algebraic construction of sparse matrices with large girth in this correspondence we present a method for constructing sparse matrices that have a compact description and whose associated bipartite graphs have large girth based on an arbitrary seed matrix of nonnegative integers a new matrix is constructed which replaces each entry of the seed matrix with a sum of permutation matrices algebraic conditions that lead to short cycles in the associated bipartite graph are analyzed and methods to achieve large girth in two special cases are presented in one all the permutation matrices are circulants in the other they are all affine permutation matrices when used to define a low density parity check ldpc code the compact description should lead to efficient implementation and the large girth to good error correction performance the method is adaptable to a variety of rates and a variety of row and column degrees

a survey on learning to hash nearest neighbor search is a problem of finding the data points from the database such that the distances from them to the query point are the smallest learning to hash is one of the major solutions to this problem and has been widely studied recently in this paper we present a comprehensive survey of the learning to hash algorithms categorize them according to the manners of preserving the similarities into pairwise similarity preserving multiwise similarity preserving implicit similarity preserving as well as quantization and discuss their relations we separate quantization from pairwise similarity preserving as the objective function is very different though quantization as we show can be derived from preserving the pairwise similarities in addition we present the evaluation protocols and the general performance analysis and point out that the quantization algorithms perform superiorly in terms of search accuracy search time cost and space cost finally we introduce a few emerging topics

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

signal space characterization of iterative decoding by tracing the flow of computations in the iterative decoders for low density parity check codes we formulate a signal space view for a finite number of iterations in a finite length code on a gaussian channel maximum a posteriori map codeword decoding or maximum likelihood decoding decodes to the codeword signal that is closest to the channel output in euclidean distance in contrast we show that iterative decoding decodes to the pseudosignal that has highest correlation with the channel output the set of pseudosignals corresponds to pseudocodewords only a vanishingly small number of which correspond to codewords we show that some pseudocodewords cause decoding errors but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords

turbo decoding on the binary erasure channel finite length analysis and turbo stopping sets this paper is devoted to the finite length analysis of turbo decoding over the binary erasure channel bec the performance of iterative belief propagation decoding of low density parity check ldpc codes over the bec can be characterized in terms of stopping sets we describe turbo decoding on the bec which is simpler than turbo decoding on other channels we then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure apply turbo decoding until the transmitted codeword has been recovered or the decoder fails to progress further then the set of erased positions that will remain when the decoder stops is equal to the unique maximum size turbo stopping set which is also a subset of the set of erased positions furthermore we present some improvements of the basic turbo decoding algorithm on the bec the proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results finally we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption and an efficient enumeration algorithm of small size turbo stopping sets for a particular interleaver the solution is based on the algorithm proposed by garello et al in 2001 to compute an exhaustive list of all low weight codewords in a turbo code

optimized product quantization product quantization pq is an effective vector quantization method a product quantizer can generate an exponentially large codebook at very low memory time cost the essence of pq is to decompose the high dimensional vector space into the cartesian product of subspaces and then quantize these subspaces separately the optimal space decomposition is important for the pq performance but still remains an unaddressed issue in this paper we optimize pq by minimizing quantization distortions w r t the space decomposition and the quantization codebooks we present two novel solutions to this challenging optimization problem the first solution iteratively solves two simpler sub problems the second solution is based on a gaussian assumption and provides theoretical analysis of the optimality we evaluate our optimized product quantizers in three applications i compact encoding for exhaustive ranking 1 ii building inverted multi indexing for non exhaustive search 2 and iii compacting image representations for image retrieval 3 in all applications our optimized product quantizers outperform existing solutions

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

the design of rate compatible protograph ldpc codes this paper presents a simple yet effective method for designing nested families of ldpc codes rate compatible codes are essential for many communication applications e g hybrid automatic repeat request harq systems and their design is nontrivial due to the difficulty of simultaneously guaranteeing the quality of several related codes puncturing can be used to generate rate compatible ldpc codes but it produces a gap to capacity that in practice often significantly exceeds the gap of the mother code we propose an alternative method based on successively extending a high rate protograph the resulting codes not only inherit the advantages of protograph codes namely low encoding complexity and efficient decoding algorithms but also cover a wide range of rates and have very good performance with iterative decoding thresholds that are within 0 2 db of their capacity limits

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

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

enumerators for protograph based ensembles of ldpc and generalized ldpc codes protograph based ldpc and generalized ldpc g ldpc codes have the advantages of a simple design procedure and highly structured encoders and decoders the design of such protograph based codes relies on what is effectively a computer based search as such following gallager it is prudent to restrict the search to a good ensemble for example an ensemble whose minimum distance grows linearly with codeword length a good ensemble can also mean one with good stopping set trapping set or pseudocodeword properties in this paper ensemble codeword weight enumerators for finite length ldpc and g ldpc codes based on protographs were derived and then the asymptotic case was considered the asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length then the codeword weight enumerator technique is adapted to yield ensemble stopping set trapping set and pseudocodeword enumerators for protograph ldpc and g ldpc codes in this case the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size trapping set size and pseudoweight grows linearly with codeword length trapping set enumerators for g ldpc code ensembles represent a more complex problem which we do not consider here

distributed adaptive binary quantization for fast nearest neighbor search hashing has been proved an attractive technique for fast nearest neighbor search over big data compared with the projection based hashing methods prototype based ones own stronger power to generate discriminative binary codes for the data with complex intrinsic structure however existing prototype based methods such as spherical hashing and k means hashing still suffer from the ineffective coding that utilizes the complete binary codes in a hypercube to address this problem we propose an adaptive binary quantization abq method that learns a discriminative hash function with prototypes associated with small unique binary codes our alternating optimization adaptively discovers the prototype set and the code set of a varying size in an efficient way which together robustly approximate the data relations our method can be naturally generalized to the product space for long hash codes and enjoys the fast training linear to the number of the training data we further devise a distributed framework for the large scale learning which can significantly speed up the training of abq in the distributed environment that has been widely deployed in many areas nowadays the extensive experiments on four large scale up to 80 million data sets demonstrate that our method significantly outperforms state of the art hashing methods with up to 58 84 performance gains relatively

multiview alignment hashing for efficient image search hashing is a popular and efficient method for nearest neighbor search in large scale data spaces by embedding high dimensional feature descriptors into a similarity preserving hamming space with a low dimension for most hashing methods the performance of retrieval heavily depends on the choice of the high dimensional feature descriptor furthermore a single type of feature cannot be descriptive enough for different images when it is used for hashing thus how to combine multiple representations for learning effective hashing functions is an imminent task in this paper we present a novel unsupervised multiview alignment hashing approach based on regularized kernel nonnegative matrix factorization which can find a compact representation uncovering the hidden semantics and simultaneously respecting the joint probability distribution of data in particular we aim to seek a matrix factorization to effectively fuse the multiple information sources meanwhile discarding the feature redundancy since the raised problem is regarded as nonconvex and discrete our objective function is then optimized via an alternate way with relaxation and converges to a locally optimal solution after finding the low dimensional representation the hashing functions are finally obtained through multivariable logistic regression the proposed method is systematically evaluated on three data sets 1 caltech 256 2 cifar 10 and 3 cifar 20 and the results show that our method significantly outperforms the state of the art multiview hashing techniques

autocorrelation values of generalized cyclotomic sequences of order two the generalized cyclotomic sequence of order two has several good randomness properties and behaves like the legendre sequence in several aspects we calculate the autocorrelation values of the generalized cyclotomic sequence of order two our result shows that this sequence could have very good autocorrelation property and pattern distributions of length two if the two primes are chosen properly

sequential compact code learning for unsupervised image hashing effective hashing for large scale image databases is a popular research area attracting much attention in computer vision and visual information retrieval several recent methods attempt to learn either graph embedding or semantic coding for fast and accurate applications in this paper a novel unsupervised framework termed evolutionary compact embedding ece is introduced to automatically learn the task specific binary hash codes it can be regarded as an optimization algorithm that combines the genetic programming gp and a boosting trick in our architecture each bit of ece is iteratively computed using a weak binary classification function which is generated through gp evolving by jointly minimizing its empirical risk with the adaboost strategy on a training set we address this as greedy optimization by embedding high dimensional data points into a similarity preserved hamming space with a low dimension we systematically evaluate ece on two data sets sift 1m and gist 1m showing the effectiveness and the accuracy of our method for a large scale similarity search

on the hardness of approximating stopping and trapping sets we prove that approximating the size of stopping and trapping sets in tanner graphs of linear block codes and more restrictively the class of low density parity check ldpc codes is np hard the ramifications of our findings are that methods used for estimating the height of the error floor of moderate and long length ldpc codes based on stopping and trapping set enumeration cannot provide accurate worst case performance predictions for most codes

asymptotic distribution of absorbing sets and fully absorbing sets for regular sparse code ensembles it is well recognized that low density parity check ldpc codes can suffer from an error floor when decoded iteratively this performance degradation is often attributed to the class of objects known as trapping sets past work has focused on characterizing the distribution of trapping sets for a variety of code ensembles including regular irregular and structured ldpc codes as a subset of the trapping set collection there exists a class of graphical structures called the absorbing sets an absorbing set is a combinatorially defined object in particular a fully absorbing set is stable under bit flipping decoding by construction there can exist trapping sets that are not stable under such a decoder as a result for finite precision iterative decoding algorithms used over additive channels absorbing sets can describe decoding errors more accurately than the broader class of trapping sets in this paper we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets including elementary fully absorbing sets the calculations are based on the trapping set enumeration method proposed by milenkovic soljanin and whiting in 1 we compare distributions of absorbing and trapping sets for representative code parameters of interest and quantify the lack of discrepancies between the two good absorbing set properties are implied for known structured ldpc codes including repeat accumulate codes and protograph based constructions establishing the distribution of fully absorbing sets especially when the discrepancy with the trapping set distribution is significant allows one to further refine the estimates of the error rates under bit flipping and related decoders

maxwell construction the hidden bridge between iterative and maximum a posteriori decoding there is a fundamental relationship between belief propagation and maximum a posteriori decoding a decoding algorithm which is called the maxwell decoder is introduced and provides a constructive description of this relationship both the algorithm itself and the analysis of the new decoder are reminiscent of the maxwell construction in thermodynamics this paper investigates in detail the case of transmission over the binary erasure channel while the extension to general binary memoryless channels is discussed in a companion paper

error correction on a tree an instanton approach we introduce a method that allows analytical or semianalytical estimating of the post error correction bit error rate ber when a forward error correction is utilized for transmitting information through a noisy channel the generic method that applies to a variety of error correction schemes in the regimes where the ber is low is illustrated using the example of a finite size code approximated by a treelike structure exploring the statistical physics formulation of the problem we find that the ber decreases with the signal to noise ratio nonuniformly i e crossing over through a sequence of phases the higher the signal to noise ratio the lower the symmetry of the phase dominating ber the last 50 years have witnessed a tremendous increase in the amount of data transmitted through various communication systems this also leads to tightening of the conditions for error free data transfer in the presence of noise and other transmission related impairments the problem of dealing with errors in information flow has a fundamental importance and has been studied extensively in information and coding theory in 1948 shannon 1 proved that applying an error corrected code can result in an error free communication in the thermodynamic limit of an infinitely long message as long as the rate of transmitted information is kept below a certain value known as the channel capacity constructing well performing capacity approaching yet practical codes has been a challenge until the discovery that some classes of codes based on random construction 2 can achieve near optimum performance when used for transmission over white additive gaussian noise channels 3 in the past few years several codes have been designed with performances very close to this limit 4 generally these codes are referred to as codes on graphs and their prime examples are low density parity check ldpc codes and turbo codes a linear block code for which ldpc codes are an example can be represented as the set

quasi cyclic ldpc codes influence of proto and tanner graph structure on minimum hamming distance upper bounds quasi cyclic qc low density parity check ldpc codes are an important instance of proto graph based ldpc codes in this paper we present upper bounds on the minimum hamming distance of qc ldpc codes and study how these upper bounds depend on graph structure parameters like variable degrees check node degrees girth of the tanner graph and of the underlying proto graph moreover for several classes of proto graphs we present explicit qc ldpc code constructions that achieve or come close to the respective minimum hamming distance upper bounds because of the tight algebraic connection between qc codes and convolutional codes we can state similar results for the free hamming distance of convolutional codes in fact some qc code statements are established by first proving the corresponding convolutional code statements and then using a result by tanner that says that the minimum hamming distance of a qc code is upper bounded by the free hamming distance of the convolutional code that is obtained by unwrapping the qc code

on the tanner graph cycle distribution of random ldpc random protograph based ldpc and random quasi cyclic ldpc code ensembles in this paper we study the cycle distribution of random low density parity check ldpc codes randomly constructed protograph based ldpc codes and random quasi cyclic qc ldpc codes we prove that for a random bipartite graph with a given irregular degree distribution the distributions of cycles of different length tend to independent poisson distributions as the size of the graph tends to infinity we derive asymptotic upper and lower bounds on the expected values of the poisson distributions that are independent of the size of the graph and only depend on the degree distribution and the cycle length for a random lift of a bi regular protograph we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical for random qc ldpc codes however we show that the cycle distribution can be quite different from the other two categories in particular depending on the protograph and the value of c the expected number of cycles of length c in this case can be either theta n or theta 1 where n is the lifting degree code length we also provide numerical results that match our theoretical derivations our results provide a theoretical foundation for emperical results that were reported in the literature but were not well justified they can also be used for the analysis and design of ldpc codes and associated algorithms that are based on cycles

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

ldpc codes over the bec bounds and decoding algorithms the performance of maximum likelihood ml decoding on the binary erasure channel for finite length low density parity check ldpc codes from two random ensembles is studied a tightened union type upper bound on the ml decoding error probability based on the precise coefficients of the average weight spectrum is presented for ldpc codes from the gallager ensemble and the richardson urbanke ensemble new upper bounds on the ml decoding performance based on computing the rank of submatrices of the code parity check matrix are derived a new lower bound on the ml decoding threshold followed from the latter error probability bound is obtained an improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well a new low complexity near ml decoding algorithm for quasi cyclic ldpc codes is proposed and simulated its performance is compared to the simulated belief propagation and ml decoding performance and simulated performance of the best known improved iterative decoding techniques as well as with the derived upper bounds on the ml decoding performance and with decoding thresholds obtained by the density evolution technique

on the stopping redundancy of reed muller codes the stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel in this paper we will consider the stopping redundancy of reed muller codes and related codes let r lscr m be the reed muller code of length 2m and order lscr schwartz and vardy gave a recursive construction of parity check matrices for the reed muller codes and asked whether the number of rows in those parity check matrices is the stopping redundancy of the codes we prove that the stopping redundancy of r m 2 m which is also the extended hamming code of length 2m is 2m 1 and thus show that the recursive bound is tight in this case we prove that the stopping redundancy of the simplex code equals its redundancy several constructions of codes for which the stopping redundancy equals the redundancy are discussed we prove an upper bound on the stopping redundancy of r 1 m this bound is better than the known recursive bound and thus gives a negative answer to the question of schwartz and vardy

generalized ldpc codes and generalized stopping sets a generalized low density parity check code gldpc is a low density parity check code in which the constraint nodes of the code graph are block codes rather than single parity checks in this paper we study gldpc codes which have bch or reed solomon codes as subcodes under bounded distance decoding bdd the performance of the proposed scheme is investigated in the limit case of an infinite length cycle free code used over a binary erasure channel bec and the corresponding thresholds for iterative decoding are derived the performance of the proposed scheme for finite code lengths over a bec is investigated as well structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of gldpc codes which yields exact bit and block error rates of the ensemble average is derived unfortunately this study shows that gldpc codes do not compare favorably with their ldpc counterpart over the bec fortunately it is also shown that under certain conditions objects identified in the analysis of gldpc codes over a bec and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of gldpc codes over a binary symmetric channel bsc simulation results show that the proposed method yields competitive performance with a good decoding complexity trade off for the bsc

on ensembles of low density parity check codes asymptotic distance distributions we derive expressions for the average distance distributions in several ensembles of regular low density parity check codes ldpc among these ensembles are the standard one defined by matrices having given column and row sums ensembles defined by matrices with given column sums or given row sums and an ensemble defined by bipartite graphs

design of regular 2 d sub c ldpc codes over gf q using their binary images in this paper a method to design regular 2 dc ldpc codes over gf q with both good waterfall and error floor properties is presented based on the algebraic properties of their binary image first the algebraic properties of rows of the parity check matrix h associated with a code are characterized and optimized to improve the waterfall then the algebraic properties of cycles and stopping sets associated with the underlying tanner graph are studied and linked to the global binary minimum distance of the code finally simulations are presented to illustrate the excellent performance of the designed codes

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

on the linear complexity of legendre sequences we determine the linear complexity of all legendre sequences and the monic feedback polynomial of the shortest linear feedback shift register that generates such a legendre sequence the result shows that legendre sequences are quite good from the linear complexity viewpoint

construction of near capacity protograph ldpc code sequences with block error thresholds density evolution for protograph low density parity check ldpc codes is considered and it is shown that the message error rate falls double exponentially with iterations whenever the degree 2 subgraph of the protograph is cycle free and noise level is below threshold conditions for stability of protograph density evolution are established and related to the structure of the protograph using large girth graphs sequences of protograph ldpc codes with block error threshold equal to bit error threshold and block error rate falling near exponentially with blocklength are constructed deterministically small sized protographs are optimized to obtain thresholds near capacity for binary erasure and binary input gaussian channels

asymptotic average multiplicity of structures within different categories of trapping sets absorbing sets and stopping sets in random regular and irregular ldpc code ensembles the performance of low density parity check ldpc codes in the error floor region is closely related to some substructures of the code s tanner graph collectively referred to as trapping sets tss in this paper we study the asymptotic average number of different types of trapping sets such as elementary tss ets leafless etss lets absorbing sets abs elementary abss eabs and stopping sets ss in random variable regular and irregular ldpc code ensembles we demonstrate that regardless of the type of the ts as the code s length tends to infinity the average number of a given structure tends to infinity to a positive constant or to zero if the structure contains no cycle only one cycle or more than one cycle respectively for the case where the structure contains a single cycle we derive the asymptotic expected multiplicity of the structure by counting the average number of its constituent cycles and all the possible ways that the structure can be constructed from the cycle this in general involves computing the expected number of cycles of a certain length with a certain given combination of node degrees or computing the expected number of cycles of a certain length expanded to the desired structure by the connection of trees to its nodes the asymptotic results obtained in this work which are independent of the block length and only depend on the code s degree distributions are shown to be accurate even for finite length codes

on generic erasure correcting sets and related problems motivated by iterative decoding techniques for the binary erasure channel hollmann and tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes a generic r s erasure correcting set generates for all codes of codimension r a parity check matrix that allows iterative decoding of all correctable erasure patterns of size s or less the problem is to derive bounds on the minimum size f r s of generic erasure correcting sets and to find constructions for such sets in this paper we continue the study of these sets we derive better lower and upper bounds hollmann and tolhuizen also introduced the stronger notion of r s sets and derived bounds for their minimum size g r s here also we improve these bounds we observe that these two conceps are closely related to so called s wise intersecting codes an area in which g r s has been studied primarily with respect to ratewise performance we derive connections finally we observed that hypergraph covering can be used for both problems to derive good upper bounds

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

generalized and doubly generalized ldpc codes with random component codes for the binary erasure channel in this paper a method for the asymptotic analysis of generalized low density parity check gldpc codes and doubly generalized low density parity check d gldpc codes over the binary erasure channel bec based on extrinsic information transfer exit chart is described this method overcomes the problem consisting of the impossibility to evaluate the exit function for the check or variable component codes in situations where the information functions or split information functions for component codes are unknown according to the proposed technique gldpc codes and d gldpc codes where the generalized check and variable component codes are random codes with minimum distance at least 2 are considered a technique is then developed which finds the exit chart for the overall gldpc or d gldpc code by evaluating the expected exit function for each check and variable component code this technique is finally combined with the differential evolution algorithm in order to generate some good gldpc and d gldpc edge distributions numerical results of long random codes are presented which confirm the effectiveness of the proposed approach they also reveal that d gldpc codes can outperform standard ldpc codes and gldpc codes in terms of both waterfall performance and error floor

counting in graph covers a combinatorial characterization of the bethe entropy function we present a combinatorial characterization of the bethe entropy function of a factor graph such a characterization being in contrast to the original analytical definition of this function we achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph analogously we give a combinatorial characterization of the bethe partition function whose original definition was also of an analytical nature as we point out our approach has similarities to the replica method but also stark differences the above findings are a natural backdrop for introducing a decoder for graph based codes that we will call symbolwise graph cover decoding a decoder that extends our earlier work on blockwise graph cover decoding both graph cover decoders are theoretical tools that help toward a better understanding of message passing iterative decoding namely blockwise graph cover decoding links max product min sum algorithm decoding with linear programming decoding and symbolwise graph cover decoding links sum product algorithm decoding with bethe free energy function minimization at temperature one in contrast to the gibbs entropy function which is a concave function the bethe entropy function is in general not concave everywhere in particular we show that every code picked from an ensemble of regular low density parity check codes with minimum hamming distance growing with high probability linearly with the block length has a bethe entropy function that is convex in certain regions of its domain

product quantization for nearest neighbor search this paper introduces a product quantization based approach for approximate nearest neighbor search the idea is to decompose the space into a cartesian product of low dimensional subspaces and to quantize each subspace separately a vector is represented by a short code composed of its subspace quantization indices the euclidean distance between two vectors can be efficiently estimated from their codes an asymmetric version increases precision as it computes the approximate distance between a vector and a code experimental results show that our approach searches for nearest neighbors efficiently in particular in combination with an inverted file system results for sift and gist image descriptors show excellent search accuracy outperforming three state of the art approaches the scalability of our approach is validated on a data set of two billion vectors

the linear complexity of related prime sequences polyphase related prime rp sequences can be constructed from the modified combination of two polyphase power residue sequences and are generalizations of the two phase modified jacobi sequences which include the twin prime sequences as a special case these sequences have also been referred to as cyclotomic sequences rp sequences have been conjectured to possess high linear complexity under certain conditions and this is now confirmed theoretically for the q phase case when q is a prime the linear complexity of an rp sequence of length r s where r and s are distinct primes is shown to depend on the relationship between the two primes used in the construction of the sequence and their selected primitive roots and on the value assigned to the initial digit of the sequence

minimum pseudoweight analysis of 3 dimensional turbo codes in this paper we consider pseudocodewords of relaxed linear programming lp decoding of 3 dimensional turbo codes 3d tcs we present a relaxed lp decoder for 3d tcs adapting the relaxed lp decoder for conventional turbo codes proposed by feldman in his thesis we show that the 3d tc polytope is proper and c symmetric and make a connection to finite graph covers of the 3d tc factor graph this connection is used to show that the support set of any pseudocodeword is a stopping set of iterative decoding of 3d tcs using maximum a posteriori constituent decoders on the binary erasure channel furthermore we compute ensemble average pseudoweight enumerators of 3d tcs and perform a finite length minimum pseudoweight analysis for small cover degrees moreover an explicit description of the fundamental cone of the 3d tc polytope is given finally we present an extensive numerical study of small to medium block length 3d tcs which shows that 1 typically i e in most cases when the minimum distance d min and or the stopping distance h min is high the minimum pseudoweight on the additive white gaussian noise channel is strictly smaller than both d min and h min and that 2 the minimum pseudoweight grows with the block length at least for small to medium block lengths

finding the exhaustive list of small fully absorbing sets and designing the corresponding low error floor decoder this work provides an efficient exhaustive search algorithm for finding all small fully absorbing sets fass of any arbitrary low density parity check ldpc code the proposed algorithm is based on the branch bound principle for solving np complete problems in particular given any ldpc code the problem of finding all fass of size less than t is formulated as an integer programming problem for which a new branch bound algorithm is devised with new node selection and tree trimming mechanisms the resulting algorithm is capable of finding all fass of size 7 for ldpc codes of length 1000 when limiting the fass of interest to those with the number of violated parity check nodes 3 the proposed algorithm is capable of finding all such fass of size 14 for ldpc codes of lengths 1000 the resulting exhaustive list of small fass is then used to devise a new efficient post processing low error floor ldpc decoder the numerical results show that by exploiting the exhaustive list of small fass the proposed post processing decoder can significantly lower the error floor performance of a given ldpc code for various example codes of length 3000 the proposed post processing decoder lowers the error floor by a couple of orders of magnitude when compared to the standard belief propagation decoder and by an order of magnitude when compared to other existing low error floor decoders

pseudocodewords of tanner graphs this paper presents a detailed analysis of pseudocodewords of tanner graphs pseudocodewords arising on the iterative decoder s computation tree are distinguished from pseudocodewords arising on finite degree lifts lower bounds on the minimum pseudocodeword weight are presented for the bec bsc and awgn channel some structural properties of pseudocodewords are examined and pseudocodewords and graph properties that are potentially problematic with min sum iterative decoding are identified an upper bound on the minimum degree lift needed to realize a particular irreducible lift realizable pseudocodeword is given in terms of its maximal component and it is shown that all irreducible lift realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure examples and different tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed the results obtained provide some insights in relating the structure of the tanner graph to the pseudocodeword distribution and suggest ways of designing tanner graphs with good minimum pseudocodeword weight

approaching capacity at high rates with iterative hard decision decoding a variety of low density parity check ldpc ensembles have now been observed to approach capacity with message passing decoding however all of them use soft i e non binary messages and a posteriori probability decoding of their component codes in this paper we show that one can approach capacity at high rates using iterative hard decision decoding hdd of generalized product codes specifically a class of spatially coupled generalized ldpc codes with bose chaudhuri hocquengham component codes is considered and it is observed that in the high rate regime they can approach capacity under the proposed iterative hdd these codes can be seen as generalized product codes and are closely related to braided block codes an iterative hdd algorithm is proposed that enables one to analyze the performance of these codes via density evolution

density evolution for deterministic generalized product codes on the binary erasure channel at high rates generalized product codes gpcs are extensions of product codes pcs where code symbols are protected by two component codes but not necessarily arranged in a rectangular array we consider a deterministic construction of gpcs as opposed to randomized code ensembles and analyze the asymptotic performance over the binary erasure channel under iterative decoding our code construction encompasses several classes of gpcs previously proposed in the literature such as irregular pcs blockwise braided codes and staircase codes it is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity we show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph using a convergence result for these graphs we derive the density evolution equations that characterize the asymptotic decoding performance as an application we discuss the design of irregular gpcs employing a mixture of component codes with different erasure correcting capabilities

the trapping redundancy of linear block codes we generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in tanner graphs of linear block codes in this context we introduce the notion of the trapping redundancy of a code which quantifies the relationship between the number of redundant rows in any parity check matrix of a given code and the size of its smallest trapping set trapping sets with certain parameter sizes are known to cause error floors in the performance curves of iterative belief propagation bp decoders and it is therefore important to identify decoding matrices that avoid such sets bounds on the trapping redundancy are obtained using probabilistic and constructive methods and the analysis covers both general and elementary trapping sets numerical values for these bounds are computed for the 2640 1320 margulis code and the class of projective geometry codes and compared with some new code specific trapping set size estimates

performance analysis of ldpc coded diversity combining on rayleigh fading channels with impulsive noise spatial diversity is an effective method to mitigate the effects of fading and when used in conjunction with low density parity check ldpc codes it can achieve excellent error correcting performance noise added at each branch of the diversity combiner is generally assumed to be additive white gaussian noise but there are many applications where the received signal is impaired by noise with a non gaussian distribution in this paper we derive the exact bit error probability of different linear combining techniques on rayleigh fading channels with impulsive noise which is modeled using symmetric alpha stable distributions the relationship for the signal to noise ratios of these linear combiners is derived and then different non linear detectors are presented a detector based on the bi parameter cauchy gaussian mixture model is used and shows near optimal performance with a significant reduction in complexity when compared with the optimal detector furthermore the threshold signal to noise ratio of ldpc codes for different combining techniques on these channels is derived using density evolution and an estimation of the waterfall performance of ldpc codes is derived that reduces the gap between simulated and asymptotic performance

weight distribution of low density parity check codes we derive the average weight distribution function and its asymptotic growth rate for low density parity check ldpc code ensembles we show that the growth rate of the minimum distance of ldpc codes depends only on the degree distribution pair it turns out that capacity achieving sequences of standard unstructured ldpc codes under iterative decoding over the binary erasure channel bec known to date have sublinearly growing minimum distance in the block length

distance distributions in ensembles of irregular low density parity check codes we derive asymptotic expressions for the average distance distributions in ensembles of irregular low density parity check ldpc codes the ensembles are defined by matrices with given profiles of column and row sums

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

on finding bipartite graphs with a small number of short cycles and large girth the problem of finding bipartite tanner graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low density parity check ldpc codes in this paper we prove that for given integers alpha beta and gamma and degree sequences pi and pi the problem of determining whether there exists a simple bipartite graph with degree sequences pi pi that has at most alpha beta and gamma cycles of length four six and eight respectively is np complete this is proved by a two step polynomial time reduction from the 3 partition problem on the other hand using connections to linear hypergraphs we prove that given the degree sequence pi a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is pi and has a girth of at least six in addition to these complexity results we devise a quasi polynomial time algorithm that can construct a bipartite graph with given irregular degree sequences and a girth that increases logarithmically with the size of the graph compared to the well known progressive edge growth peg algorithm the proposed method in some cases results in tanner graphs with larger girth particularly for scenarios where the degree sequences of the graph are strictly enforced

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

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

pseudo codeword analysis of tanner graphs from projective and euclidean planes we consider coded data transmission over a binary input output symmetric memoryless channel using a binary linear code in order to understand the performance of maximum likelihood ml decoding one studies the codewords in particular the minimal codewords and their hamming weights in the context of linear programming lp decoding one s attention needs to be shifted to the pseudo codewords in particular to the minimal pseudo codewords and their pseudo weights in this paper we investigate some families of codes that have good properties under lp decoding namely certain families of low density parity check ldpc codes that are derived from projective and euclidean planes we study the structure of their minimal pseudo codewords and give lower bounds on their pseudo weight besides this main focus we also present some results that hold for pseudo codewords and minimal pseudo codewords of any tanner graph and we highlight how the importance of minimal pseudo codewords under lp decoding varies depending on which binary input output symmetric memoryless channel is used

polyphase power residue sequences polyphase versions of the well known binary legendre or quadratic residue sequences were first described and analysed by sidelnikov over thirty years ago but have since received very little attention here it is shown that these q phase sequences of prime length l can also be constructed from the index sequence of length l or equivalently from the cosets of q th power residues and non residues these sequences are also shown to fall into two classes each of which has well defined periodic autocorrelation functions and merit factors class i sequences exist for prime l q 1 mod 2 q and q even whereas class ii sequences correspond to primes of the form l 1 mod 2 q and are available for all values of q class i sequences have complex out of phase correlation values with magnitudes less than or equal to 5 whereas for class ii sequences these are purely real with magnitude less than or equal to 3 some other properties are also investigated

quasi cyclic generalized ldpc codes with low error floors in this paper a novel methodology for designing structured generalized low density parity check g ldpc codes is presented the proposed design results in quasi cyclic g ldpc codes for which efficient encoding is feasible through shift register based circuits the structure imposed on the bipartite graphs together with the choice of simple component codes leads to a class of codes suitable for fast iterative decoding a pragmatic approach to the construction of g ldpc codes is proposed the approach is based on the substitution of check nodes in the protograph of a low density parity check code with stronger nodes based for instance on hamming codes such a design approach which we call low density parity check ldpc code doping leads to low rate quasi cyclic g ldpc codes with excellent performance in both the error floor and waterfall regions on the additive white gaussian noise channel

design of repeat accumulate codes for iterative detection and decoding extrinsic information transfer exit charts are used to design systematic and nonsystematic repeat accumulate ra codes for iterative detection and decoding the convergence problems of nonsystematic ra codes are solved by introducing a biregular or doped layer of check nodes as examples such nonsystematic codes are designed for multi input multi output mimo fading channels and are shown to operate close to capacity

doubly generalized ldpc codes stability bound over the bec the iterative decoding threshold of low density parity check ldpc codes over the binary erasure channel bec fulfills an upper bound depending only on the variable and check nodes with minimum distance 2 this bound is a consequence of the stability condition and is here referred to as stability bound in this paper a stability bound over the bec is developed for doubly generalized ldpc codes where variable and check nodes can be generic linear block codes assuming maximum a posteriori erasure correction at each node it is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance 2 a condition is also developed namely the derivative matching condition under which the bound is achieved with equality the stability bound leads to consider single parity check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes

systematic design of low density parity check code ensembles for binary erasure channels we propose a systematic method to design irregular low density parity check ldpc codes for binary erasure channels bec compared to the existing methods which are based on the application of asymptotic analysis tools such as density evolution or extrinsic information transfer exit charts in an optimization process the proposed method is much simpler and faster through a number of examples we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization an important property of the proposed designs is the flexibility to select the number of constituent variable node degrees p the proposed designs include existing systematic designs as a special case with p n 1 where n is the maximum variable node degree compared to the existing systematic designs for a given rate and a given 0 the designed ensembles can have a threshold in neighborhood of the capacity upper bound with smaller values of p and n they can also achieve the capacity of the bec as n and correspondingly p and the maximum check node degree tend to infinity

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

an efficient pseudocodeword search algorithm for linear programming decoding of ldpc codes in linear programming lp decoding of a low density parity check ldpc code one minimizes a linear functional with coefficients related to log likelihood ratios over a relaxation of the polytope spanned by the codewords in order to quantify lp decoding it is important to study vertexes of the relaxed polytope so called pseudocodewords we propose a technique to heuristcally create a list of pseudocodewords close to the zero codeword and their distances our pseudocodeword search algorithm starts by randomly choosing configuration of the noise the configuration is modified through a discrete number of steps each step consists of two substeps one applies an lp decoder to the noise configuration deriving a pseudocodeword and then finds configuration of the noise equidistant from the pseudocodeword and the zero codeword the resulting noise configuration is used as an entry for the next step the iterations converge rapidly to a pseudocodeword neighboring the zero codeword repeated many times this procedure is characterized by the distribution function of the pseudocodeword effective distance the efficiency of the procedure is demonstrated on examples of the tanner code and margulis codes operating over an additive white gaussian noise awgn channel

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

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

distance enumerator analysis for interleave division multi user codes consider an interleave division multi user coding model for a gaussian multiple access channel mac each user s message is encoded by a separate channel encoder followed by a user interleaver which is employed for user separation all the users codewords are jointly regarded as one multi user codeword and the distance between two multi user codewords is defined as the euclidean distance after the inter user superposition caused by the mac a distance enumerator analysis estimates the multi user maximum likelihood decoding performance by introducing user scramblers a multi user code ensemble is defined and its distance enumerator is calculated as concatenated multiple single user codes and a superposition code both finite length and asymptotic analyses are given and it is shown that interleave division multi user codes have much better distance properties than conventional random direct sequence code division multiple access in fact the codes achieve almost the same asymptotic uniquely decodable performance as a multi user random code

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

weight distributions of non binary multi edge type ldpc code ensembles analysis and efficient evaluation non binary multi edge type ensembles of low density parity check codes are analyzed in terms of non binary codeword weight distribution and its growth rate in particular an exact expression of the growth rate for small weights is developed as a side result the stopping set distributions of these ensembles are developed examples of weight distributions are provided showing that the derived closed form expressions can be easily evaluated the obtained results can thus be exploited to analyze and design non binary low density parity check codes that fall within the multi edge type framework such as but not limited to protograph based codes

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

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

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

new nonasymptotic channel coding theorems for structured codes new nonasymptotic random coding theorems with error probability epsilon and finite block length n based on gallager parity check ensemble and general parity check ensembles are derived in this paper the resulting nonasymptotic achievability bounds when combined with nonasymptotic equipartition properties developed in this paper can be easily computed analytically these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or subexponentially decreasing epsilon in the case of gallager parity check ensemble and to imply that low density parity check ldpc codes be capacity achieving in the case of ldpc ensembles numerically they are also compared favorably for finite n and epsilon of practical interest with existing nonasymptotic achievability bounds in the literature

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

weight distributions of regular low density parity check codes over finite fields the average weight distribution of a regular low density parity check ldpc code ensemble over a finite field is thoroughly analyzed in particular a precise asymptotic approximation of the average weight distribution is derived for the small weight case and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved based on this analysis a general result including all previous results as special cases is established for the minimum distance of individual codes in a regular ldpc code ensemble

trace representation of legendre sequences of mersenne prime period in this correspondence it is shown that legendre sequences of period p can be explicitly represented using the trace function defined on the finite field with 2 sup n elements whenever p 2 sup n 1 is prime for some n spl ges 3

multiple feature kernel hashing for large scale visual search recently hashing has become attractive in large scale visual search owing to its theoretical guarantee and practical success however most of the state of the art hashing methods can only employ a single feature type to learn hashing functions related research on image search clustering and other domains has proved the advantages of fusing multiple features in this paper we propose a novel multiple feature kernel hashing framework where hashing functions are learned to preserve certain similarities with linearly combined multiple kernels corresponding to different features the framework is not only compatible with general types of data and diverse types of similarities indicated by different visual features but also general for both supervised and unsupervised scenarios we present efficient alternating optimization algorithms to learn both the hashing functions and the optimal kernel combination experimental results on three large scale benchmarks cifar 10 nus wide and a trecvid show that the proposed approach can achieve superior accuracy and efficiency over state of the art methods highlightswe propose a generic multiple feature hashing framework using multiple kernels visual features are implicitly mapped and concatenated to reduce complexity we formulate both supervised and unsupervised hashing problems in the framework alternating optimization ways efficiently learn hashing functions and the kernel space experiments validate the superior performances and efficiency of the proposed approach

query adaptive hash code ranking for large scale multi view visual search hash based nearest neighbor search has become attractive in many applications however the quantization in hashing usually degenerates the discriminative power when using hamming distance ranking besides for large scale visual search existing hashing methods cannot directly support the efficient search over the data with multiple sources and while the literature has shown that adaptively incorporating complementary information from diverse sources or views can significantly boost the search performance to address the problems this paper proposes a novel and generic approach to building multiple hash tables with multiple views and generating fine grained ranking results at bitwise and tablewise levels for each hash table a query adaptive bitwise weighting is introduced to alleviate the quantization loss by simultaneously exploiting the quality of hash functions and their complement for nearest neighbor search from the tablewise aspect multiple hash tables are built for different data views as a joint index over which a query specific rank fusion is proposed to rerank all results from the bitwise ranking by diffusing in a graph comprehensive experiments on image search over three well known benchmarks show that the proposed method achieves up to 17 11 and 20 28 performance gains on single and multiple table search over the state of the art methods

bounds on the minimum distance of punctured quasi cyclic ldpc codes recent work by divsalar has shown that properly designed protograph based low density parity check codes typically have minimum hamming distance linearly increasing with block length this fact rests on ensemble arguments over all possible expansions of the base protograph however when implementation complexity is considered the expansions are frequently selected from a smaller class of structured expansions for example protograph expansion by cyclically shifting connections generates a quasi cyclic qc code other recent work by smarandache and vontobel has provided upper bounds on the minimum distance of qc codes in this paper we generalize these bounds to punctured qc codes and then show how to tighten these for certain classes of codes we then evaluate these upper bounds for the family of protograph codes known as ar4ja codes that have been recommended for use in deep space communications in a standard established by the consultative committee for space data systems at block lengths larger than 4400 bits these upper bounds fall well below the ensemble lower bounds

spectral shape of doubly generalized ldpc codes efficient and exact evaluation this paper analyzes the asymptotic exponent of the weight spectrum for irregular doubly generalized ldpc d gldpc codes in the process an efficient numerical technique for its evaluation is presented involving the solution of a 4 4 system of polynomial equations the expression is consistent with previous results including the case where the normalized weight or stopping set size tends to zero the spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree a case which includes tanner codes for this case it is also shown how certain symmetry properties of the local weight distribution at the cns induce a symmetry in the overall weight spectral shape function finally using these new results weight and stopping set size spectral shapes are evaluated for some example generalized and doubly generalized ldpc code ensembles

asymptotic spectra of trapping sets in regular and irregular ldpc code ensembles we evaluate the asymptotic normalized average distributions of a class of combinatorial configurations in random regular and irregular binary low density parity check ldpc code ensembles among the configurations considered are trapping and stopping sets these sets represent subsets of variable nodes in the tanner graph of a code that play an important role in determining the height and point of onset of the error floor in its performance curve the techniques used for deriving the spectra include large deviations theory and statistical methods for enumerating binary matrices with prescribed row and column sums these techniques can also be applied in a setting that involves more general structural entities such as subcodes and or minimal codewords that are known to characterize other important properties of soft decision decoders of linear block codes

bounds on the expansion properties of tanner graphs this work focuses on the expansion properties of a tanner graph because they are known to be related to the performance of associated iterative message passing algorithms over various channels by analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a tanner graph lower bounds on these expansion properties are derived specifically for the binary erasure channel these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference set codes type i 2 d projective geometry low density parity check ldpc codes

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

woven graph codes asymptotic performances and examples constructions of woven graph codes based on constituent block and convolutional codes are studied it is shown that within the random ensembles of such codes based on s partite s uniform hypergraphs where s depends only on the code rate there exist codes satisfying the gilbert varshamov gv and the costello lower bound on the minimum distance and the free distance respectively a connection between regular bipartite graphs and tailbiting tb codes is shown some examples of woven graph codes are presented among them an example of a rate rwg 1 3 woven graph code with dfree 32 based on heawood s bipartite graph containing n 7 constituent rate rc 2 3 convolutional codes with overall constraint lengths c 5 is given

hashing on nonlinear manifolds learning based hashing methods have attracted considerable attention due to their ability to greatly increase the scale at which existing algorithms may operate most of these methods are designed to generate binary codes preserving the euclidean similarity in the original space manifold learning techniques in contrast are better able to model the intrinsic structure embedded in the original high dimensional data the complexities of these models and the problems with out of sample data have previously rendered them unsuitable for application to large scale embedding however in this paper how to learn compact binary embeddings on their intrinsic manifolds is considered in order to address the above mentioned difficulties an efficient inductive solution to the out of sample data problem and a process by which nonparametric manifold learning may be used as the basis of a hashing method are proposed the proposed approach thus allows the development of a range of new hashing techniques exploiting the flexibility of the wide variety of manifold learning approaches available it is particularly shown that hashing on the basis of t distributed stochastic neighbor embedding outperforms state of the art hashing methods on large scale benchmark data sets and is very effective for image classification with very short code lengths it is shown that the proposed framework can be further improved for example by minimizing the quantization error with learned orthogonal rotations without much computation overhead in addition a supervised inductive manifold hashing framework is developed by incorporating the label information which is shown to greatly advance the semantic retrieval performance