discrete optimal control with eigenvalue assigned inside a circular region a discrete time optimal control that guarantees that all the closed loop poles will lie inside a circle centered at beta 0 with radius is formulated it is shown how the exposed problem can be reduced to a standard discrete time linear quadratic regulator problem furthermore a quantitative measure of the robustness of linear quadratic state feedback design in the presence of a perturbation is obtained bounds are derived for allowable nonlinear perturbations such that the resultant closed loop is stable

compact riemannian manifolds with positive curvature operators m is said to have positive curvature operators if the eigenvalues of z are positive at each point p m meyer used the theory of harmonic forms to prove that a compact oriented n dimensional riemannian manifold with positive curvature operators must have the real homology of an n dimensional sphere gm proposition 2 9 using the theory of minimal two spheres we will outline a proof of the following stronger result

deriving good ldpc convolutional codes from ldpc block codes low density parity check ldpc convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity in this paper we discuss several graph cover based methods for deriving families of time invariant and time varying ldpc convolutional codes from ldpc block codes and show how earlier proposed ldpc convolutional code constructions can be presented within this framework some of the constructed convolutional codes significantly outperform the underlying ldpc block codes we investigate some possible reasons for this convolutional gain and we also discuss the mostly moderate decoder cost increase that is incurred by going from ldpc block to ldpc convolutional codes

non fragile pd state h control for a class of uncertain descriptor systems abstract this paper is concerned with the problem of non fragile proportional plus derivative pd state h control for a class of uncertain descriptor systems the parameter uncertainties are assumed to be time varying norm bounded appearing not only in the state matrix but also in the derivative matrix a new version of bounded real lemma for descriptor systems is proposed which guarantees a descriptor system to be normal and stable ns with disturbance rejection based on this we address the problem of designing non fragile pd state h controllers with additive and multiplicative controller uncertainties respectively sufficient conditions are presented to synthesize two classes of such non fragile controllers in terms of lmis finally a numerical example is provided to illustrate the effectiveness of the proposed design method

pole assignment robustness in a specified disk abstract a sufficient condition is proposed to guarantee robust pole location within a specified circular region for a linear uncertain system the explicit bounds on linear time invariant perturbations with highly structured information are obtained under these allowable highly structured perturbations both stability robustness and certain performance robustness will thus be ensured the merit of the proposed new approach is demonstrated by a given example where the results achieved are much better than the ones reported recently

robust stability conditions for large scale interconnected systems with structured and unstructured uncertainties the paper is mainly concerned with the problem of decentralized robust stability of large scale interconnected systems with structured and unstructured uncertainties a simple method is presented whereby some sufficient conditions are derived so that asymptotic stability of large scale interconnected systems can be guaranteed in the presence of uncertain perturbations the method is also extended to large scale discrete time systems and the corresponding robust stability conditions are established for uncertain large scale discrete time systems finally two numerical examples are given to demonstrate that our robust stability conditions are less conservative than those reported in the control literature

robust non fragile control for uncertain discrete delay large scale systems with a class of controller gain variations this paper considers the problems of robust non fragile control for uncertain discrete delay large scale systems under state feedback gain variations two classes of controller gain variations are considered based on the lyapunov method the state feedback control design for robust stability is given in terms of solutions to a linear matrix inequality lmi the solutions of the lmi can be easily obtained using efficient convex optimization techniques a numerical example is included to illustrate the design procedures

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

decentralized stabilization of large scale systems with multiple delays in the interconnections in this paper the problem of robust stabilization of a class of time varying large scale systems subject to external disturbances uncertain parameters and multiple time varying delays in the interconnections and non linear inputs is considered linear memory less controllers are developed to guarantee that all solutions of the class of systems considered are uniformly ultimately bounded also non linear memoryless controllers are synthesized to guarantee that the solutions are uniformly ultimately bounded with respect to a bound with any prespecified convergence rate towards it finally an example is included to illustrate the results developed in this paper

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

distance bounds for periodically time varying and tail biting ldpc convolutional codes existence type lower bounds on the free distance of periodically time varying ldpc convolutional codes and on the minimum distance of tail biting ldpc convolutional codes are derived it is demonstrated that the bound on free distance of periodically time varying ldpc convolutional codes approaches the bound on free distance of general nonperiodic time varying ldpc convolutional codes as the period increases the proof of the bound is based on lower bounding the minimum distance of corresponding tail biting ldpc convolutional codes which is of interest in its own right

decentralized stabilization of large scale interconnected systems with delays in this note we study the decentralized stabilization problem of linear time invariant large scale interconnected systems with delays without any assumption of system structure our main result here is the sufficient condition for the decentralized stabilization one application of the procedures to the decentralized observers is also presented

performance comparison of ldpc block and spatially coupled codes over gf q in this paper we compare the finite length performance of protograph based spatially coupled low density paritycheck sc ldpc codes and ldpc block codes ldpc bcs over gf q to reduce computational complexity and latency a sliding window decoder with a stopping rule based on a soft belief propagation bp estimate is used for the q ary sc ldpc codes two regimes are considered one when the constraint length of q ary sc ldpc codes is equal to the block length of q ary ldpc bcs and the other when the two decoding latencies are equal simulation results confirm that in both regimes 3 6 3 9 and 3 12 regular non binary sc ldpc codes can significantly outperform both binary and non binary ldpc bcs and binary sc ldpc codes finally we present a computational complexity comparison of q ary sc ldpc codes and q ary ldpc bcs under equal decoding latency and equal decoding performance assumptions

real stability radii of linear time invariant time delay systems in this paper the real stability radii for perturbed linear differential equations of the retarded and neutral type are investigated we present the corresponding readily computable formulae with respect to an arbitrary stability region in the complex plane the results are further extended to the case of linear discrete time delay systems

non fragile asynchronous h control for uncertain stochastic memory systems with bernoulli distribution abstract this study constructs and investigates the non fragile asynchronous h control for uncertain stochastic memory systems with bernoulli distribution the system not only contains the randomly occurring uncertainties of all parameters rouaps and stochastic disturbances in the system model but also includes randomly gain perturbations in the controller by introducing the stochastic variables a new model structure is built obeying bernoulli distribution to describe the system and the controller moreover a modified lyapunov krasovskii function lkf is constructed combining ito s differential formula and free weighting matrix method less conservative results are presented in terms of the linear matrix inequality lmi furthermore an observer based non fragile asynchronous h controller is designed without any limits on the system parameters finally three numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed results

non fragile robust h control for uncertain spacecraft rendezvous system with pole and input constraints this article investigates the robust h control problem for a class of spacecraft rendezvous systems with parameter uncertainties and subject to input constraint pole constraint and external and controller perturbations based on the lyapunov theory a sufficient condition for the existence of the non fragile robust state feedback controller is given in terms of linear matrix inequalities lmis then proper non fragile controller design can be cast as a convex optimisation problem subject to lmi constraints with the obtained controller the spacecraft rendezvous mission can be accomplished an illustrative example is provided to show the effectiveness of the proposed control design method

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

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

pseudocodeword performance analysis for ldpc convolutional codes message passing iterative decoders for low density parity check ldpc block codes are known to be subject to decoding failures due to so called pseudocodewords these failures can cause the large signal to noise ratio snr performance of message passing iterative decoding to be worse than that predicted by the maximum likelihood ml decoding union bound in this paper we address the pseudocodeword problem from the convolutional code perspective in particular we compare the performance of ldpc convolutional codes with that of their wrapped quasi cyclic block versions and we show that the minimum pseudoweight of an ldpc convolutional code is at least as large as the minimum pseudoweight of an underlying quasi cyclic code this result which parallels a well known relationship between the minimum hamming weight of convolutional codes and the minimum hamming weight of their quasi cyclic counterparts is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code s pseudocodeword this difference in the weight spectra leads to improved performance at low to moderate snrs for the convolutional code a conclusion supported by simulation results

robustness of disk stability for perturbed large scale systems in this paper a sufficient condition of robust disk stability for perturbed large scale systems is presented if all the eigenvalues of each nominal subsystem are assigned within the disk d r then the proposed sufficient condition guarantees that all the eigenvalues of the overall uncertain large scale system remain inside the same disk the proposed technique for robust pole assignment in a specified circular region is available both for continuous and discrete time systems the application of the proposed method is illustrated by two examples

brief paper robust finite word length controller design a novel finite word length fwl controller design is proposed in the framework of a mixed m theory a robust fwl controller performance measure is first developed which takes into account the standard robust control requirements as well as the fwl implementation considerations and the corresponding controller design problem is naturally reformulated as a mixed m problem which can be treated effectively with the results of the mixed m theory

robust non fragile lq controllers the static state feedback case this paper describes the synthesis of non fragile or resilient regulators for linear systems a general framework for fragility is described using state space methodologies and the lq h2 static state feedback problem is examined in detail we discuss the multiplicative structured uncertainties case and propose remedies of the fragility problem using an optimization programming framework via matrix inequalities a special case that leads to a convex optimization framework via linear matrix inequalities lmis will be considered the benchmark problem is taken as an example to show how special controller gain variations can affect the performance of the closed loop system

d stability of continuous time delay systems subjected to a class of highly structured perturbations the d stability testing problem for continuous time delay systems subjected to a class of highly structured parametric perturbations is addressed in this note by means of the lyapunov stability theorem razumikhin type theorem gersgorin theorem concept of spectral radius and norm and matrix measure techniques the author has developed several new sufficient conditions for guaranteeing that all characteristic roots of the above systems are located inside a specified disk d spl alpha r with center at spl alpha j0 and radius r in the left half complex plane for the present results it is not necessary to solve any lyapunov equation which may be unsolvable though the lyapunov stability theorem is utilized

design of spatially coupled ldpc codes over gf q for windowed decoding in this paper we study spatially coupled low density parity check sc ldpc codes over finite fields gf q q geq 2 and develop design rules for q ary sc ldpc code ensembles based on their iterative belief propagation decoding thresholds with particular emphasis on low latency windowed decoding wd we consider transmission over both the binary erasure channel bec and the binary input additive white gaussian noise channel biawgnc and present results for a variety of j k regular sc ldpc code ensembles constructed over gf q using protographs thresholds are calculated using the protograph versions of q ary density evolution for the bec and the q ary extrinsic information transfer analysis for the biawgnc we show that the wd of q ary sc ldpc codes provides significant threshold gains compared with corresponding uncoupled q ary ldpc block code ldpc bc ensembles when the window size w is large enough and that these gains increase as the finite field size q 2 m increases moreover we demonstrate that the new design rules provide wd thresholds that are close to capacity even when both m and w are relatively small thereby reducing decoding complexity and latency the analysis further shows that compared with standard flooding schedule decoding the wd of q ary sc ldpc code ensembles results in significant reductions in both the decoding complexity and the decoding latency and that these reductions increase as m increases for the applications with a near threshold performance requirement and a constraint on decoding latency we show that using q ary sc ldpc code ensembles with moderate q 2 instead of their binary counterparts results in reduced decoding complexity

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

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

distance bounds for an ensemble of ldpc convolutional codes an ensemble of j k regular low density parity check ldpc convolutional codes is introduced and existence type lower bounds on the minimum distance dl of code segments of finite length l and on the free distance dfree are derived for sufficiently large constraint lengths v the distances are shown to grow linearly with v and the ratio dl v approaches the ratio dfee v for large l moreover the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for gallager s ensemble of j k regular ldpc block codes

an analysis of the block error probability performance of iterative decoding asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes n and the number of iterations i go to infinity three classes of codes are considered these are gallager s regular low density parity check ldpc codes tanner s generalized ldpc gldpc codes and the turbo codes due to berrou et al it is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability p sub b but also the block frame error probability p sub b goes to zero as n and i go to infinity

minimum distance and trapping set analysis of protograph based ldpc convolutional codes low density parity check ldpc convolutional codes have been shown to be capable of achieving capacity approaching performance with iterative message passing decoding in the first part of this paper using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio we show that several ensembles of regular and irregular ldpc convolutional codes derived from protograph based ldpc block codes have the property that the free distance grows linearly with respect to the constraint length i e the ensembles are asymptotically good in particular we show that the free distance to constraint length ratio of the ldpc convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding ldpc block code ensembles a large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal to noise ratios under maximum likelihood decoding when suboptimal decoding methods are employed there are many factors that affect the performance of a code recently it has been shown that so called trapping sets are a significant factor affecting decoding failures of ldpc codes over the additive white gaussian noise channel with iterative message passing decoding in the second part of this paper we study the trapping sets of the asymptotically good protograph based ldpc convolutional codes considered earlier by extending the theory presented in part one and using similar bounding techniques we show that the size of the smallest non empty trapping set grows linearly with the constraint length for these ensembles

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

stabilization of continuous and discrete linear systems subjected to control structure constraints in this paper we present a simple procedure for obtaining stabilization of a class of continuous and discrete linear systems subjected to control structure constraints giving more attention to the so called decentralized control contrary to the continuous case where much work has been done the solution presented here for the discrete case is one of the first to be presented some examples are treated and for the continuous case some comparisons are made with other methods found in the literature

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

time invariant quasi cyclic spatially coupled ldpc codes based on packings this paper presents two packings derived from balanced incomplete block designs to construct quasi cyclic spatially coupled ldpc convolutional codes sc ldpc ccs the construction gives time invariant codes since the sub blocks corresponding to each time instant of the parity check matrix are identical moreover it provides flexible design rates and constraint lengths simulation results show that the proposed packing based sc ldpc ccs outperform the existing time invariant codes and perform closely to the time varying protograph based codes

quasi cyclic ldpc codes for fast encoding in this correspondence we present a special class of quasi cyclic low density parity check qc ldpc codes called block type ldpc b ldpc codes which have an efficient encoding algorithm due to the simple structure of their parity check matrices since the parity check matrix of a qc ldpc code consists of circulant permutation matrices or the zero matrix the required memory for storing it can be significantly reduced as compared with randomly constructed ldpc codes we show that the girth of a qc ldpc code is upper bounded by a certain number which is determined by the positions of circulant permutation matrices the b ldpc codes are constructed as irregular qc ldpc codes with parity check matrices of an almost lower triangular form so that they have an efficient encoding algorithm good noise threshold and low error floor their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

pole assignment in a specified disk the problem of assigning all poles of a closed loop system in a specified disk by state feedback is considered for both continuous and discrete systems a state feedback control law is determined by using a discrete riccati equation this kind of pole assignment problem is named d pole assignment and its relation to the optimal control problem and its robustness properties are discussed the gain and phase margins for all closed loop poles to stay inside the specified disk d are determined for the proposed control

spatially coupled ensembles universally achieve capacity under belief propagation we investigate spatially coupled code ensembles for transmission over the binary erasure channel it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble this explains why convolutional ldpc ensembles originally introduced by felstrom and zigangirov perform so well over this channel we show that the equivalent result holds true for transmission over general binary input memoryless output symmetric channels more precisely given a desired error probability and a gap to capacity we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding in fact most codes in this ensemble have this property the quantifier universal refers to the single ensemble code that is good for all channels but we assume that the channel is known at the receiver the key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble we conclude by discussing some interesting open problems

spatially coupled ldpc codes constructed from protographs in this paper we construct protograph based spatially coupled low density parity check ldpc codes by coupling together a series of l disjoint or uncoupled ldpc code tanner graphs into a single coupled chain by varying l we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary l we demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design capacity approaching iterative belief propagation bp decoding thresholds and linear growth of minimum distance with block length in particular we show that for sufficiently large l the bp thresholds on both the binary erasure channel and the binary input additive white gaussian noise channel saturate to a particular value significantly better than the bp decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled ldpc code when all variable nodes in the coupled chain have degree greater than two asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good ldpc codes with fast convergence rates and bp thresholds close to the shannon limit further the gap to capacity decreases as the density of the graph increases opening up a new way to construct capacity achieving codes on memoryless binary input symmetric output channels with low complexity bp decoding

multiple access demodulation in the lifted signal graph with spatial coupling demodulation in a random multiple access channel is considered where the signals are chosen uniformly randomly with unit energy it is shown that by lifting replicating the graph of this system and randomizing the graph connections a simple iterative cancellation demodulator achieves the same performance as an optimal symbol by symbol detector of the original system the iterative detector has a complexity that is linear in the number of users while the direct optimal approach is known to be np hard however the maximal system load of this lifted graph is limited to alpha even for large signal to noise ratios snrs the system is interference limited spatial coupling between subsequent lifted graphs is introduced and anchoring the initial graphs the achievable system load alpha can go to infinity as the snr goes to infinity our results apply to several well documented system proposals such as interleave division multiple access partitioned spreading and certain forms of multiple input multiple output communications

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

stability of harmonic maps of kahler manifolds soit m n une application harmonique stable d une surface de riemann fermee dans un espace symetrique hermitien simple compact alors est holomorphe

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