effectiveness of a loudness model for time varying sounds in equating the loudness of sentences subjected to different forms of signal processing a model for the loudness of time varying sounds glasberg and moore 2012 j audio eng soc 50 331 342 was assessed for its ability to predict the loudness of sentences that were processed to either decrease or increase their dynamic fluctuations in a paired comparison task subjects compared the loudness of unprocessed and processed sentences that had been equalized in 1 root mean square rms level 2 the peak long term loudness predicted by the model 3 the mean long term loudness predicted by the model method 2 was most effective in equating the loudness of the original and processed sentences

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

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

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

edge coloring and stopping sets analysis in product codes with mds components we consider non binary product codes with mds components and their iterative row column algebraic decoding on the erasure channel both independent and block erasures are considered in this paper a compact graph representation is introduced on which we define double diversity edge colorings via the rootcheck concept an upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size m then we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols the complexity of this algorithm per iteration is o m aleph for a given differential evolution parameter aleph where m aleph itself is small with respect to the huge cardinality of the coloring ensemble stopping sets of a product code are defined in the context of mds components and a relationship is established with the graph representation a full characterization of these stopping sets is given up to a size d 1 2 where d is the minimum hamming distance of the mds component code the performance of mds based product codes with and without double diversity coloring is analyzed in presence of both the block and the independent erasures in the latter case ml and iterative decoding are proven to coincide at small channel erasure probability furthermore numerical results show excellent performance in presence of unequal erasure probability due to double diversity colorings

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

improved upper bounds on stopping redundancy for a linear block code with minimum distance d its stopping redundancy is the minimum number of check nodes in a tanner graph representation of the code such that all nonempty stopping sets have size d or larger we derive new upper bounds on stopping redundancy for all linear codes in general and for maximum distance separable mds codes specifically and show how they improve upon previous results for mds codes the new bounds are found by upper bounding the stopping redundancy by a combinatorial quantity closely related to turan numbers the turan number t v k t is the smallest number of t subsets of a v set such that every k subset of the v set contains at least one of the t subsets asymptotically we show that the stopping redundancy of mds codes with length n and minimum distance d 1 is t n d 1 d 2 1 o n 1 for fixed d and is at most t n d 1 d 2 3 o n 1 for fixed code dimension k n d 1 for d 3 4 we prove that the stopping redundancy of mds codes is equal to t n d 1 d 2 for which exact formulas are known for d 5 we show that the stopping redundancy of mds codes is either t n 4 3 or t n 4 3 1

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

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

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

statistical mechanics of error exponents for error correcting codes error exponents characterize the exponential decay when increasing message length of the probability of error of many error correcting codes to tackle the long standing problem of computing them exactly we introduce a general thermodynamic formalism that we illustrate with maximum likelihood decoding of low density parity check codes on the binary erasure channel and the binary symmetric channel in this formalism we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents which differ by the procedure used to select the codes from specified ensembles when decreasing the noise intensity we find that two phase transitions take place at two different levels a glass to ferromagnetic transition in the space of codewords and a paramagnetic to glass transition in the space of codes

the nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

on the pseudocodeword redundancy of binary linear codes for a binary linear code the pseudocodeword redundancy with respect to the additive white gaussian noise channel the binary symmetric channel or the max fractional weight is defined to be the smallest number of rows in a parity check matrix such that the corresponding minimum pseudoweight is equal to the minimum hamming distance of the code it is shown that most codes do not have a finite pseudocodeword redundancy also upper bounds on the pseudocodeword redundancy for some families of codes including codes based on designs are provided the pseudocodeword redundancies for all codes of small length at most 9 are computed furthermore comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of vontobel and koetter is sharp

lexicographic codes error correcting codes from game theory lexicographic codes or lexicodes are defined by various versions of the greedy algorithm the theory of these codes is closely related to the theory of certain impartial games which leads to a number of surprising properties for example lexicodes over an alphabet of size b 2 a are closed under addition while if b 2 2 a the lexicodes are closed under multiplication by scalars where addition and multiplication are in the nim sense explained in the text hamming codes and the binary golay codes are lexieodes remarkably simple constructions are given for the steiner systems s 5 6 12 and s 5 8 24 several record breaking constant weight codes are also constructed

finite length scaling for iteratively decoded ldpc ensembles we investigate the behavior of iteratively decoded low density parity check ldpc codes over the binary erasure channel in the so called ldquowaterfall region rdquo we show that the performance curves in this region follow a simple scaling law we conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture the scaling law together with the error floor expressions developed previously can be used for a fast finite length optimization

stopping set distributions of some reed muller codes stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel bec let c be a binary n k linear code with parity check matrix h where the rows of h may be dependent a stopping set s of c with parity check matrix h is a subset of column indices of h such that the restriction of h to s does not contain a row of weight one the stopping set distribution ti h i 0n enumerates the number of stopping sets with size i of c with parity check matrix h note that stopping sets and stopping set distribution are related to the parity check matrix h of c let h be the parity check matrix of c which is formed by all the nonzero codewords of its dual code c a parity check matrix h is called bec optimal if ti h ti h i 0 1 n and h has the smallest number of rows in this paper we study stopping sets stopping set distributions and bec optimal parity check matrices of binary linear codes using finite geometry in combinatorics we obtain bec optimal parity check matrices and then determine the stopping set distributions for the simplex codes the hamming codes the first order reed muller codes and the extended hamming codes which are some reed muller codes or their shortening or puncturing versions

on the pairwise error probability of linear programming decoding on independent rayleigh flat fading channels in this paper we consider the pairwise error probability pep of a linear programming lp decoder for a general binary linear code as formulated by feldman et al ieee trans inf theory mar 2005 on an independent or memoryless rayleigh flat fading channel with coherent detection and perfect channel state information csi at the receiver let h be a parity check matrix of a binary linear code and consider lp decoding based on h the output of the lp decoder is always a pseudocode word we will show that the pep of decoding to a pseudocodeword w when the all zero codeword is transmitted on the above mentioned channel behaves asymptotically as k omega ldr es n0 chi omega where chi omega is the support set of omega i e the set of nonzero coordinates es n0 is the average signal to noise ratio snr and k omega is a constant independent of the snr note that the support set chi omega of omega is a stopping set thus the asymptotic decay rate of the error probability with the average snr is determined by the size of the smallest nonempty stopping set in the tanner graph of h as an example we analyze the well known 155 64 tanner code and present performance curves on the independent rayleigh flat fading channel

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

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

stopping set elimination by parity check matrix extension via integer linear programming error rate floor phenomenon is known to be a serious impediment to the use of low density parity check ldpc codes for some practical applications that demand high data reliability in the case of binary erasure channels becs certain error prone patterns known as stopping sets are proven to cause this performance degradation a possible approach to diminish this drawback over becs is to eliminate stopping sets by parity check matrix extension given a parity check matrix h and a list mathcal l of its stopping sets we present an integer linear programming ilp formulation to find a parity check equation which eliminates the maximum number of stopping sets in mathcal l one of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as limiting the weight of the new parity check row making the new row redundant or linearly independent 4 cycle avoidance and taking into account the sizes of stopping sets armed with these adjustments the method can provide good performance improvements as evidenced by simulation results furthermore for a given varrho in mathbb n by extending the basic formulation we provide an ilp formulation for finding a set of size varrho of parity check equations which can best eliminate the stopping sets in mathcal l among all such sets

improved probabilistic bounds on stopping redundancy for a linear code c the stopping redundancy of c is defined as the minimum number of check nodes in a tanner graph t for c such that the size of the smallest stopping set in t is equal to the minimum distance of c han and siegel recently proved an upper bound on the stopping redundancy of general linear codes using probabilistic analysis for most code parameters this bound is the best currently known in this correspondence we present several improvements upon this bound

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

optimal information bit decoding of linear block codes we consider the problem of decoding a linear block code used over a binary symmetric channel with crossover probability p when the goal of the decoding is to minimize the probability of an information bit error in general there will be several different decoding strategies each corresponding to the optimal rule over some range p spl isin p sub i 1 p sub i each of these strategies can be implemented by standard array with the appropriate choice of coset leaders that are not in general of minimum weight we give optimal rules for selecting coset leaders in the first region 0 p sub 1 and the region p sub max 1 2 for any linear mapping between information bits and codewords the determination of the optimal rule has been an open problem for small p the rule for p p sub max is important because for many codes when p is fixed the optimal rule for any coset corresponds to either the optimal rule for small p or the optimal rule for p near 1 2 the optimal decoding strategy varies with the generator matrix used in general but it is demonstrated that the optimal generator matrix is separable when d sup spl perp 2 the results are shown to extend in a natural way to the case where the goal is to minimize the probability that arbitrary collections of information bits are in error we examine some specific codes and compare the performance under optimal decoding to the performance of previously suggested strategies

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

a new method for creating sparse design velocity fields we present a novel method for the computation of mesh node sensitivities with respect to the boundary node movement the sensitivity field is sparse in a sense that movement of each boundary node affects only given amount of inner mesh nodes which can result in considerable savings in the storage space the method needs minimal control from the user and it does not place any restrictions such as block structure on the mesh use of the method is demonstrated with a shape optimization problem using cad free parametrization a solution to the classical die swell free boundary problem by coupling the boundary node locations with the state variables is also presented in that case sparsity of the sensitivity field directly affects the sparsity of the jacobian of the residual and results in significant savings in the computation time

average stopping set weight distributions of redundant random ensembles in this paper redundant random ensembles are defined and their average stopping set ss weight distributions are analyzed a redundant random ensemble consists of a set of binary matrices with linearly dependent rows these linearly dependent rows significantly reduce the number of stopping sets ss of small size upper and lower bounds on the average ss weight distribution of the redundant random ensembles are proved based on a combinatorial argument asymptotic forms of these bounds reveal asymptotic behavior of the average ss weight distributions from these bounds a tradeoff between the number of redundant rows corresponding to decoding complexity of belief propagation on binary erasure channel and the average ss weight distribution corresponding to decoding performance can be derived

speech reception in quiet and in noisy conditions by individuals with noise induced hearing loss in relation to their tone audiogram tone thresholds and speech reception thresholds were measured in 200 individuals 400 ears with noise induced hearing loss the speech reception thresholds were measured in a quiet condition and in noise with a speech spectrum at levels of 35 50 65 and 80 dba the tone audiograms could be described by three principal components hearing loss in the regions above 3 khz from 1 to 3 khz and below 1 khz the speech thresholds could be described by two components speech reception in quiet and speech reception in noise at 50 80 dba hearing loss above 1 khz was related to speech reception in noise hearing loss at and below 1 khz to speech reception in quiet the correlation between the speech thresholds in quiet and in noise was only r 0 45 an adequate predictor of the speech threshold in noise the primary factor in the hearing handicap was the pure tone average at 2 and 4 khz pta2 4 r 0 72 the minimum value of the prediction error for any tone audiometric predictor of this speech threshold was 1 2 db standard deviation the prediction could not be improved by taking into account the critical ratio for low frequency noise nor by its upward spread of masking the prediction error is due to measurement error and to a factor common to both ears the latter factor is ascribed to cognitive skill in speech reception hearing loss above 10 to 15 db hl hearing level already shows an effect on the speech threshold in noise a noticeable handicap is found at pta2 4 30 db hl

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

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

effect of loudness recruitment on the perception of amplitude modulation people with hearing loss of cochlear origin usually display loudness recruitment the rate of growth of loudness level with increasing sound level is greater than for a normally hearing person loudness recruitment has usually been studied with steady sounds of relatively long duration the present study examines how recruitment affects the perception of dynamically varying sounds namely amplitude modulated sinusoids the modulation rates used 4 8 16 and 32 hz were chosen to span the range of the most prominent modulations present in the envelope of speech three subjects with unilateral cochlear hearing loss were used in experiment 1 subjects were required to make loudness matches between 1 khz tones presented alternately to the two ears this was done over a wide range of sound levels experiment 2 used 1 khz carriers that were amplitude modulated the modulation was sinusoidal on a db scale the modulated tones were presented alternately to the two ears and were approximately equally loud in the two ears the modulation depth was fixed in one ear and the subject was required to adjust the modulation depth in the other ear so that the modulation depth appeared equal in the two ears this was done for a range of modulation depths and with the fixed tone presented to both the normal and the impaired ear a given modulation depth in the impaired ear was matched by a greater modulation depth in the normal ear to a first approximation the modulation matching functions were independent of modulation rate furthermore the functions could be predicted reasonably well from the loudness matching results of experiment 1 obtained with steady tones the results are consistent with the idea that loudness recruitment results from the loss of a fast acting compressive nonlinearity that operates in the normal peripheral auditory system possible implications of the results for the use of fast acting compression in hearing aids are discussed

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

evaluation of near end speech enhancement under equal loudness constraint for listeners with normal hearing and mild to moderate hearing loss four algorithms designed to enhance the intelligibility of speech when noise is added after processing were evaluated under the constraint that the speech should have the same loudness before and after processing as determined using a loudness model the algorithms applied spectral modifications and two of them included dynamic range compression on average the methods with dynamic range compression required the least level adjustment to equate loudness for the unprocessed and processed speech subjects with normal hearing experiment 1 and mild to moderate hearing loss experiment 2 were tested using unmodified and enhanced speech presented in speech shaped noise ssn and a competing speaker cs the results showed a the algorithms with dynamic range compression yielded the largest intelligibility gains in both experiments and for both types of background b the algorithms without dynamic range compression either yielded benefit only with the ssn or yielded no consistent benefit c speech rec

tree structure expectation propagation for ldpc decoding over the bec we present the tree structure expectation propagation tree ep algorithm to decode low density parity check ldpc codes over discrete memoryless channels dmcs expectation propagation generalizes belief propagation bp in two ways first it can be used with any exponential family distribution over the cliques in the graph second it can impose additional constraints on the marginal distributions we use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the ldpc code s tanner graph thanks to these additional constraints the tree ep marginal estimates for each variable in the graph are more accurate than those provided by bp we also reformulate the tree ep algorithm for the binary erasure channel bec as a peeling type algorithm tep and we show that the algorithm has the same computational complexity as bp and it decodes a higher fraction of errors we describe the tep decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters the solution of these equations is used to predict the tep decoder performance in both the asymptotic regime and the finite length regimes over the bec while the asymptotic threshold of the tep decoder is the same as the bp decoder for regular and optimized codes we propose a scaling law for finite length ldpc codes which accurately approximates the tep improved performance and facilitates its optimization

multiple bases belief propagation decoding of high density cyclic codes we introduce a new method for decoding short and moderate length linear block codes with dense parity check matrix representations of cyclic form this approach is termed multiple bases belief propagation the proposed iterative scheme makes use of the fact that a code has many structurally diverse parity check matrices capable of detecting different error patterns we show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief propagation decoding furthermore we describe how to choose sets of parity check matrices of cyclic form amenable for multiple bases decoding based on analytical studies performed for the binary erasure channel for several cyclic and extended cyclic codes the multiple bases belief propagation decoding performance can be shown to closely follow that of the maximum likelihood decoder

parity check matrices separating erasures from errors most decoding algorithms of linear codes in general are designed to correct or detect errors however many channels cause erasures in addition to errors in principle decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code for any given linear code and any given maximum number of correctable erasures parity check matrices are introduced that yield parity check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures these matrices allow for the separation of erasures from errors to facilitate decoding several constructions of such separating parity check matrices are presented to reduce decoding complexity separating parity check matrices with small number of rows are preferred the minimum number of rows in a parity check matrix separating a given maximum number of erasures is called the separating redundancy upper and lower bounds on the separating redundancies are derived in particular it is shown that the separating redundancies tend to grow linearly with the number of rows in full rank parity check matrices of codes the separating redundancies of some classes of codes are determined for some maximum numbers of erasures

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

complete enumeration of stopping sets of full rank parity check matrices of hamming codes stopping sets and in particular their numbers and sizes play an important role in determining the performance of iterative decoders of linear codes over binary erasure channels in the 2004 shannon lecture mceliece presented an expression for the number of stopping sets of size three for a full rank parity check matrix of the hamming code in this correspondence we derive an expression for the number of stopping sets of any given size for the same parity check matrix

successive maximization for systematic design of universally capacity approaching rate compatible sequences of ldpc code ensembles over binary input output symmetric memoryless channels a systematic construction of capacity achieving low density parity check ldpc code ensemble sequences over the binary erasure channel bec has been proposed by saeedi et al based on a method here referred to as successive maximization sm in sm the fraction of degree i nodes are successively maximized starting from i 2 with the constraint that the ensemble remains convergent over the channel in this paper we propose sm to design universally capacity approaching rate compatible ldpc code ensemble sequences over the general class of binary input output symmetric memoryless biosm channels this is achieved by first generalizing the sm method to other biosm channels to design a sequence of capacity approaching ensembles called the parent sequence the sm principle is then applied to each ensemble within the parent sequence this time to design rate compatible puncturing schemes as part of our results we extend the stability condition which was previously derived for degree 2 variable nodes to other variable node degrees as well as to the case of rate compatible codes consequently we prove that using the sm principle one is able to design universally capacity achieving rate compatible ldpc code ensemble sequences over the bec unlike the previous results in the literature the proposed sm approach is naturally extendable to other biosm channels the performance of the rate compatible schemes designed based on our method is comparable to those designed by optimization

more on the stopping and minimum distances of array codes for q an odd prime and 1 m q two specific binary qm q2 parity check matrices denoted by hp m q and hi m q are considered the corresponding binary codes cp m q and ci m q respectively are called proper and improper array codes with parameters m and q given a parity check matrix h representing a binary code c let s h denote the stopping distance of h and d c be the minimum hamming distance of c it is known that that s hi m q s hp m q d ci m q d cp m q for m 3 in this paper we show that these equalities do not hold for all values of m and q in particular although s hp 4 7 d cp 4 7 8 we have s hi 4 7 9 and d ci 4 7 10 it is also shown that s hp 5 1 dcp 5 11 10 while s hi 5 11 11 and d ci 5 11 12 this suggests that in many cases the improper array codes would perform better than the proper array codes over the awgn and binary erasure channels performance results are given which confirm this claim the combinatorial structure of the eight element stopping sets for h m 4 q 5 is also determined

transactions papers evaluation and design of irregular ldpc codes using ace spectrum the construction of finite length irregular ldpc codes with low error floors is currently an attractive research problem in particular for the binary erasure channel bec the problem is to find the elements of selected irregular ldpc code ensembles with the size of their minimum stopping set being maximized due to the lack of analytical solutions to this problem a simple but powerful heuristic design algorithm the approximate cycle extrinsic message degree ace constrained design algorithm has recently been proposed building upon the ace metric associated with a cycle in a code graph we introduce the ace spectrum of ldpc codes as a useful tool for evaluation of codes from selected irregular ldpc code ensembles using the ace spectrum we generalize the ace constrained design algorithm making it more flexible and efficient we justify the ace spectrum approach through examples and simulation results

analysis of message passing decoding of finite length concatenated codes we analyze the performance of message passing decoding of finite length concatenated codes we first show that the message passing decoder is closely related to a dual optimization decoder the connections between these two decoders are further elucidated by proving that both of them attain the same objective function value of a generalized linear programming decoder in the limit as the signal to noise ratio snr goes to infinity consequently the framework of pseudo weight analysis which was originally proposed for analyzing the linear programming decoder can be extended to analyze the performance of the message passing decoder for finite length codes we then derive lower bounds to the pseudo weights of general concatenated codes by utilizing the special structure of their parity check matrices we finally present a method to increase the max fractional weight by adding redundant parity check constraints and thereby improving the decoding performance simulation studies are carried out to assess the performance of the proposed algorithms and substantiate the theoretic claims

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

permutation decoding and the stopping redundancy hierarchy of cyclic and extended cyclic codes we introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel we derive lower and upper bounds for the stopping redundancy hierarchy via lovasz s local lemma lll and bonferroni type inequalities and specialize them for codes with cyclic parity check matrices based on the observed properties of parity check matrices with good stopping redundancy characteristics we develop a novel decoding technique termed automorphism group decoding that combines iterative message passing and permutation decoding we also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size simulation results demonstrate that for a large number of algebraic codes the performance of the new decoding method is close to that of maximum likelihood ml decoding

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

rate gains in block coded modulation systems with interblock memory this paper examines the performance gains achievable by adding interblock memory to and altering the mapping of coded bits to symbols in block coded modulation systems the channel noise considered is additive gaussian and the twin design goals are to maximize the asymptotic coding gain and to minimize the number of apparent nearest neighbors in the case of the additive white gaussian noise channel these goals translate into the design of block codes of a given weighted or normalized distance whose rate is as high as possible and whose number of codewords at minimum normalized distance is low the effect of designing codes for normalized distance rather than hamming distance is to ease the problem of determining the best codes for given parameters in the cases of greatest interest and many such best codes are given

stopping sets of algebraic geometry codes stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code let c be an n k linear code over fq with parity check matrix h where the rows of h may be dependent let n 1 2 n denote the set of column indices of h a stopping set s of c with parity check matrix h is a subset of n such that the restriction of h to s does not contain a row of weight 1 the stopping set distribution ti h i 0n enumerates the number of stopping sets with size i of c with parity check matrix h denote h the parity check matrix consisting of all the nonzero codewords in the dual code c in this paper we study stopping sets and stopping set distributions of some residue algebraic geometry ag codes with parity check matrix h first we give two descriptions of stopping sets of residue ag codes for the simplest ag codes i e the generalized reed solomon codes it is easy to determine all the stopping sets then we consider the ag codes from elliptic curves we use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets then the stopping sets the stopping set distribution and the stopping distance of the ag code from an elliptic curve are reduced to the search counting and decision versions of the subset sum problem in the group of rational points of the elliptic curve respectively finally for some special cases we determine the stopping set distributions of the ag codes from elliptic curves

on the stopping distance of array code parity check matrices for q an odd prime and 1 les m les q we study two binary qm times q2 parity check matrices for binary array codes for both parity check matrices we determine the stopping distance and the minimum distance of the associated code for 2 les m les 3 and for m q 4 5 in the case m q 4 7 the stopping distance and the related minimum distance are also determined for one of the given parity check matrices moreover we give a lower bound on the stopping distances for m 3 and q 3

stopping sets of hermitian codes combinatorial structures called stopping sets are useful in analyzing the performance of a linear code when coupled with an iterative decoding algorithm over an erasure channel in this paper we consider stopping sets of hermitian codes

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

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

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

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

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

on the stopping distance and the stopping redundancy of codes it is now well known that the performance of a linear code copf under iterative decoding on a binary erasure channel and other channels is determined by the size of the smallest stopping set in the tanner graph for copf several recent papers refer to this parameter as the stopping distance s of copf this is somewhat of a misnomer since the size of the smallest stopping set in the tanner graph for copf depends on the corresponding choice of a parity check matrix it is easy to see that s les d where d is the minimum hamming distance of copf and we show that it is always possible to choose a parity check matrix for copf with sufficiently many dependent rows such that s d we thus introduce a new parameter the stopping redundancy of copf defined as the minimum number of rows in a parity check matrix h for copf such that the corresponding stopping distance s h attains its largest possible value namely s h d we then derive general bounds on the stopping redundancy of linear codes we also examine several simple ways of constructing codes from other codes and study the effect of these constructions on the stopping redundancy specifically for the family of binary reed muller codes of all orders we prove that their stopping redundancy is at most a constant times their conventional redundancy we show that the stopping redundancies of the binary and ternary extended golay codes are at most 34 and 22 respectively finally we provide upper and lower bounds on the stopping redundancy of mds codes

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 pseudoweight and minimum pseudocodewords of ldpc codes in this correspondence we study the minimum pseudoweight and minimum pseudocodewords of low density parity check ldpc codes under linear programming lp decoding first we show that the lower bound of kelley sridhara xu and rosenthal on the pseudoweight of a nonzero pseudocodeword of an ldpc code whose tanner graph has girth greater than is tight if and only if this pseudocodeword is a real multiple of a codeword then the lower bound of kashyap and vardy on the stopping distance of an ldpc code is proved to be also a lower bound on the pseudoweight of a nonzero pseudocodeword of an ldpc code whose tanner graph has girth and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword using these results we further obtain that for some ldpc codes there are no other minimum pseudocodewords except the real multiples of minimum weight codewords this means that the lp decoding for these ldpc codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of lp decoding and maximum likelihood decoding approaches as the signal to noise ratio snr tends to infinity finally some ldpc codes are listed to illustrate these results

finding all small error prone substructures in ldpc codes it is proven in this work that it is np complete to exhaustively enumerate small error prone substructures in arbitrary finite length low density parity check ldpc codes two error prone patterns of interest include stopping sets for binary erasure channels becs and trapping sets for general memoryless symmetric channels despite the provable hardness of the problem this work provides an exhaustive enumeration algorithm that is computationally affordable when applied to codes of practical short lengths n ap 500 by exploiting the sparse connectivity of ldpc codes the stopping sets of size les 13 and the trapping sets of size les11 can be exhaustively enumerated the central theorem behind the proposed algorithm is a new provably tight upper bound on the error rates of iterative decoding over becs based on a tree pruning technique this upper bound can be iteratively sharpened until its asymptotic order equals that of the error floor this feature distinguishes the proposed algorithm from existing non exhaustive ones that correspond to finding lower bounds of the error floor the upper bound also provides a worst case performance guarantee that is crucial to optimizing ldpc codes when the target error rate is beyond the reach of monte carlo simulation numerical experiments on both randomly and algebraically constructed ldpc codes demonstrate the efficiency of the search algorithm and its significant value for finite length code optimization

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

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

on parity check collections for iterative erasure decoding that correct all correctable erasure patterns of a given size recently there has been interest in the construction of small parity check sets for iterative decoding of the hamming code with the property that each uncorrectable or stopping set of size three is the support of a codeword and hence uncorrectable anyway here we reformulate and generalize the problem and improve on this construction we show that a parity check collection that corrects all correctable erasure patterns of size m for the hamming code with codimension r provides in fact for all codes of codimension r a corresponding generic parity check collection with this property this leads in a natural way to a necessary and sufficient condition for such generic parity check collections we use this condition to construct a generic parity check collection for codes of codimension r correcting all correctable erasure patterns of size at most m for all r and mlesr thus generalizing the known construction for m 3 then we discuss optimality of our construction and show that it can be improved for mges3 and r large enough finally we discuss some directions for further research

efficient algorithm for finding dominant trapping sets of ldpc codes this paper presents an efficient algorithm for finding the dominant trapping sets of a low density parity check ldpc code the algorithm can be used to estimate the error floor of ldpc codes or as a tool to design ldpc codes with low error floors for regular codes the algorithm is initiated with a set of short cycles as the input for irregular codes in addition to short cycles variable nodes with low degree and cycles with low approximate cycle extrinsic message degree ace are also used as the initial inputs the initial inputs are then expanded recursively to dominant trapping sets of increasing size at the core of the algorithm lies the analysis of the graphical structure of dominant trapping sets and the relationship of such structures to short cycles low degree variable nodes and cycles with low ace the algorithm is universal in the sense that it can be used for an arbitrary graph and that it can be tailored to find a variety of graphical objects such as absorbing sets and zyablov pinsker trapping sets known to dominate the performance of ldpc codes in the error floor region over different channels and for different iterative decoding algorithms simulation results on several ldpc codes demonstrate the accuracy and efficiency of the proposed algorithm in particular the algorithm is significantly faster than the existing search algorithms for dominant trapping sets

on the covering radius of z sub 4 codes and their lattices in this correspondence we investigate the covering radius of codes over z sub 4 for the lee and euclidean distances in relation with those of binary nonlinear codes and lattices obtained by the gray map and construction a sub 4 respectively we give several upper and lower bounds on covering radii including z sub 4 analogs of the sphere covering bound the packing radius bound the delsarte bound and the redundancy bound we show that any euclidean optimal type ii code of length 24 has covering radius 8 with respect to the euclidean distance we determine the covering radius of the klemm codes with respect to the lee distance we derive lower bounds on the covering radii of the niemeier lattices

tree based construction of ldpc codes having good pseudocodeword weights we present a tree based construction of low density parity check ldpc codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance and perform well with iterative decoding the construction involves enumerating a d regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal latin squares to close the tree methods are presented for degrees d ps and d ps 1 for p a prime one class corresponds to the well known finite geometry and finite generalized quadrangle ldpc codes the other codes presented are new we also present some bounds on pseudocodeword weight for p ary ldpc codes treating these codes as p ary ldpc codes rather than binary ldpc codes improves their rates minimum distances and pseudocodeword weights thereby giving a new importance to the finite geometry ldpc codes where p 2

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

arbitrary lagrangian eulerian and free surface methods in fluid mechanics abstract a computational procedure is developed to solve problems of viscous incompressible flows under large free surface motions the arbitrary lagrangian eulerian ale method is used to move the free surface nodes as well as the internal nodes the coupling of the mesh motion equations and the fluid equations is essentially done through the free surface boundary conditions an elastic model with free surface boundary conditions is used to model the mesh motion the boundary condition at the free surface is deduced from the fact that no fluid particle can cross the free surface the fluid problem is solved using a galerkin least square gls finite element formulation

on enhancement of spectral contrast in speech for hearing impaired listeners a digital processing method is described for altering spectral contrast the difference in amplitude between spectral peaks and valleys in natural utterances speech processed with programs implementing the contrast alteration procedure was presented to listeners with moderate to severe sensorineural hearing loss the task was a three alternative b d or g stop consonant identification task for consonants at a fixed location in short nonsense utterances overall tokens with enhanced contrast showed moderate gains in percentage correct stop consonant identification when compared to unaltered tokens conversely reducing spectral contrast generally reduced percent correct stop consonant identification contrast alteration effects were inconsistent for utterances containing d the observed contrast effects also interacted with token intelligibility

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

results on parity check matrices with optimal stopping and or dead end set enumerators the performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying tanner graph or equivalently the parity check matrix representing the code in this correspondence we introduce the notion of dead end sets to explicitly demonstrate this dependency the choice of the parity check matrix entails a tradeoff between performance and complexity we give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity check matrices further we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator

single exclusion number and the stopping redundancy of mds codes for a linear block code c its stopping redundancy is defined as the smallest number of check nodes in a tanner graph for c such that there exist no stopping sets of size smaller than the minimum distance of c schwartz and vardy conjectured that the stopping redundancy of a maximum distance separable mds code should only depend on its length and minimum distance we define the n t single exclusion number s n t as the smallest number of i subsets of an n set such that for each i subset of the n set i 1 t 1 there exists a i subset that contains all but one element of the i subset new upper bounds on the single exclusion number are obtained via probabilistic methods recurrent inequalities as well as explicit constructions the new bounds are used to better understand the stopping redundancy of mds codes in particular it is shown that for n k n d 1 d mds codes as n rarr infin the stopping redundancy is asymptotic to s n d 2 if d o radic n or if k o radic n k rarr infin thus giving partial confirmation of the schwartz vardy conjecture in the asymptotic sense

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

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

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

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