design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

an efficient algorithm to find all small size stopping sets of low density parity check matrices in this work we introduce an efficient algorithm to find all stopping sets of size less than some threshold of a fixed low density parity check ldpc matrix the solution is inspired by the algorithm proposed by rosnes and ytrehus in 2005 to find an exhaustive list of all small size turbo stopping sets in a turbo code the efficiency of the proposed algorithm is demonstrated by several numerical examples for instance we have applied the algorithm to the well known 3 5 regular 155 64 tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer also we have verified that the minimum stopping set size of the 4896 2474 ramanujan margulis code is indeed 24 and that the corresponding multiplicity is exactly 204 furthermore we have applied the algorithm to the ieee 802 16e ldpc codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes finally as an application we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity check matrix in order to remove all stopping sets in the corresponding tanner graph of size less than the minimum distance an extensive case study of the 155 64 tanner code illustrates the usefulness of the algorithm and we present a 110 times 155 redundant parity check matrix for this code with no stopping sets of size less than the minimum distance simulation results of iterative decoding on the binary erasure channel show performance improvements for low to medium erasure probabilities when this redundant parity check matrix is used for decoding

spectral collocation method for solving fredholm integral equations on the half line the main purpose of this paper is to present a spectral collocation method to approximate the solution of fredholm integral equations of the second kind on the half line in this approach interpolation operator based on scaled laguerre functions is proposed also convergence theorems are established under suitable assumptions finally two numerical examples are presented and commented

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

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

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

sumis near optimal soft in soft out mimo detection with low and fixed complexity the fundamental problem of interest here is soft input soft output multiple input multiple output mimo detection we propose a method referred to as subspace marginalization with interference suppression sumis that yields unprecedented performance at low and fixed deterministic complexity our method provides a well defined tradeoff between computational complexity and performance apart from an initial sorting step consisting of selecting channel matrix columns the algorithm involves no searching nor algorithmic branching hence the algorithm has a completely predictable run time and allows for a highly parallel implementation we numerically assess the performance of sumis in different practical settings full partial channel state information sequential iterative decoding and low high rate outer codes we also comment on how the sumis method performs in systems with a large number of transmit antennas

nearly optimal algorithms for canonical matrix forms a las vegas type probabilistic algorithm is presented for finding the frobenius canonical form of an n times n matrix t over any field kk the algorithm requires softo mm n mm n cdot log n o 1 operations in kk where o mm n operations in kk are sufficient to multiply two n times n matrices over kk this nearly matches the lower bound of omega mm n operations in kk for this problem and improves on the o n 4 operations in kk required by the previously best known algorithms a fast parallel implementation of the algorithm is also demonstrated for the frobenius form which is processor efficient on a pram as an application we give an algorithm to evaluate a polynomial g in kk x at t which requires only softo mm n operations in kk when deg g leq n 2 other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix the rational jordan form of a matrix for testing whether two matrices are similar and for matrix powering which are substantially faster than those previously known

constructing short length irregular ldpc codes with low error floor trapping sets tss are known to cause error floors in regular and irregular low density parity check ldpc codes by avoiding major error contributing tss during the code construction process codes with low error floors can effectively be built in it has been shown that tss labeled as w u are considered as being equivalent under the automorphism of the graph and are therefore contributing equally to the error floor however tss with the same label w u are not identical in general particularly for the case of irregular ldpc codes in this paper we introduce a parameter e that can identify the number of distinguishable cycles in the connected subgraph induced by an elementary trapping set further we propose a code construction algorithm namely the progressive edge growth approximate minimum cycle set extrinsic message degree peg acse method that aims to avoid small elementary trapping sets etss particularly detrimental etss we also develop theorems evaluating the minimum possible etss formed by peg construction algorithms in general we compare the characteristics of the codes built using the proposed method and those built using peg only or peg approximate minimum cycle extrinsic message degree peg ace methods results from simulations show that the codes constructed using the proposed peg acse method produce lower error rates particularly at the high signal to noise snr region compared with codes constructed using other peg based algorithms

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

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

minimum distance bounds by graph analysis the parity check matrix of a linear code is used to define a bipartite code constraint tanner graph in which bit nodes are connected to parity check nodes the connectivity properties of this graph are analyzed using both local connectivity and the eigenvalues of the associated adjacency matrix a simple lower bound on the minimum distance of the code is expressed in terms of the two largest eigenvalues for a more powerful bound local properties of the subgraph corresponding to a minimum weight word in the code are used to create an optimization problem whose solution is a lower bound on the code s minimum distance linear programming gives one bound the technique is illustrated by applying it to sparse block codes with parameters 7 3 4 and 42 23 6

an iterative algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation abstract the generalized coupled sylvester matrix equations ay zb cy zd e f with unknown matrices y z are encountered in many systems and control applications also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils in this work we construct an iterative algorithm to solve the generalized coupled sylvester matrix equations over reflexive matrices y z and when the matrix equations are consistent for any initial matrix pair y 0 z 0 a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors and the least frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair also we obtain the optimal approximation reflexive solution pair to a given matrix pair y z in the reflexive solution pair set of the generalized coupled sylvester matrix equations ay zb cy zd e f moreover several numerical examples are given to show the efficiency of the presented iterative algorithm

an efficient formula for linear recurrences the solutions to a scalar homogeneous constant coefficient linear recurrence are expressible in terms of the powers of a companion matrix we show how to compute these powers efficiently via polynomial multiplication the result is a simple expression for the solution which does not involve the characteristic roots and which is valid for any module over any commutative ring the formula yields the nth term of the solution to a kth order recurrence with o mu k cdot log n arithmetic operations where mu k is the total number of arithmetic operations required to multiply two polynomials of degree k 1 thus if the ring supports a fast fourier transform then o k cdot log k cdot log n operations are sufficient to compute the nth term

a generalized laguerre fourier hermite pseudospectral method for computing the dynamics of rotating bose einstein condensates a time splitting generalized laguerre fourier hermite pseudospectral method is proposed for computing the dynamics of rotating bose einstein condensates becs in two and three dimensions the new numerical method is based on the following i the use of a time splitting technique for decoupling the nonlinearity ii the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient and iii the construction of eigenfunctions for the linear operator by properly scaling the generalized laguerre fourier and hermite functions the new method enjoys the following properties i it is explicit time reversible and time transverse invariant ii it conserves the position density and is spectrally accurate in space and second order or fourth order accurate in time and iii it solves the problem in the original whole space instead of in a truncated artificial computational domain the method is also extend

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

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

fast algorithms for the sylvester equation ax xb t c for given matrices a fm m b fn n and c fm n over an arbitrary field f the matrix equation ax xbt c has a unique solution x fm n if and only if a and b have disjoint spectra we describe an algorithm that computes the solution x for m n n with o n logn arithmetic operations in f where 2 is such that m m matrices can be multiplied with o m arithmetic operations e g 2 376 it seems that before no better bound than o m3 n3 arithmetic operations was known the state of the art in numerical analysis is o n3 m3 flops but these algorithms due to bartels stewart and golub nash van loan involve schur decompositions i e they compute the eigenvalues of at least one of a and b and can hence not be transferred for general f

correctness of local probability propagation in graphical models with loops graphical models such as bayesian networks and markov networks represent joint distributions over a set of variables by means of a graph when the graph is singly connected local propagation rules of the sort proposed by pearl 1988 are guaranteed to converge to the correct posterior probabilities recently a number of researchers have empirically demonstrated good performance of these same local propagation schemes on graphs with loops but a theoretical understanding of this performance has yet to be achieved for graphical models with a single loop we derive an analytical relationship between the probabilities computed using local propagation and the correct marginals using this relationship we show a category of graphical models with loops for which local propagation gives rise to provably optimal maximum a posteriori assignments although the computed marginals will be incorrect we also show how nodes can use local information in the messages they receive in order to correct their computed marginals we discuss how these results can be extended to graphical models with multiple loops and show simulation results suggesting that some properties of propagation on single loop graphs may hold for a larger class of graphs specifically we discuss the implication of our results for understanding a class of recently proposed error correcting codes known as turbo codes

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

the use of generalized laguerre polynomials in spectral methods for nonlinear differential equations the expansion of products of generalized laguerre polynomials l n x in terms of a series of generalized laguerre polynomials is considered the expansion coefficients which are equal to triple product integrals of generalized laguerre polynomials are expressed in terms of a three index recurrence relation this is reduced to a one index relation which facilitates computation of the expansion coefficients the results are useful in the solution of nonlinear differential equations when it is desired to express products of generalized laguerre polynomials as a linear series of these functions as an application we use the results to compute a spectral solution of a nonlinear boundary value problem namely the blasius equation on a semi infinite interval by using a truncated series containing the first eight polynomials l1 2n x a solution is obtained within 4 accuracy

sor for ax xb c we consider a new approach to the block sor method applied to linear systems of equations which can be written as a matrix equation ax xb c such systems arise for example from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains on one hand this gives us an iterative method for the solution of such matrix equations e g lyapunov s matrix equation where b at and on the other hand the problem of choosing appropriate parameters for the block sor method can be written in a more compact form which may be helpful especially for non self adjoint problems i e if a and b are nonsymmetric using this technique we determine under more general assumptions than those of chin and manteuffel the optimal sor parameters for the model problem of a convection diffusion equation

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

laguerre collocation method for the flow and heat transfer due to a permeable stretching surface embedded in a porous medium with a second order slip and viscous dissipation a numerical method is given for studying the effect of viscous dissipation on the steady flow with heat transfer of newtonian fluid towards a permeable stretching surface embedded in a porous medium with a second order slip the governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations by using similarity variables the proposed method is based on replacement of the unknown function by truncated series of well known laguerre expansion of functions an approximate formula of the derivative is introduced special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula the introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with laguerre coefficients thus by solving this system of equations the laguerre coefficients are obtained graphically results are shown for non dimensional velocities and temperature the effects of the porous parameter the suction injection parameter eckert number first and second order velocity slip parameter and the prandtl number on the flow and temperature profiles are presented moreover the local skin friction and nusselt numbers are presented a comparison of numerical results is made with the earlier published results under limiting cases

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

application of fixed point method to obtain semi analytical solution to blasius flow and its variation the classical blasius flow and its variation sakiadis flow are investigated by the fixed point method fpm and the semi analytical solutions are obtained different from early perturbation series solutions the present semi analytical solutions are valid in the whole region without using matching procedure moreover these solutions can correctly provide the asymptotic property of the governing equations the comparison of the present results with numerical results is carried out which reveals that fpm is a convenient and effective tool for nonlinear problems

design of low rate irregular ldpc codes using trellis search a simple design method using trellis search is proposed for good low density parity check ldpc codes with relatively low code rates by applying a trellis search technique to the design of a pre assigned part of the parity check matrix that allows a simple encoding we improve the distribution of cycles formed by the entries contained in the parity check part of the parity check matrix in addition we extend the proposed algorithm to a class of structured ldpc codes which have been recently preferred in many practical applications simulation results show that the codes designed by the proposed method outperform those constructed by conventionally used greedy design algorithms

explicit solution of the operator equation a x x a b in this paper we find the explicit solution of the equationa x x a bfor linear bounded operators on hilbert spaces where x is the unknown operator this solution is expressed in terms of the moore penrose inverse of the operator a thus results of j h hodges some matrix equations over a finite field ann mat pura appl 44 1957 245 550 are extended to the infinite dimensional settings

design of efficiently encodable moderate length high rate irregular ldpc codes this paper presents a new class of irregular low density parity check ldpc codes of moderate length 10 sup 3 spl les n spl les 10 sup 4 and high rate r spl ges 3 4 codes in this class admit low complexity encoding and have lower error rate floors than other irregular ldpc code design approaches it is also shown that this class of ldpc codes is equivalent to a class of systematic serial turbo codes and is an extension of irregular repeat accumulate codes a code design algorithm based on the combination of density evolution and differential evolution optimization with a modified cost function is presented moderate length high rate codes with no error rate floors down to a bit error rate of 10 sup 9 are presented although our focus is on moderate length high rate codes the proposed coding scheme is applicable to irregular ldpc codes with other lengths and rates

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

low density parity check codes from transversal designs with improved stopping set distributions this paper examines the construction of low density parity check ldpc codes from transversal designs based on sets of mutually orthogonal latin squares mols by transferring the concept of configurations in combinatorial designs to the level of latin squares we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes based on large sets of simple structured mols we derive powerful constraints for the choice of suitable subsets leading to improved stopping set distributions for the corresponding codes we focus on ldpc codes with column weight 4 but the results are also applicable for the construction of codes with higher column weights finally we show that a subclass of the presented codes has quasi cyclic structure which allows low complexity encoding

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance