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

a new construction of structured binary regular ldpc codes based on steiner systems with parameter t 2 this paper presents a novel method for constructing structured regular low density parity check ldpc codes based on a special type of combinatorial designs known as steiner systems this code design approach can be considered as a generalization of the well known method which uses the point block incidence matrix of a steiner 2 design for the code construction though the given method can be applied on any steiner system s t k v in this paper we focus only on steiner systems with t 3 furthermore we show that not only a steiner system x b itself but also its residual design with respect to an arbitrary point x x can be employed for code construction we also present a technique for constructing binary and non binary qc ldpc codes from steiner systems the tanner graph of the constructed codes is free of 4 cycles and hence the codes have girth at least six simulation results show that the so constructed codes perform well over the awgn channel with iterative message passing decoding

nonlinear variable structure systems in sliding mode the general case the problem of inducing local sliding regimes on smooth state space surfaces of nonlinear single input single output controlled systems is addressed in full generality the notion of relative degree in its more general form is used to establish the most salient features of nonlinear controlled systems undergoing sliding motions including their characteristic disturbance rejection properties it is shown that the simplest possible structure at infinity must be exhibited by nonlinear systems undergoing sliding motions on the zero level set of the output feedback function

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

construction of nonbinary cyclic quasi cyclic and regular ldpc codes a finite geometry approach this paper presents five methods for constructing nonbinary ldpc codes based on finite geometries these methods result in five classes of nonbinary ldpc codes one class of cyclic ldpc codes three classes of quasi cyclic ldpc codes and one class of structured regular ldpc codes experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the awgn channel and they achieve significant coding gains over reed solomon codes of the same lengths and rates with either algebraic hard decision decoding or kotter vardy algebraic soft decision decoding at the expense of a larger decoding computational 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

sliding mode control in discrete state and hybrid systems the sliding mode control approach is developed for a class of discrete state systems in a metric space a concept of trajectory continuity is introduced which allows the use of sliding mode design techniques similar to that in continuous state systems this approach is then applied to a class of hybrid systems which consists of a continuous plant an interface and a discrete state controller the introduction of the sliding mode concept leads to a design which is invariant to disturbances and also the solution of the stabilization problem

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

structured invariant spaces of vector valued functions sesquilinear forms and a generalization of the lohvidov laws abstract vector spaces of pairs of rational vector valued functions which are 1 invariant under the generalized backward shift and 2 endowed with a sesquilinear form which is subject to a structural identity are studied it is shown that any matrix can be viewed as the gram matrix of a suitably defined basis for such a space this identification is used to show that a rule due to iohvidov for evaluating the rank of certain subblocks of a toeplitz or hankel matrix is applicable to a wider class of matrices with appropriately defined displacement rank equal to two enroute a theory of reproducing kernel spaces is developed for nondegenerate spaces of the type mentioned above

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

non binary protograph based ldpc codes enumerators analysis and designs this paper provides a comprehensive analysis of nonbinary low density parity check ldpc codes built out of protographs we consider both random and constrained edge weight labeling and refer to the former as the unconstrained nonbinary protograph based ldpc codes u nbpb codes and to the latter as the constrained nonbinary protograph based ldpc codes c nbpb codes equipped with combinatorial definitions extended to the nonbinary domain ensemble enumerators of codewords trapping sets stopping sets and pseudocodewords are calculated the exact enumerators are presented in the finite length regime and the corresponding growth rates are calculated in the asymptotic regime an exit chart tool for computing the iterative decoding thresholds of protograph based ldpc codes is presented followed by several examples of finite length u nbpb and c nbpb codes with high performance throughout this paper we provide accompanying examples which demonstrate the advantage of nonbinary protograph based ldpc codes over their binary counterparts and over random constructions the results presented in this paper advance the analytical toolbox of nonbinary graph based codes

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

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

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

a revolving iterative algorithm for decoding algebraic cyclic and quasi cyclic ldpc codes cyclic and quasi cyclic algebraic ldpc codes constructed based on finite fields finite geometries and combinatorial designs can achieve excellent performance in terms of error rate error floor and rate of decoding convergence with iterative decoding however the relatively high density of the parity check matrix of an algebraic cyclic or quasi cyclic ldpc code makes the hardware implementation complexity of the decoder quite large which may be a critical issue in practical applications this paper presents an effective reduced complexity algorithm for decoding algebraic cyclic and quasi cyclic ldpc codes based on the block cyclic structure and cyclic grouping of the rows of their parity check matrices the decoding of a code is carried out based on a single small submatrix of the parity check matrix of the code in a revolving manner the proposed decoding algorithm significantly reduces the hardware implementation complexity and the size of memory required to store information

a geometric approach to pulse width modulated control in nonlinear dynamical systems the author demonstrates under the assumption of high frequency control switchings the existence of an ideal equivalence among sliding regimes of variable structure feedback control and pulse width modulated pwm control responses in nonlinear dynamical systems this equivalence constitutes the basis for a geometric approach approach to pwm control loops design an illustrate example of energy conversion in a lossless switched controlled bilinear network is presented

on dynamic feedback linearization abstract we address the problem of transforming a nonlinear multi input system into a linear controllable one via nonsingular dynamic feedback and extended state space diffeomorphism we show that in the single input case if a system is dynamic feedback linearizable it is also static feedback linearizable we then give sufficient conditions for a class of multi input systems to be dynamic feedback linearizable all three dimensional systems with two controls and a controllable linear approximation about the origin are shown to belong to such a class

input linearization of nonlinear systems via pulse width control in this note it is shown that a general nonlinear system can be transformed to an affine nonlinear system by virtue of the use of pulsewidth control therefore the combined system from the pulse width control input to the nonlinear system output behaves as an affine linear in control system by this way a cumbersome nonlinear system model can be transformed to a simpler linear in control form without increasing the system dimension this in turn enables simpler control design furthermore using this methodology some control design methods developed only for affine systems can be adapted to general nonlinear systems

bounds on the belief propagation threshold of non binary ldpc codes we consider low density parity check ldpc code ensembles over non binary galois fields when used for transmission over arbitrary discrete memoryless channels belief propagation decoding for these codes has been shown to achieve excellent results however computing the decoding threshold using density evolution is usually impractical since one needs to propagate multi dimensional probability distributions and monte carlo simulations are required instead by considering the evolution of the message bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm this bound applies for both regular and irregular non binary ldpc ensembles

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

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

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

4 cycle free ldpc codes based on difference sets difference sets are used to give a novel method for constructing structured regular quasi cyclic low density parity check codes let fq be a field with q elements and assume that d d1 dk is a v k 1 difference set for zv with d1 d2 dk depending on v q 1 v 2dk or v 2dk 1 three code construction methods are given that produce regular 4 cycle free codes that is codes with girth at least 6 simulation results show that the constructed codes perform well over the awgn channel with an iterative sum product message passing decoding algorithm

transactions papers constructions of nonbinary quasi cyclic ldpc codes a finite field approach this paper is concerned with construction of efficiently encodable nonbinary quasi cyclic ldpc codes based on finite fields four classes of nonbinary quasi cyclic ldpc codes are constructed experimental results show that codes constructed perform well with iterative decoding using a fast fourier transform based q ary sum product algorithm and they achieve significant coding gains over reed solomon codes of the same lengths and rates decoded with either algebraic hard decision berlekamp massey algorithm or algebraic soft decision kotter vardy algorithm

periodic sliding motions a general geometric characterization is given for the global existence of sliding regimes on compact manifolds in nonlinear variable structure feedback systems the characterization involves a set theoretic inclusion condition to be satisfied by the control dependent flow map acting on the compact region contained by the sliding manifold a sign condition is derived on the volume integral of the divergence of the generating controlled vector field the condition is a necessary but not sufficient condition for the existence of a sliding regime the manifold invariance conditions or ideal sliding conditions are characterized in terms of volume preserving evolution of the flow map associated with the sliding dynamics an application of the general results to periodic sliding motions in r sup 2 was illustrated using some simple examples

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

searching for binary and nonbinary block and convolutional ldpc codes a unified approach to search for and optimize codes determined by their sparse parity check matrices is presented replacing the nonzero elements of a binary parity check matrix the base or parent matrix either by circulants or by companion matrices of elements from a finite field gf 2 m we obtain quasi cyclic low density parity check ldpc block codes and binary images of nonbinary ldpc block codes respectively by substituting monomials of a formal variable d we obtain the polynomial description of an ldpc convolutional code a set of performance measures applicable to different classes of ldpc codes is considered and a greedy algorithm for code performance optimization is presented the heart of the new optimization algorithm is a fast procedure for searching for ldpc codes with large girth of their tanner graphs for a few classes of ldpc codes examples of codes combining good error correcting performance with compact representation are obtained in particular we present optimized convolutional ldpc codes and conclude that the ldpc block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered moreover a specific channel model can easily be embedded into the optimization loop thereby the code can be optimized for a specific channel the efficiency of such an optimization is demonstrated via an example of faster than nyquist ftn signaling using ldpc codes the ftn strategy combined with a rate r 1 2 ldpc code of length 64 800 optimized for effective data rate r 3 4 gains more than 0 5 db compared with the standard ldpc codes of the same rate and length the obtained gain corresponds to transmission at the capacity of the binary input additive white gaussian noise channel in most numerical examples we consider codes with bidiagonal structure of the parity check matrix this restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards

a general approach to input output pseudolinearization for nonlinear systems a nonlinear system is said to be input output pseudolinearized if its family of linearizations about constant operating points has input output behavior that is independent of the particular operating point the problem of constructing static state feedback laws that achieve input output pseudolinearization for a general class of nonlinear systems is considered a generalization of the well known structure algorithm to the case of parameterized families of linear systems plays an important role

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

a unified approach to the construction of binary and nonbinary quasi cyclic ldpc codes based on finite fields a unified approach for constructing binary and nonbinary quasi cyclic ldpc codes under a single framework is presented six classes of binary and nonbinary quasi cyclic ldpc codes are constructed based on primitive elements additive subgroups and cyclic subgroups of finite fields numerical results show that the codes constructed perform well over the awgn channel with iterative decoding