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

lagrange multipliers in elastic plastic torsion problem for nonlinear monotone operators abstract the existence of lagrange multipliers as a radon measure is ensured for an elastic plastic torsion problem associated to a nonlinear strictly monotone operator a regularization of this result namely the existence of l p lagrange multipliers is obtained under strong monotonicity assumption on the operator moreover the relationships between elastic plastic torsion problem and the obstacle problem are investigated finally an example of the so called von mises functions is provided namely of solutions of the elastic plastic torsion problem associated to nonlinear monotone operators which are not obtained by means of the obstacle problem in the case f constant

quasi cyclic ldpc codes an algebraic construction rank analysis and codes on latin squares quasi cyclic ldpc codes are the most promising class of structured ldpc codes due to their ease of implementation and excellent performance over noisy channels when decoded with message passing algorithms as extensive simulation studies have shown in this paper an approach for constructing quasi cyclic ldpc codes based on latin squares over finite fields is presented by analyzing the parity check matrices of these codes combinatorial expressions for their ranks and dimensions are derived experimental results show that with iterative decoding algorithms the constructed codes perform very well over the awgn and the binary erasure channels

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

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

quasivariational solutions for first order quasilinear equations with gradient constraint we prove the existence of solutions for a quasi variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself the only required assumption on the nonlinearity of this constraint is its continuity and positivity the method relies on an appropriate parabolic regularization and suitable a priori estimates we also obtain the existence of stationary solutions by studying the asymptotic behaviour in time in the variational case corresponding to a constraint independent of the solution we also give uniqueness results

area formula for centered hausdorff measures in metric spaces abstract motivated by an example in magnani in press we study inside a separable metric space x d the relations between centered and non centered m dimensional densities of a radon measure in x and their relations with spherical and centered spherical m dimensional hausdorff measures eventually we give an application to finite perimeter sets in carnot groups

duality theory and optimality conditions for generalized complementarity problems abstract in this paper generalized complementarity problems are expressed in terms of suitable optimization problems and some necessary optimality conditions are given the infinite dimensional lagrangean and duality theories play an important role in order to achieve the main results

two low complexity reliability based message passing algorithms for decoding non binary ldpc codes this paper presents two low complexity reliability based message passing algorithms for decoding ldpc codes over non binary finite fields these two decoding algorithms require only finite field and integer operations and they provide effective trade off between error performance and decoding complexity compared to the non binary sum product algorithm they are particularly effective for decoding ldpc codes constructed based on finite geometries and finite fields

the infinite dimensional lagrange multiplier rule for convex optimization problems abstract in this paper an infinite dimensional generalized lagrange multipliers rule for convex optimization problems is presented and necessary and sufficient optimality conditions are given in order to guarantee the strong duality furthermore an application is presented in particular the existence of lagrange multipliers associated to the bi obstacle problem is obtained

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

quasi cyclic ldpc codes an algebraic construction this paper presents two new large classes of qc ldpc codes one binary and one non binary codes in these two classes are constructed by array dispersions of row distance constrained matrices formed based on additive subgroups of finite fields experimental results show that codes constructed perform very well over the awgn channel with iterative decoding based on belief propagation codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry ldpc codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low complexity reliability based iterative decoding algorithms such as binary message passing decoding algorithms non binary codes decoded with a fast fourier transform based sum product algorithm achieve significantly large coding gains over reed solomon codes of the same lengths and rates decoded with either the hard decision berlekamp massey algorithm or the algebraic soft decision kotter vardy algorithm they have potential to replace reed solomon codes in some communication or storage systems where combinations of random and bursts of errors or erasures occur

hybrid linear programming based decoding algorithm for ldpc codes this paper presents a hybrid decoding algorithm for low density parity check ldpc codes based on the interior point method with barrier function for linear programming lp decoding introduced by wadayama first an efficient implementation of wadayama s algorithm is presented the main idea behind the modification is to approximate the barrier function for the fundamental polytope defining the code so that it contains only one linear constraint for each of the parity check constraints a two stage hybrid decoding which combines the interior point decoding and a low complexity decoding algorithm for ldpc codes is then proposed simulation results show that the approximations introduced in the proposed algorithms do not result in any performance degradation while considerably reducing the decoding complexity and latency

two reliability based iterative majority logic decoding algorithms for ldpc codes this paper presents two novel reliability based iterative majority logic decoding algorithms for ldpc codes both algorithms are binary message passing algorithms and require only logical operations and integer additions consequently they can be implemented with simple combinational logic circuits they either outperform or perform just as well as the existing weighted bit flipping or other reliability based iterative decoding algorithms for ldpc codes in error performance with a faster rate of decoding convergence and less decoding complexity compared to the sum product algorithm for ldpc codes they offer effective trade offs between performance and decoding complexity

a measure type lagrange multiplier for the elastic plastic torsion abstract the existence of lagrange multiplier as a radon measure is ensured for an elastic plastic torsion problem associated to a linear operator the result is obtained by means of strong duality theory

quasi cyclic ldpc codes on cyclic subgroups of finite fields a new class of quasi cyclic ldpc codes whose parity check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented this class of codes contains several known classes of algebraic quasi cyclic ldpc codes as subclasses experimental results show that the codes constructed perform very well over the awgn channel when decoded with iterative decoding based on belief propagation this class of new qc ldpc codes contains a subclass of codes which have large minimum distances combinatorial expressions for the ranks of the parity check matrices of a subclass of codes constructed based on fields of characteristic two are given

lagrange multipliers and non constant gradient constrained problem abstract the aim of the paper is to study a gradient constrained problem associated with a linear operator two types of problems are investigated the first one is the equivalence between a non constant gradient constrained problem and a suitable obstacle problem where the obstacle solves a hamilton jacobi equation in the viscosity sense the equivalence result is obtained under a condition on the gradient constraint the second problem is the existence of lagrange multipliers we prove that the non constant gradient constrained problem admits a lagrange multiplier which is a radon measure if the free term of the equation f l p p 1 if f is a positive constant we regularize the result namely we prove that the lagrange multipliers belong to l 2

variational and quasivariational inequalities with first order constraints abstract we study the existence of solutions of stationary variational and quasivariational inequalities with curl constraint neumann type boundary condition and a p curl type operator these problems are studied in bounded not necessarily simply connected domains with a special geometry and the functional framework is the space of divergence free functions with curl in l p and null tangential or normal traces the analogous variational or quasivariational inequalities with gradient constraint are also studied considering neumann or dirichlet non homogeneous boundary conditions the existence of a generalized solution for a lagrange multiplier problem with homogeneous dirichlet boundary condition and the equivalence with the variational inequality is proved in the linear case for an arbitrary gradient constraint

iterative algorithms for decoding a class of two step majority logic decodable cyclic codes codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority logic decoding in one or multiple steps in 2001 kou lin and fossorier showed that the one step majority logic decodable finite geometry codes form a class of cyclic ldpc codes whose tanner graphs are free of cycles of length 4 these cyclic finite geometry ldpc codes perform very well over the awgn channel using iterative decoding based on belief propagation idbp and have very low error floors however the standard idbp is not effective for decoding other cyclic finite geometry codes because their tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance this paper investigates iterative decoding of two step majority logic decodable finite geometry codes three effective algorithms for decoding these codes are proposed these algorithms are devised based on the orthogonal structure of the parity check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4 these decoding algorithms provide significant coding gains over the standard idbp using either the sum product or the min sum algorithms

a new necessary and sufficient condition for the strong duality and the infinite dimensional lagrange multiplier rule abstract throughout this paper the authors introduce a new condition defined by assumption s which establishes a necessary and sufficient condition for the validity of the strong duality between a convex optimization problem and its lagrange dual this work will be focused on the context of emptiness of the interior of the ordering cone and convexity of the equality constraints moreover this new condition will be necessary and sufficient for the infinite dimensional lagrange multiplier rule this new principle will find application to the elastic plastic torsion problem to the continuum model of transportation and to a problem with quadratic equality constraint with connected to evolutionary illumination and visibility problems

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

comparisons between reliability based iterative min sum and majority logic decoding algorithms for ldpc codes a modified reliability based iterative majority logic decoding mrbi mlgd algorithm for two classes of structured ldpc codes is presented based on a recent work by huang et al compared with the original one the modified algorithm has better performance with slightly increased complexity then a reliability based iterative min sum decoding rbi msd algorithm is presented for the presented rbi msd algorithm reliability based integer messages are processed and exchanged between variable nodes and check nodes the main computations include only binary logical operations and integer additions different from the conventional min sum algorithm the variable nodes pass full messages rather than extrinsic messages to check nodes this can reduce the memory loads and the computational complexity but with a little or negligible performance degradation simulation results show that compared with the m rbi mlgd algorithms the presented rbi msd algorithm achieves better error performance faster decoding convergence rate and fewer quantization bits with moderate increased computational complexity furthermore the rbi msd algorithm is also applicable to decoding random ldpc codes a distinct difference from the m rbi mlgd algorithms finally we point out that the scaling factors employed in the mrbi mlgd algorithm and the rbi msd algorithm can be optimized using discretized density evolution

lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem abstract this paper is focused on the study of a dynamic competitive equilibrium by using lagrangean multipliers this mathematical formulation allows us the improve the walrasian model by considering the common possibility of an uncharged delayed payment in a given time for example by using a credit card firstly the economic equilibrium problem is reformulated as an evolutionary variational problem then the lagrangean theory in infinite dimensional spaces is applied thanks to the application of this theory we obtain the existence of lagrangean multipliers which allows us to give a computational procedure for the equilibrium solutions

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

infinite dimensional duality and applications the usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear in this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi relative interior which in all the concrete problems considered is nonempty we apply this theory to solve the until now unsolved problem of finding in the infinite dimensional case the lagrange multipliers associated to optimization problems or to variational inequalities as an example we find the lagrange multiplier associated to a general elastic plastic torsion problem

on a measure theoretic area formula we review some classical differentiation theorems for measures showing how they can be turned into an integral representation of a borel measure with respect to a fixed caratheodory measure we focus our attention on the case when this measure is the spherical hausdorff measure giving a metric measure area formula our aim is to use certain covering derivatives as generalized densities some consequences for the sub riemannian heisenberg group are also pointed out