using quaternary phase shift keying and 8 phase shift keying respectively over the awgn channel it is shown that for some group structured ldpc codes the error correcting performance of the nonbinary lclp decoding algorithms is similar to or better than that of the min sum decoding algorithm

improved linear programming decoding of ldpc codes and bounds on the minimum and fractional distance we examine ldpc codes decoded using linear programming lp four contributions to the lp framework are presented first a new method of tightening the lp relaxation and thus improving the lp decoder is proposed second we present an algorithm which calculates a lower bound on the minimum distance of a specific code this algorithm exhibits complexity which scales quadratically with the block length third we propose a method to obtain a tight lower bound on the fractional distance also with quadratic complexity and thus less than previously existing methods finally we show how the fundamental lp polytope for generalized ldpc codes and nonbinary ldpc codes can be obtained

single parity check product codes this paper considers the performance of iteratively decoded single parity check spc multidimensional product codes in an additive white gaussian noise channel asymptotic performance bounds are compared to simulation results a new code structure based on spc product codes is introduced this structure involves interleaving between the encoding of each dimension in the product code an analysis of the weight distribution is used to explain the good performance results for these randomly interleaved spc product codes

a separation algorithm for improved lp decoding of linear block codes maximum likelihood ml decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming ip problem feldman relaxed this ip problem and presented linear programming lp based decoding in this paper we propose a new separation algorithm to improve the error correcting performance of lp decoding for binary linear block codes we use an ip formulation with indicator variables that help in detecting the violated parity checks we derive gomory cuts from the ip and use them in our separation algorithm an efficient method of finding cuts induced by redundant parity checks rpc is also proposed under certain circumstances we can guarantee that these rpc cuts are valid and cut off the fractional optimal solutions of lp decoding it is demonstrated on three ldpc codes and two bch codes that our separation algorithm performs significantly better than lp decoding and belief propagation bp decoding

rate laws of the self induced aggregation kinetics of brownian particles abstract in this paper we have studied the self induced aggregation kinetics of brownian particles in the presence of both multiplicative and additive noises in addition to the drift due to the self aggregation process the environment may induce a drift term in the presence of a multiplicative noise then there would be an interplay between the two drift terms it may account qualitatively the appearance of the different laws of aggregation process at low strength of white multiplicative noise the cluster number decreases as a gaussian function of time if the noise strength becomes appreciably large then the variation of cluster number with time is fitted well by the mono exponentially decaying function of time for additive noise driven case the decrease of cluster number can be described by the power law but in case of multiplicative colored driven process cluster number decays multi exponentially however we have explored how the rate constant in the mono exponentially cluster number decaying case depends on strength of interference of the noises and their intensity we have also explored how the structure factor at long time depends on the strength of the cross correlation cc between the additive and the multiplicative noises

analysis of message passing decoding of finite length concatenated codes we analyze the performance of message passing decoding of finite length concatenated codes we first show that the message passing decoder is closely related to a dual optimization decoder the connections between these two decoders are further elucidated by proving that both of them attain the same objective function value of a generalized linear programming decoder in the limit as the signal to noise ratio snr goes to infinity consequently the framework of pseudo weight analysis which was originally proposed for analyzing the linear programming decoder can be extended to analyze the performance of the message passing decoder for finite length codes we then derive lower bounds to the pseudo weights of general concatenated codes by utilizing the special structure of their parity check matrices we finally present a method to increase the max fractional weight by adding redundant parity check constraints and thereby improving the decoding performance simulation studies are carried out to assess the performance of the proposed algorithms and substantiate the theoretic claims

mathematical programming decoding of binary linear codes theory and algorithms mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches challenging established but often heuristic algorithms based on iterative message passing concepts from mathematical programming used in the context of decoding include linear integer and nonlinear programming network flows notions of duality as well as matroid and polyhedral theory this paper reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary input memoryless symmetric channels

efficient admm decoding of ldpc codes using lookup tables linear programming decoding with the alternating direction method of multipliers admm is a promising decoding technique for low density parity check ldpc codes where the computational complexity of euclidean projections onto check polytopes becomes a prominent problem in this paper the problem is circumvented by building lookup tables luts and quantizing the inputs to approach approximate euclidean projections at low computational complexities to challenge the huge memory cost of luts we first propose two commutative compositions of euclidean projection and self map and show the existence of a small quantization range which does not alter the euclidean projection then we investigate the design and simplification of the luts by exploiting the commutative compositions and check node decomposition techniques an efficient algorithm for the lut based projection is demonstrated by using one simplification method simulation results show that for both the regular and irregular ldpc codes the admm decoding using lut based projection can substantially reduce the decoding time while maintaining the error rate performance at a comparatively large memory cost

stochastic resonance driven by two different kinds of colored noise in a bistable system the phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time ensuremath tau 1 and ensuremath tau 2 combining the functional analysis and unified colored noise approximation the two different kinds of colored noise in the nonlinear system can be simplified the signal to noise ratio is calculated when a weakly periodic signal is added to the system it is found that there appears a transition between one peak and two peaks in the curve of the signal to noise ratio when either the noise correlation time ensuremath tau 1 and ensuremath tau 2 or the coupling strength ensuremath lambda between additive and multiplicative noise is increased the transition between one and two peaks depending on ensuremath tau 1 and ensuremath lambda is more complex than that depending on ensuremath tau 2

stopping set elimination by parity check matrix extension via integer linear programming error rate floor phenomenon is known to be a serious impediment to the use of low density parity check ldpc codes for some practical applications that demand high data reliability in the case of binary erasure channels becs certain error prone patterns known as stopping sets are proven to cause this performance degradation a possible approach to diminish this drawback over becs is to eliminate stopping sets by parity check matrix extension given a parity check matrix h and a list mathcal l of its stopping sets we present an integer linear programming ilp formulation to find a parity check equation which eliminates the maximum number of stopping sets in mathcal l one of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as limiting the weight of the new parity check row making the new row redundant or linearly independent 4 cycle avoidance and taking into account the sizes of stopping sets armed with these adjustments the method can provide good performance improvements as evidenced by simulation results furthermore for a given varrho in mathbb n by extending the basic formulation we provide an ilp formulation for finding a set of size varrho of parity check equations which can best eliminate the stopping sets in mathcal l among all such sets

dynamics of the langevin model subjected to colored noise functional integral method abstract we have discussed the dynamics of langevin model subjected to colored noise by using the functional integral method fim combined with equations of motion for mean and variance of the state variable two sets of colored noise have been investigated a one additive and one multiplicative colored noise and b one additive and two multiplicative colored noise the case b is examined with relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate and fire model including stochastic excitatory and inhibitory synapses and a noisy input we have studied the stationary probability distribution and dynamical responses to time dependent pulse and sinusoidal inputs of the linear langevin model model calculations have shown that results of the fim are in good agreement with those of direct simulations dss a comparison is made among various approximate analytic solutions such as the universal colored noise approximation ucna it has been pointed out that dynamical responses to pulse and sinusoidal inputs calculated by the ucna are rather different from those of ds and the fim although they yield the same stationary distribution