interference alignment with finite extensions in partially connected networks in this paper we investigate the interference alignment ia for a class of partially connected multi input multi output heterogeneous networks which consist of l fully connected pico cells j partially connected pico cells and one macro cell in particular we investigate the feasibility of ia conditions with no channel extensions t independent channel extensions and t dependent channel extensions respectively for the single beam case we present three theorems to characterize some necessary conditions which l j t and the channel diversity order overline l must jointly satisfy when ia conditions are feasible these necessary conditions also implicitly impose the constraints on the upper bounds of achievable degrees of freedom with different channel extensions

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

correction of mismatched l values in bicm receivers in this work we analyze the problem of correction of the reliability metrics l values in bit interleaved coded modulation bicm receivers first we propose a method for finding the linear correction factors that minimize the probability of error of a maximum likelihood decoder that uses the corrected l values to this end we use the efficient approximation of the pairwise error probability in the domain of the cumulant generating functions cgf of the l values and conclude that the optimal correction factors are equal to the twice of value of the saddlepoint of the cgf next a simple extension of the proposed method to the non linear correction is presented we provide numerical examples demonstrating improvements attainable with the proposed method comparing it to the competitive solutions

dynamics and performance analysis of analog iterative decoding for low density parity check ldpc codes conventional iterative decoding with flooding or parallel schedule can be formulated as a fixed point problem solved iteratively by a successive substitution ss method in this paper we investigate the dynamics of a continuous time asynchronous analog implementation of iterative decoding and show that it can be approximated as the application of the well known successive relaxation sr method for solving the fixed point problem we observe that sr with the optimal relaxation factor can considerably improve the error rate performance of iterative decoding for short low density parity check ldpc codes compared with ss our simulation results for the application of sr to belief propagation sum product and min sum algorithms demonstrate improvements of up to about 0 7 db over the standard ss for randomly constructed ldpc codes the improvement in performance increases with the maximum number of iterations and by accordingly reducing the relaxation factor the asymptotic result corresponding to an infinite maximum number of iterations and infinitesimal relaxation factor represents the steady state performance of analog iterative decoding this means that under ideal circumstances continuous time asynchronous analog decoders can outperform their discrete time synchronous digital counterparts by a large margin our results also indicate that with the assumption of a truncated gaussian distribution for the random delays among computational modules the error rate performance of the analog decoder particularly in steady state is rather independent of the variance of the distribution the proposed simple model for analog decoding and the associated performance curves can be used as an ideal analog decoder benchmark for performance evaluation of analog decoding circuits

iterative decoding using eigenmessages the eigenmessage approach to iterative decoding introduces a degree of nonlocality into a belief propagation decoder by representing an entire set of messages around a cycle of the tanner graph as a linear operator the eigenvector for the operator represents a fixed point of the belief propagation algorithm around a cycle with incident messages fixed a multiple eigenmessage approach is also presented in which messages around several cycles are simultaneously expressed the eigenmessage approach may be applied to any graph with cycles but we demonstrate its use on ldpc decoding in this setting computational results compare eigenmessage methods with conventional belief propagation decoding showing using simulation and exit charts that the eigenmessage approaches slightly reduce the number of decoder iterations compared to belief propagation decoding while preserving the probability of error of conventional decoding

minimization of csi feedback dimension for interference alignment in mimo interference multicast networks it is well known that interference alignment ia can achieve substantial theoretical gains in multiple input multiple output mimo networks however the conventional works usually assume the full channel state information csi is available at the transmitter side which would create a overwhelming csi feedback overhead for practical frequency division duplexing fdd wireless systems to implement ia in practice it is highly desirable to reduce the amount of required csi feedback overhead in this paper we consider ia in mimo interference multicast networks under partial csi feedback and we attempt to minimize the csi feedback cost subject to ia feasibility constraints with a given degree of freedom dof requirements first we propose a csi feedback profile to embrace two important csi feedback reduction strategies and we use the metric of csi feedback dimension to quantify the associated csi feedback cost we then formulate the ia conditions under partial csi feedback in mimo interference multicast networks and we derive new ia feasibility conditions under the proposed partial csi feedback framework based on these results we consider the csi feedback dimension minimization subject to the ia feasibility constraints with a given dof requirements in mimo interference multicast networks which is formulated as a combinatorial optimization problem based on the specific problem structure we derive an asymptotically optimal solution for a category of network topologies and we further obtain closed form tradeoff results between dofs and the csi feedback cost for mimo interference multicast networks

self compensation technique for simplified belief propagation algorithm the min sum algorithm is the most common method to simplify the belief propagation algorithm for decoding low density parity check ldpc codes however there exists a performance gap between the min sum and belief propagation algorithms due to nonlinear approximation in this paper a self compensation technique using dynamic normalization is thus proposed to improve the approximation accuracy the proposed scheme scales the min sum algorithm by a dynamic factor that can be derived theoretically from order statistics moreover applying the proposed technique to several ldpc codes for dvb s2 system the average signal to noise ratio degradation which results from approximation inaccuracy and quantization error is reduced to 0 2 db not only does it enhance the error correcting capability of the min sum algorithm but the proposed self compensation technique also preserves a modest hardware cost after realized with 0 13 mum standard cell library the dynamic normalization requires about 100 additional gates for each check node unit in the min sum algorithm

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

reversible low density parity check codes low density parity check ldpc codes may be decoded using a circuit implementation of the sum product algorithm which maps the factor graph of the code by reusing the decoder for encoding both tasks can be performed using the same circuit thus reducing area and verification requirements motivated by this iterative encoding techniques based upon the graphical representation of the code are proposed code design constraints which ensure encoder convergence are presented and then used to design iteratively encodable codes while also preventing 4 cycle creation we show how the jacobi method for iterative matrix inversion can be applied to finite field matrices viewed as message passing and employed as the core of an iterative encoder we present an algebraic construction of 4 cycle free iteratively encodable codes using circulant matrices analysis of these codes identifies a weakness in their structure due to a repetitive pattern in the factor graph the graph supports pseudo codewords of low pseudo weight in order to remove the repetitive pattern in the graph we propose a recursive technique for generating iteratively encodable codes the new codes offer flexibility in the choice of code length and rate and performance that compares well to randomly generated quasi cyclic and extended euclidean geometry codes

generation of tetrahedral mesh of variable element size by sphere packing over an unbounded 3d domain this paper describes an algorithm for the generation of tetrahedral mesh of specified element size over an unbounded three dimensional domain starting from an arbitrary point in space defined as the origin and guided by the concept of advancing front spheres of size compatible with the specified element size are packed tightly together one by one to form a cluster of spheres of different sizes the compactness of the cluster of spheres is achieved by packing spheres at a site closest to the origin in a densest manner with tangent but no overlapping with as many other spheres as possible in view of these criteria a rotational mechanism between spheres is innovated which allows the newly inserted sphere to follow the path by rotation between existing spheres until the lowest point is reached the centres of the packing spheres provide ideal locations for delaunay point insertion to form a triangulation of tetrahedral elements of size compatible with the specified value spheres of random size distribution or of size specified by a node spacing function are packed by the proposed algorithm in all test examples high quality tetrahedral elements of size consistent with the specified node spacing are generated the process is fast and robust and the time complexity for mesh generation is expected to be almost linear however in the present implementation a quasi linear time relationship is observed as a search for the nearest node on the front is proposed in the sphere packing process the memory requirement has been kept to a minimum as no additional data structure other than the adjacency relationship of the tetrahedral elements as required by the delaunay triangulation is stored the structure of the advancing frontal surface needs not be explicitly constructed nor updated as it is simply not required in the sphere packing process

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 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

optimization of scaling soft information in iterative decoding via density evolution methods density evolution has recently been used to analyze iterative decoding and explain many characteristics of iterative decoding including convergence of performance and preferred structures for the constituent codes the scaling of extrinsic information messages has been heuristically used to enhance the performance in the iterative decoding literature particularly based on the min sum message passing algorithm in this paper it is demonstrated that density evolution can be used to obtain the optimal scaling factor and also estimate the maximum achievable scaling gain for low density parity check ldpc codes and serially concatenated convolutional codes sccc with two state constituent codes the analytic density evolution technique is used while the signal to noise ratio snr evolution technique and the exit chart technique is used for sccc with more than 2 state constituent codes simulation results show that the scaling gain predicted by density evolution or snr evolution matches well with the scaling gain observed by simulation

finite alphabet iterative decoders part i decoding beyond belief propagation on the binary symmetric channel we introduce a new paradigm for finite precision iterative decoding on low density parity check codes over the binary symmetric channel the messages take values from a finite alphabet and unlike traditional quantized decoders which are quantized versions of the belief propagation bp decoder the proposed finite alphabet iterative decoders faids do not propagate quantized probabilities or log likelihoods and the variable node update functions do not mimic the bp decoder rather the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code thereby rendering these decoders capable of outperforming the bp in the error floor region on certain column weight three codes of practical interest we show that there exist faids that surpass the floating point bp decoder in the error floor region while requiring only three bits of precision for the representation of the messages hence faids are able to achieve a superior performance at much lower complexity we also provide a methodology for the selection of faids that is not code specific but gives a set of candidate faids containing potentially good decoders in the error floor region for any column weight three code we validate the code generality of our methodology by providing particularly good three bit precision faids for a variety of codes with different rates and lengths

iterative decoding of one step majority logic deductible codes based on belief propagation previously the belief propagation bp algorithm has received a lot of attention in the coding community mostly due to its near optimum decoding for low density parity check ldpc codes and its connection to turbo decoding in this paper we investigate the performance achieved by the bp algorithm for decoding one step majority logic decodable osmld codes the bp algorithm is expressed in terms of likelihood ratios rather than probabilities as conventionally presented the proposed algorithm fits better the decoding of osmld codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding ldpc codes although it has been believed that osmld codes are far inferior to ldpc codes we show that for medium code lengths say between 200 1000 bits the bp decoding of osmld codes can significantly outperform bp decoding of their equivalent ldpc codes the reasons for this behavior are elaborated

interference alignment with partial csi feedback in mimo cellular networks interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks however most existing ia designs require full channel state information csi at the transmitters which could lead to significant csi signaling overhead there are two techniques namely csi quantization and csi feedback filtering that reduce csi feedback overhead in this paper we consider ia processing with csi feedback filtering in mimo cellular networks we introduce a novel metric namely the feedback dimension to quantify the first order csi feedback cost associated with the csi feedback filtering the csi feedback filtering poses several important challenges in ia processing first there is a hidden partial csi knowledge constraint in ia precoder design which cannot be handled using conventional ia design methodology furthermore existing results on the feasibility conditions of ia cannot be applied due to the partial csi knowledge finally it is very challenging to find out how much csi feedback is actually needed to support ia processing we shall address the above challenges and propose a new ia feasibility condition under partial csit knowledge in mimo cellular networks based on this we consider the csi feedback dimension minimization subject to the degrees of freedom requirements and further propose an asymptotically optimal solution and derive closed form trade off results between the csi feedback cost and ia performance in mimo cellular networks

error rate estimation of low density parity check codes decoded by quantized soft decision iterative algorithms this paper describes a combinatorial approach to estimate the error rate performance of low density parity check ldpc codes decoded by quantized soft decision iterative decoding algorithms the method is based on efficient enumeration of input vectors with small distances to a reference vector whose elements are selected to be the most reliable values from the input alphabet several techniques including modified cycle enumeration and the efficient derivation of problematic inputs for finer quantizers from those of coarser ones are employed to reduce the complexity of the enumeration the error rate estimate is derived by testing the input vectors of small distances followed by estimating the contribution of larger distance vectors we demonstrate by a number of examples that the proposed method provides accurate estimates of error rate with computational complexity much lower than that of monte carlo simulations especially at the error floor region

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

on the largest principal angle between random subspaces formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p dirnensional subspaces of r n chosen from the uniform distribution on the grassmann manifold which is the unique distribution invariant by orthogonal transformations of r n the formulas involve the gamma function and the hypergeometric function of a matrix argument c 2005 elsevier inc all rights reserved

high rate quasi cyclic low density parity check codes derived from finite affine planes this paper shows that several attractive classes of quasi cyclic qc low density parity check ldpc codes can be obtained from affine planes over finite fields one class of these consists of duals of one generator qc codes presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum weight codewords further it is shown that the minimum hamming distance of a code in this class is equal to its minimum additive white gaussian noise awgn pseudoweight also discussed is a class consisting of codes from circulant permutation matrices and an explicit formula for the rank of the parity check matrix is presented for these codes additionally it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2 the construction is similar to the derivation of reed solomon rs based ldpc codes presented by chen and djurdjevic experimental results show that a number of high rate qc ldpc codes with excellent error performance are contained in these classes

the three dimensional knapsack problem with balancing constraints abstract in this paper we introduce a new packing problem the three dimensional knapsack problem with balancing constraints 3bkp the extension of the three dimensional knapsack problem 3kp where additional constraints related to the packing center of mass are given the 3bkp consists in orthogonally packing a subset of three dimensional weighted items into a knapsack in order to maximize the total profit of the loaded items the items must not overlap and the packing center of mass must lie into a predefined boxed domain inside the knapsack we assume that items can be rotated we give a mip model for the problem upper bounds and an efficient heuristic to solve large size instances the computational results show that the mip model cannot find optimal solutions except for small size instances but it can be used to calculate upper and lower bounds it is shown that our heuristic outperforms the solution quality both of the mip model and the heuristics available in the literature explicitly designed to solve the 3kp

message passing algorithms and improved lp decoding linear programming lp decoding for low density parity check codes and related domains such as compressed sensing has received increased attention over recent years because of its practical performance coming close to that of iterative decoding algorithms and its amenability to finite blocklength analysis several works starting with the work of feldman showed how to analyze lp decoding using properties of expander graphs this line of analysis works for only low error rates about a couple of orders of magnitude lower than the empirically observed performance it is possible to do better for the case of random noise as shown by daskalakis and koetter and vontobel building on work of koetter and vontobel we obtain a novel understanding of lp decoding which allows us to establish a 0 05 fraction of correctable errors for rate codes this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for lp decoding our analysis exploits an explicit connection between lp decoding and message passing algorithms and unlike other techniques directly works with the primal linear program an interesting byproduct of our method is a notion of a locally optimal solution that we show to always be globally optimal i e it is the nearest codeword such a solution can in fact be found in near linear time by a reweighted version of the min sum algorithm obviating the need for lp our analysis implies in particular that this reweighted version of the min sum decoder corrects up to a 0 05 fraction of errors

error floor approximation for ldpc codes in the awgn channel this paper addresses the prediction of error floors of low density parity check codes transmitted over the additive white gaussian noise channel using a linear state space model to estimate the behavior of the sum product algorithm spa decoder in the vicinity of trapping sets tss we study the performance of the spa decoder in the log likelihood ratio llr domain as a function of the llr saturation level when applied to several widely studied codes the model accurately predicts a significant decrease in the error floor as the saturation level is allowed to increase for nonsaturating decoders however we find that the state space model breaks down after a small number of iterations due to the strong correlation of llr messages we then revisit richardson s importance sampling methodology for estimating error floors due to tss when those floors are too low for monte carlo simulation we propose modifications that account for the behavior of a nonsaturating decoder and present the resulting error floor estimates for the margulis code these estimates are much lower significantly steeper and more sensitive to iteration count than those previously reported

quantized iterative message passing decoders with low error floor for ldpc codes the error floor phenomenon observed with ldpc codes and their graph based iterative message passing mp decoders is commonly attributed to the existence of error prone substructures variously referred to as near codewords trapping sets absorbing sets or pseudocodewords in a tanner graph representation of the code many approaches have been proposed to lower the error floor by designing new ldpc codes with fewer such substructures or by modifying the decoding algorithm using a theoretical analysis of iterative mp decoding in an idealized trapping set scenario we show that a contributor to the error floors observed in the literature may be the imprecise implementation of decoding algorithms and in particular the message quantization rules used we then propose a new quantization method q 1 bit quasi uniform quantization that efficiently increases the dynamic range of messages thereby overcoming a limitation of conventional quantization schemes finally we use the quasi uniform quantizer to decode several ldpc codes that suffer from high error floors with traditional fixed point decoder implementations the performance simulation results provide evidence that the proposed quantization scheme can for a wide variety of codes significantly lower error floors with minimal increase in decoder complexity

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

mean field and mixed mean field iterative decoding of low density parity check codes in this paper the mean field mf and mixed mean field mmf algorithms for decoding low density parity check ldpc codes are considered the mf principle is well established in statistical physics and artificial intelligence instead of using a single completely factorized approximated distribution as in the mf approach the mixed mf algorithm forms a weighted average of several mf distributions as an approximation of the true posterior probability distribution the mf decoding algorithm for linear block codes is derived and shown to be an approximation of the a posteriori probability app decoding algorithm the mf approach is then developed in the context of iterative decoding and presented as an approximation of the popular belief propagation decoding method these results are extended to iterative decoding with the mmf algorithm simulation results show that the mf and mmf decoding algorithms yield a good performance complexity tradeoff especially when employed for decoding ldpc codes based on finite geometries

ad 3 alternating directions dual decomposition for map inference in graphical models we present ad3 a new algorithm for approximate maximum a posteriori map inference on factor graphs based on the alternating directions method of multipliers like other dual decomposition algorithms ad3 has a modular architecture where local subproblems are solved independently and their solutions are gathered to compute a global update the key characteristic of ad3 is that each local subproblem has a quadratic regularizer leading to faster convergence both theoretically and in practice we provide closed form solutions for these ad3 subproblems for binary pairwise factors and factors imposing first order logic constraints for arbitrary factors large or combinatorial we introduce an active set method which requires only an oracle for computing a local map configuration making ad3 applicable to a wide range of problems experiments on synthetic and real world problems show that ad3 compares favorably with the state of the art

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

new classes of partial geometries and their associated ldpc codes the use of partial geometries to construct parity check matrices for binary low density parity check ldpc codes has resulted in the design of successful codes with a probability of error on the awgn channel close to the shannon capacity at bit error rate down to 10 15 such considerations have motivated this further investigation a new and simple construction of a type of partial geometries with a quasi cyclic qc structure is given and their properties are investigated two new classes of this type of partial geometries one based on prime fields and the other based on cyclic subgroups of prime orders of finite fields are constructed qc ldpc codes with good error performances are constructed based on these two new classes of partial geometries the trapping sets of the partial geometry codes were previously considered using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets this topic is further considered here finally there is a natural relationship between partial geometries and strongly regular graphs the eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the tanner graphs derived from the partial geometries are good expanders for certain parameter sets since it can be argued that codes with good geometric and expansion properties might perform well on the awgn channel under message passing decoding

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

algebraic construction of quasi cyclic ldpc codes for the awgn and erasure channels this paper is concerned with construction of quasi cyclic qc low density parity check ldpc codes for three different types of channels the additive white gaussian noise the binary random erasure and the binary burst erasure channels two algebraic methods for systematic construction of qc ldpc codes are presented codes constructed perform well over all three types of channels

computing the stopping distance of a tanner graph is np hard two decision problems related to the computation f stopping sets in tanner graphs are shown to be np complete it follows as a consequence that there exists no polynomial time algorithm for computing the stopping distance of a tanner graph unless p np

hierarchical and high girth qc ldpc codes we present an approach to designing capacity approaching high girth low density parity check ldpc codes that are friendly to hardware implementation and compatible with some desired input code structure defined using a protograph the approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi cyclic hqc ldpc codes whereas the parity check matrices of standard quasi cyclic qc ldpc codes are composed of circulant submatrices those of hqc ldpc codes are composed of a hierarchy of circulant submatrices that are in turn constructed from circulant submatrices and so on through some number of levels next we present a girth maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high girth hqc ldpc code subject to bounds imposed by the fact that hqc codes are still quasi cyclic finally we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a squashing procedure that results in a high girth qc ldpc code although not a hierarchical one we illustrate our approach with three design examples of qc ldpc codes two girth 10 codes of rates 1 3 and 0 45 and one girth 8 code of rate 0 7 all of which are obtained from protographs of one sided spatially coupled codes

reed solomon based quasi cyclic ldpc codes designs girth cycle structure and reduction of short cycles designs and constructions of quasi cyclic qc ldpc codes for the awgn channel are presented the codes are constructed based on the conventional parity check matrices of reed solomon rs codes and are referred to as rs qc ldpc codes several classes of rs qc ldpc codes are given cycle structural properties of the tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented the designed codes perform well in both waterfall and low error rate regions

reduced complexity iterative decoding of low density parity check codes based on belief propagation two simplified versions of the belief propagation algorithm for fast iterative decoding of low density parity check codes on the additive white gaussian noise channel are proposed both versions are implemented with real additions only which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed also these two algorithms do not require any knowledge about the channel characteristics both algorithms yield a good performance complexity trade off and can be efficiently implemented in software as well as in hardware with possibly quantized received values

svd iterative decision feedback demodulation and detection of coded space time unitary differential modulation a new suboptimal demodulator based on iterative decision feedback demodulation dfd and a singular value decomposition svd for estimation of unitary matrices is introduced noncoherent communication over the rayleigh flat fading channel with multiple transmit and receive antennas where no channel state information csi is available at the receiver is investigated with four transmit antennas codes achieving bit error rate ber lower than 10 sup 4 at bit energy over the noise spectral density ratio e sub b n sub o of 0 25 db up to 3 5 db with coding rates of 1 6875 to 5 06 bits per channel use were found the performance is compared to the mutual information upper bound of the capacity attaining isotropically random ir unitary transmit matrices the codes achieve ber lower than 10 sup 4 at e sub b n sub o of 3 2 db to 5 8 db from this bound system performance including the iterative dfd algorithm is compared to the one using euclidean distance as a reliability measure for demodulation the dfd system presents a performance gain of up to 1 5 db uncoded systems doing iterative dfd demodulation and idealized pilot sequence assisted modulation psam detection are compared iterative dfd introduces a gain of more than 1 2 db the coded system comprises a serial concatenation of turbo code and a unitary matrix differential modulation code the receiver employs the high performance coupled iterative decoding of the turbo code and the modulation code information theoretic arguments are harnessed to form guidelines for code design and to evaluate performance of the iterative decoder

svd iterative detection of turbo coded multiantenna unitary differential modulation a new suboptimal demodulator based on a singular value decomposition for estimation of unitary matrices is introduced noncoherent communication over the rayleigh flat fading channel with multiple transmit and receive antennas where no channel state information is available at the receiver is investigated codes achieving bit error rate ber lower than 10 sup 4 at bit energy over the noise spectral density ratio e sub b n sub 0 of 1 6 1 9 db from code restricted capacity limit were found at higher data rates computation of code restricted capacity is impractical therefore the mutual information upper bound of the capacity attaining isotropically random unitary transmit matrices was used the codes achieve ber lower than 10 sup 4 at e sub b n sub 0 of 3 2 6 db from this bound with coding rates of 1 125 5 06 bits per channel use and different modulation decoding complexities the codes comprise a serial concatenation of turbo code and a unitary matrix differential modulation code the receiver employs the high performance coupled iterative decoding of the turbo code and the modulation code information theoretic arguments are harnessed to form guidelines for code design and to evaluate performance of the iterative decoder

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

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

ldpc codes from generalized polygons we use the theory of finite classical generalized polygons to derive and study low density parity check ldpc codes the tanner graph of a generalized polygon ldpc code is highly symmetric inherits the diameter size of the parent generalized polygon and has minimum one half diameter to girth ratio we show formally that when the diameter is four or six or eight all codewords have even hamming weight when the generalized polygon has in addition an equal number of points and lines we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length a new minimum distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines finally we prove that all codes derived from finite classical generalized quadrangles are quasi cyclic and we give the explicit size of the circulant blocks in the parity check matrix our simulation studies of several generalized polygon ldpc codes demonstrate powerful bit error rate ber performance when decoding is carried out via low complexity variants of belief propagation

an efficient instanton search algorithm for lp decoding of ldpc codes over the bsc we consider linear programming lp decoding of a fixed low density parity check ldpc code over the binary symmetric channel bsc the lp decoder fails when it outputs a pseudo codeword which is not equal to the transmitted codeword we design an efficient algorithm termed the instanton search algorithm isa which generates an error vector called the bsc instanton we prove that a the lp decoder fails for any error pattern with support that is a superset of the support of an instanton b for any input the isa outputs an instanton in the number of steps upper bounded by twice the number of errors in the input error vector we then find the number of unique instantons of different sizes for a given ldpc code by running the isa sufficient number of times

new insights into weighted bit flipping decoding a natural relationship between weighted bit flipping wbf decoding and belief propagation like bp like decoding is explored this understanding can help us develop wbf algorithms from bp like algorithms for min sum decoding one can find that its wbf algorithm is the algorithm proposed by jiang et al for bp decoding we propose a new wbf algorithm and show its performance advantage the proposed wbf algorithms are parallelized to achieve rapid convergence two efficient simulation based procedures are proposed for the optimization of the associated thresholds

milnor numbers and the topology of polynomial hypersurfaces letf n 1 be a polynomial the problem of determining the homology groupsh q f 1 c c in terms of the critical points off is considered in the best case it is shown for a certain generic class of polynomials tame polynomials that for allc f 1 c has the homotopy type of a bouquet of c n spheres here is the sum of all the milnor numbers off at critical points off and c is the corresponding sum for critical points lying onf 1 c a second best case is also discussed and the homology groupsh q f 1 c are calculated for genericc this case gives an example in which the critical points at infinity off must be considered in order to determine the homology groupsh q f 1 c

lp decoding of regular ldpc codes in memoryless channels we study error bounds for linear programming decoding of regular low density parity check ldpc codes for memoryless binary input output symmetric channels we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph for memoryless binary input awgn channel we prove lower bounds on the threshold for regular ldpc codes whose factor graphs have logarithmic girth under lp decoding specifically we prove a lower bound of 0 735 upper bound of eb n0 2 67 db on the threshold of 3 6 regular ldpc codes whose factor graphs have logarithmic girth our proof is an extension of a recent paper of arora daskalakis and steurer stoc 2009 who presented a novel probabilistic analysis of lp decoding over a binary symmetric channel their analysis is based on the primal lp representation and has an explicit connection to message passing algorithms we extend this analysis to any mbios channel

design of ldpc decoders for improved low error rate performance quantization and algorithm choices many classes of high performance low density parity check ldpc codes are based on parity check matrices composed of permutation submatrices we describe the design of a parallel serial decoder architecture that can be used to map any ldpc code with such a structure to a hardware emulation platform high throughput emulation allows for the exploration of the low bit error rate ber region and provides statistics of the error traces which illuminate the causes of the error floors of the 2048 1723 reed solomon based ldpc rs ldpc code and the 2209 1978 array based ldpc code two classes of error events are observed oscillatory behavior and convergence to a class of non codewords termed absorbing sets the influence of absorbing sets can be exacerbated by message quantization and decoder implementation in particular quantization and the log tanh function approximation in sum product decoders strongly affect which absorbing sets dominate in the errorfloor region we show that conventional sum product decoder implementations of the 2209 1978 array based ldpc code allow low weight absorbing sets to have a strong effect and as a result elevate the error floor dually quantized sum product decoders and approximate sum product decoders alleviate the effects of low weight absorbing sets thereby lowering the error floor

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

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

interference alignment for partially connected mimo cellular networks in this paper we propose an iterative interference alignment ia algorithm for mimo cellular networks with partial connectivity which is induced by heterogeneous path losses and spatial correlation such systems impose several key technical challenges in the ia algorithm design namely the overlapping between the direct and interfering links due to the mimo cellular topology as well as how to exploit the partial connectivity we shall address these challenges and propose a three stage ia algorithm as illustration we analyze the achievable degree of freedom dof of the proposed algorithm for a symmetric partially connected mimo cellular network we show that there is significant dof gain compared with conventional ia algorithms due to partial connectivity the derived dof bound is also backward compatible with that achieved on fully connected k pair mimo interference channels

augmented belief propagation decoding of low density parity check codes we propose an augmented belief propagation bp decoder for low density parity check ldpc codes which can be utilized on memoryless or intersymbol interference channels the proposed method is a heuristic algorithm that eliminates a large number of pseudocodewords that can cause nonconvergence in the bp decoder the augmented decoder is a multistage iterative decoder where at each stage the original channel messages on select symbol nodes are replaced by saturated messages the key element of the proposed method is the symbol selection process which is based on the appropriately defined subgraphs of the code graph and or the reliability of the information received from the channel we demonstrate by examples that this decoder can be implemented to achieve substantial gains compared to the standard locally operating bp decoder for short ldpc codes decoded on both memoryless and intersymbol interference gaussian channels using the margulis code example we also show that the augmented decoder reduces the error floors finally we discuss types of bp decoding errors and relate them to the augmented bp decoder

on implementation of min sum algorithm and its modifications for decoding low density parity check ldpc codes the effects of clipping and quantization on the performance of the min sum algorithm for the decoding of low density parity check ldpc codes at short and intermediate block lengths are studied it is shown that in many cases only four quantization bits suffice to obtain close to ideal performance over a wide range of signal to noise ratios moreover we propose modifications to the min sum algorithm that improve the performance by a few tenths of a decibel with just a small increase in decoding complexity a quantized version of these modified algorithms is also studied it is shown that when optimized modified quantized min sum algorithms perform very close to and in some cases even slightly outperform the ideal belief propagation algorithm at observed error rates

near optimum universal belief propagation based decoding of low density parity check codes in this paper we propose a belief propagation bp based decoding algorithm which utilizes normalization to improve the accuracy of the soft values delivered by a previously proposed simplified bp based algorithm the normalization factors can be obtained not only by simulation but also importantly theoretically this new bp based algorithm is much simpler to implement than bp decoding as it requires only additions of the normalized received values and is universal i e the decoding is independent of the channel characteristics some simulation results are given which show this new decoding approach can achieve an error performance very close to that of bp on the additive white gaussian noise channel especially for low density parity check ldpc codes whose check sums have large weights the principle of normalization can also be used to improve the performance of the max log map algorithm in turbo decoding and some coding gain can be achieved if the code length is long enough

a differential binary message passing ldpc decoder in this paper we propose a binary message passing algorithm for decoding low density parity check ldpc codes the algorithm substantially improves the performance of purely hard decision iterative algorithms with a small increase in the memory requirements and the computational complexity we associate a reliability value to each nonzero element of the code s parity check matrix and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes for the tested random and finite geometry ldpc codes the proposed algorithm can perform as close as about 1 db and 0 5 db to belief propagation bp at the error rates of interest respectively this is while unlike bp the algorithm does not require the estimation of channel signal to noise ratio low memory and computational requirements and binary message passing make the proposed algorithm attractive for high speed low power applications

algebraic quasi cyclic ldpc codes construction low error floor large girth and a reduced complexity decoding scheme this paper presents a simple and very flexible method for constructing quasi cyclic qc low density paritycheck ldpc codes based on finite fields the code construction is based on two arbitrary subsets of elements from a given field some well known constructions of qc ldpc codes based on finite fields and combinatorial designs are special cases of the proposed construction the proposed construction in conjunction with a technique known as masking results in codes whose tanner graphs have girth 8 or larger experimental results show that codes constructed using the proposed construction perform well and have low error floors also presented in the paper is a reduced complexity iterative decoding scheme for qc ldpc codes based on the section wise cyclic structure of their parity check matrices the proposed decoding scheme is an improvement of an earlier proposed reduced complexity iterative decoding scheme

dynamic weighted bit flipping decoding algorithms for ldpc codes bit flipping bf decoding of low density parity check codes is of low complexity but gives inferior performance in general to improve performance and provide new bf decoder options for complexity performance tradeoffs we propose new designs for the flipping function ff the flipped bit selection fbs rule and the checksum weight updating schedule the new ff adjusts the checksum weights in every iteration while our fbs rules take more information into account these two modifications represent efforts to track more closely the evolutions of both check and variable nodes reliabilities two selective update schedules are proposed to offer more performance and complexity tradeoffs the combinations of the new fbs rule and known ffs result in new bf decoders with improved performance and a modest complexity increase on the other hand combining the new ff and fbs rule gives a new decoder with performance comparable to that of the normalized min sum algorithm while if we use a much simpler fbs rule instead the decoder suffers little performance loss with reduced complexity we also present a simple decision theoretical argument to justify the new checksum weight formula and a time expanded factor graph model to explain the proposed selective weight updating schedules

the dimensional family approach in hyper sphere packing a typological study for new patterns structures and interdimensional functions powerful topological invariants associated with sphere packing in euclidean spaces are investigated and new interdimensional functions and structures are delineated that enable those invariants to be readily determined the fundamental invariants discussed are tangency or contact number and the density of centres in the class of recursive packings those whose process of formation is uniquely propagated throughout the entire dimensional continuum an important property of recursivity as here defined is the maximizing of tangency contact or kissing number for a given dimension interdimensional functions are sought and found that can select maximal contact numbers even in dimensions that have many choices thus in 25 space there are 23 packings compatible with t 2 4 196560 but only one of these is maximal and that one our function selects similar properties are found with dimensions 26 where there are at least three thousand more choices than in 25 space through 32 in each case the function selects the maximum t n as given in table 1 this article is part of a book in preparation for publication

performance of ldpc codes under faulty iterative decoding departing from traditional communication theory where decoding algorithms are assumed to perform without error a system where noise perturbs both computational devices and communication channels is considered here this paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low density parity check ldpc codes concentration of decoding performance around its average is shown to hold when noise is introduced into message passing and local computation density evolution equations for simple faulty iterative decoders are derived in one model computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases but arbitrarily small probability of error is not achievable probability of error may be driven to zero in another system model the decoding threshold again decreases smoothly with decoder noise as an application of the methods developed an achievability result for reliable memory systems constructed from unreliable components is provided

two bit flipping decoding algorithms for low density parity check codes in this letter a low complexity decoding algorithm for binary linear block codes is applied to low density paritycheck ldpc codes and improvements are described namely an extension to soft decision decoding and a loop detection mechanism for soft decoding only one real valued addition per code symbol is needed while the remaining operations are only binary as in the hard decision case the decoding performance is considerably increased by the loop detection simulation results are used to compare the performance with other known decoding strategies for ldpc codes with the result that the presented algorithms offer excellent performances at smaller complexity

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

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

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

iterative decoding algorithms for a class of non binary two step majority logic decodable cyclic codes this paper presents two iterative decoding algorithms for a class of non binary two step majority logic nb ts mlg decodable cyclic codes a partial parallel decoding scheme is also introduced to provide a balanced trade off between decoding speed and storage requirements unlike non binary one step mlg decodable cyclic codes the tanner graphs of which are 4 cycle free nb ts mlg decodable cyclic codes contain a large number of short cycles of length 4 which tend to degrade decoding performance the proposed algorithms utilize the orthogonal structure of the parity check matrices of the codes to resolve the degrading effects of the short cycles of length 4 simulation results demonstrate that the nb ts mlg decodable cyclic codes decoded with the proposed algorithms offer coding gains as much as 2 5 db over reed solomon codes of the same lengths and rates decoded with either hard decision or algebraic soft decision decoding

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

source coding with side information at the decoder and uncertain knowledge of the correlation this paper considers the problem of lossless source coding with side information at the decoder when the correlation model between the source and the side information is uncertain four parametrized models representing the correlation between the source and the side information are introduced the uncertainty on the correlation appears through the lack of knowledge on the value of the parameters for each model we propose a practical coding scheme based on non binary low density parity check codes and able to deal with the parameter uncertainty at the encoder the choice of the coding rate results from an information theoretical analysis then we propose decoding algorithms that jointly estimate the source vector and the parameters as the proposed decoder is based on the expectation maximization algorithm which is very sensitive to initialization we also propose a method to produce first a coarse estimate of the parameters

an improved decoding algorithm for finite geometry ldpc codes in this letter an improved bit flipping decoding algorithm for high rate finite geometry low density parity check fg ldpc codes is proposed both improvement in performance and reduction in decoding delay are observed by flipping multiple bits in each iteration our studies show that the proposed algorithm achieves an appealing tradeoff between performance and complexity for fg ldpc codes

blind frame synchronization for error correcting codes having a sparse parity check matrix we present in this letter a blind frame synchronization method based on a maximum a posteriori probability map approach applied to coded communication systems this method is based on the calculation of the log likelihood ratios llr of the syndrome obtained according to the parity check matrix of the code after presenting the proposed method we compare it to an existing blind synchronization method previously introduced simulation results show that the proposed method outperforms the existing one and presents good performance when applied to codes having a sparse parity check matrix such as low density parity check ldpc codes and convolutional codes

reduced complexity decoding of ldpc codes various log likelihood ratio based belief propagation llr bp decoding algorithms and their reduced complexity derivatives for low density parity check ldpc codes are presented numerically accurate representations of the check node update computation used in llr bp decoding are described furthermore approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check node update or symbol node update or both in particular two main approaches for simplified check node updates are presented that are based on the so called min sum approximation coupled with either a normalization term or an additive offset term density evolution is used to analyze the performance of these decoding algorithms to determine the optimum values of the key parameters and to evaluate finite quantization effects simulation results show that these reduced complexity decoding algorithms for ldpc codes achieve a performance very close to that of the bp algorithm the unified treatment of decoding techniques for ldpc codes presented here provides flexibility in selecting the appropriate scheme from performance latency computational complexity and memory requirement perspectives

distance properties of expander codes the minimum distance of some families of expander codes is studied as well as some related families of codes defined on bipartite graphs the weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the gilbert varshamov gv bound a lower bound on the minimum distances of constructive families of expander codes is derived the relative minimum distance of the expander code is shown to exceed the product bound i e the quantity spl delta sub 0 spl delta sub 1 where spl delta sub 0 and spl delta sub 1 are the minimum relative distances of the constituent codes as a consequence of this a polynomially constructible family of expander codes is obtained whose relative distance exceeds the zyablov bound on the distance of serial concatenations

transactions letters random redundant iterative soft in soft out decoding this letter presents an iterative soft in soft out siso decoding algorithm based on redundant tanner graphs that is applicable to arbitrary linear block codes the proposed algorithm utilizes the permutation group of a code in order to efficiently and randomly generate redundant parity checks