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

wave like solutions of general 1 d spatially coupled systems we establish the existence of wave like solutions to spatially coupled graphical models which in the large size limit can be characterized by a 1 d real valued state this is extended to a proof of the threshold saturation phenomenon for all such models which includes spatially coupled irregular low density parity check codes over the binary erasure channel bec but also addresses hard decision decoding for transmission over general channels the code division multiple access problem compressed sensing and some statistical physics models for traditional uncoupled iterative coding systems with two components and transmission over the bec the asymptotic convergence behavior is completely characterized by the exit curves of the components in particular the system converges to the desired fixed point which is the one corresponding to perfect decoding if and only if the two exit functions describing the components do not cross for spatially coupled systems whose state is 1 d a closely related graphical criterion applies now the curves are allowed to cross but not by too much more precisely we show that the threshold saturation phenomenon is related to the positivity of the signed area enclosed by two exit like functions associated to the component systems a very intuitive and easy to use graphical characterization in the spirit of exit functions and gaussian approximations we also show how to apply the technique to higher dimensional and even infinite dimensional cases in these scenarios the method is no longer rigorous but it typically gives accurate predictions to demonstrate this application we discuss transmission over general channels using both the belief propagation as well as the min sum decoder

isospectral flows related to frobenius stickelberger thiele polynomials the isospectral deformations of the frobenius stickelberger thiele fst polynomials introduced in spiridonov et al commun math phys 272 139 165 2007 are studied for a specific choice of the deformation of the spectral measure one is led to an integrable lattice fst lattice which is indeed an isospectral flow connected with a generalized eigenvalue problem in the second part of the paper the spectral problem used previously in the study of the modified camassa holm mch peakon lattice is interpreted in terms of the fst polynomials together with the associated fst polynomials resulting in a map from the mch peakon lattice to a negative flow of the finite fst lattice furthermore it is pointed out that the degenerate case of the finite fst lattice unexpectedly maps to the interlacing peakon ode system associated with the two component mch equation studied in chang et al adv math 299 1 35 2016

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

code design for the noisy slepian wolf problem we consider a noisy slepian wolf problem where two correlated sources are separately encoded using codes of fixed rate and transmitted over two independent binary memoryless symmetric channels the capacity of each channel is characterized by a single parameter that is not known at the transmitter system performance is evaluated by computing the set of channel parameters for which the system can successfully decode this set is called the achievable channel parameter region acpr the goal is to design systems whose acprs are as large as possible the main result is the design of irregular low density parity check ldpc ensembles whose acprs are significantly larger than previous designs some previous attempts to achieve large acprs with ldpc codes failed because systematic codes were used in this work we start with systematic encoders but puncture all the systematic bits before transmission we also show that additional gains are possible using a staggered structure which enables codes optimized for single user channels to perform well under symmetric channel conditions the main analysis tool is a generic density evolution framework for the analysis of joint iterative decoding for this problem

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

toda type differential equations for the recurrence coefficients of orthogonal polynomials and freud transformation transformations of the measure of orthogonality for orthogonal polynomials namely freud transformations are considered jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations dynamics of the coefficients are described by generalized toda equations the classical toda lattice equations are the simplest special case of dynamics of the coefficients under the freud transformation of the measure of orthogonality also dynamics of hankel determinants its minors and other notions corresponding to the orthogonal polynomials are studied

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

deformed laguerre hahn orthogonal polynomials on the real line abstract one derives discrete dynamical systems related to laguerre hahn orthogonal polynomials one studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t dependence on the coefficients of the riccati differential equation for the related stieltjes function

orthogonal polynomials and measures on the unit circle the geronimus transformations in this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle this perturbation is the inverse of the christoffel transformation and is called the geronimus transformation we study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated hessenberg matrices finally we show an example of this kind of transformation

inversion of a tridiagonal jacobi matrix abstract a formula for the inverse of a general tridiagonal matrix is given in terms of the principal minors

schur flow for orthogonal polynomials on the unit circle and its integrable discretization a one parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered the corresponding dynamics of the schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi discrete modified kdv equation namely the schur flow a discrete analogue of the miura transformation is found an integrable discretization of the schur flow enables us to compute a pade approximation of the caratheodory functions or equivalently to compute a perron caratheodory continued fraction in a polynomial time

characterizations of δ volterra lattice a symmetric orthogonal polynomials interpretation abstract in this paper we introduce the volterra lattice which is interpreted in terms of symmetric orthogonal polynomials it is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like 1 x 2 1 t x where is a given positive borel measure moreover the volterra lattice is related to the toda lattice from miura or backlund transformations the main ingredients are orthogonal polynomials which satisfy an appell condition with respect to the forward difference operator and the characterization of the point spectrum of a jacobian operator that satisfies a volterra equation lax type theorem we also provide an explicit example of solutions of volterra and toda lattices and connect this example with the results presented in the paper

parity check density versus performance of binary linear block codes new bounds and applications the moderate complexity of low density parity check ldpc codes under iterative decoding is attributed to the sparseness of their parity check matrices it is therefore of interest to consider how sparse parity check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity this issue was addressed by sason and urbanke and it is revisited in this paper the remarkable performance of ldpc codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum likelihood ml decoding and the additional loss which is imposed by the suboptimality of the decoder these issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes and lower bounds on the asymptotic density of their parity check matrices as a function of the gap between their achievable rates and the channel capacity these bounds are valid under ml decoding and hence they are valid for any suboptimal decoding algorithm the new bounds improve on previously reported results by burshtein and by sason and urbanke and they hold for the case where the transmission takes place over an arbitrary memoryless binary input output symmetric mbios channel the significance of these information theoretic bounds is in assessing the tradeoff between the asymptotic performance of ldpc codes and their decoding complexity per iteration under message passing decoding they are also helpful in studying the potential achievable rates of ensembles of ldpc codes under optimal decoding by comparing these thresholds with those calculated by the density evolution technique one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms

dynamics and interpretation of some integrable systems via multiple orthogonal polynomials high order non symmetric difference operators with complex coefficients are considered the correspondence between dynamics of the coefficients of the operator defined by a lax pair and its resolvent function is established the method of investigation is based on the analysis of the moments for the operator the solution of a discrete dynamical system is studied we give explicit expressions for the resolvent function and under some conditions the representation of the vector of functionals associated with the solution for the integrable systems

systematic ldpc convolutional codes asymptotic and finite length anytime properties here we propose an ensemble of non terminated systematic ldpc convolutional codes with increasing memory and show that over the binary erasure channel bec these codes achieve anytime reliability asymptotically when decoded with an expanding window message passing decoder the corresponding anytime exponents are determined through protograph based extrinsic information transfer charts fundamental complications arising when transmitting with finite block lengths are identified and a combinatorial performance analysis when transmitting over a static bec with a fixed number of erasures per codeword block is developed based on the performance analysis we explore the use of feedback for achieving anytime behavior with constraints on block length to meet complexity constraints with or without feedback the code memory can be limited at the cost of an error floor emerging with a delay proportional to the memory constraint although the analysis is developed for a static bec we show numerically that we can design efficient low complexity finite length codes with anytime properties even for the conventional bec

fuzzy multiresolution signal representation we present a multiresolution signal representation technique which uses fuzzy rules a fuzzy tree is used to connect different levels of resolution we apply our representation to several random processes and demonstrate some useful properties of our representation when applied to fractals

tree structured expectation propagation for ldpc decoding over bms channels in this paper we put forward the tree structured expectation propagation tep algorithm for decoding block and convolutional low density parity check codes over any binary channel we have already shown that tep improves belief propagation bp over the binary erasure channel bec by imposing marginal constraints over a set of pairs of variables that form a tree or a forest the tep decoder is a message passing algorithm that sequentially builds a tree forest of erased variables to capture additional information disregarded by the standard bp decoder which leads to a noticeable reduction of the error rate for finite length codes in this paper we show how the tep can be extended to any channel specifically to binary memoryless symmetric bms channels we particularly focus on how the tep algorithm can be adapted for any channel model and more importantly how to choose the tree forest to keep the gains observed for block and convolutional ldpc codes over the bec

windowed decoding of spatially coupled codes spatially coupled codes have been of interest recently owing to their superior performance over memoryless binary input channels the performance is good both asymptotically since the belief propagation thresholds approach the shannon limit as well as for finite lengths since degree 2 variable nodes that result in high error floors can be completely avoided however to realize the promised good performance one needs large blocklengths this in turn implies a large latency and decoding complexity for the memoryless binary erasure channel we consider the decoding of spatially coupled codes through a windowed decoder that aims to retain many of the attractive features of belief propagation while trying to reduce complexity further we characterize the performance of this scheme by defining thresholds on channel erasure rates that guarantee a target erasure rate we give analytical lower bounds on these thresholds and show that the performance approaches that of belief propagation exponentially fast in the window size we give numerical results including the thresholds computed using density evolution and the erasure rate curves for finite length spatially coupled codes

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

approaching the rate distortion limit with spatial coupling belief propagation and decimation we investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled low density generator matrix codes the degree of the check nodes is regular and the one of code bits is poisson distributed with an average depending on the compression rate the performance of a low complexity belief propagation guided decimation algorithm is excellent the algorithmic rate distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows moreover as the check degree grows both curves approach the ultimate shannon rate distortion limit the belief propagation guided decimation encoder is based on the posterior measure of a binary symmetric test channel this measure can be interpreted as a random gibbs measure at a temperature directly related to the noise level of the test channel we investigate the links between the algorithmic performance of the belief propagation guided decimation encoder and the phase diagram of this gibbs measure the phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds in particular the dynamical and condensation phase transition temperatures equivalently test channel noise thresholds are computed we observe that 1 the dynamical temperature of the spatially coupled construction saturates toward the condensation temperature and 2 for large degrees the condensation temperature approaches the temperature i e noise level related to the information theoretic shannon test channel noise parameter of rate distortion theory this provides heuristic insight into the excellent performance of the belief propagation guided decimation algorithm this paper contains an introduction to the cavity method

lebesgue perturbation of a quasi definite hermitian functional the positive definite case abstract in this work we study the problem of orthogonality with respect to a sum of measures or functionals first we consider the case where one of the functionals is arbitrary and quasi definite and the other one is the lebesgue normalized functional next we study the sum of two positive measures the first one is arbitrary and the second one is the lebesgue normalized measure and we obtain some relevant properties concerning the new measure finally we consider the sum of a bernstein szeg measure and the lebesgue measure in this case we obtain more simple explicit algebraic relations as well as the relation between the corresponding szeg s functions

a scaling law to predict the finite length performance of spatially coupled ldpc codes spatially coupled low density parity check sc ldpc codes are known to have excellent asymptotic properties much less is known regarding their finite length performance we propose a scaling law to predict the error probability of finite length spatially coupled code ensembles when transmission takes place over the binary erasure channel we discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters the ultimate goal of this line of research is to develop analytic tools for the design of sc ldpc codes under practical constraints

a characterization of classical and semiclassical orthogonal polynomials from their dual polynomials we study properties of the dual orthogonal polynomials op introduced by de boor and saff and present a new characterization of the classical and semiclassical op

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

difference and differential equations for deformed laguerre hahn orthogonal polynomials on the unit circle sequences of orthogonal polynomials on the unit circle whose caratheodory function satisfies a riccati type differential equation with polynomial coefficients are studied we deduce discrete lax equations which lead to difference equations for the corresponding sequences of reflection parameters and we analyze the continuous differential equations that arise when deformations through a dependence on a parameter t occur

towards the evaluation of the relevant degrees of freedom in nonlinear partial differential equations we investigate an operator renormalization group method to extract and describe the relevant degrees of freedom in the evolution of partial differential equations the proposed renormalization group approach is formulated as an analytical method providing the fundamental concepts of a numerical algorithm applicable to various dynamical systems we examine dynamical scaling characteristics in the short time and the long time evolution regime providing only a reduced number of degrees of freedom to the evolution process

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

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

on universal properties of capacity approaching ldpc code ensembles this paper is focused on the derivation of some universal properties of capacity approaching low density parity check ldpc code ensembles whose transmission takes place over memoryless binary input output symmetric mbios channels properties of the degree distributions graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information theoretic bounds these bounds are expressed in terms of the target block bit error probability and the gap in rate to capacity most of the bounds are general for any decoding algorithm and some others are proved under belief propagation bp decoding proving these bounds under a certain decoding algorithm validates them automatically also under any suboptimal decoding algorithm a proper modification of these bounds makes them universal for the set of all mbios channels which exhibit a given capacity bounds on the degree distributions and graphical complexity apply to finite length ldpc codes and to the asymptotic case of an infinite block length the bounds are compared with capacity approaching ldpc code ensembles under bp decoding and they are shown to be informative and are easy to calculate finally some interesting open problems are considered

block markov superposition transmission construction of big convolutional codes from short codes a construction of big convolutional codes from short codes called block markov superposition transmission bmst is proposed the bmst is very similar to superposition block markov encoding sbme which has been widely used to prove multiuser coding theorems the bmst codes can also be viewed as a class of spatially coupled codes where the generator matrices of the involved short codes referred to as basic codes are coupled the encoding process of bmst can be as fast as that of the basic code while the decoding process can be implemented as an iterative sliding window decoding algorithm with a tunable delay more importantly the performance of bmst can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available numerical results show that 1 the lower bounds can be matched with a moderate decoding delay in the low bit error rate ber region implying that the iterative sliding window decoding algorithm is near optimal 2 bmst with repetition codes and single parity check codes can approach the shannon limit within 0 5 db at the ber of 10 5 for a wide range of code rates and 3 bmst can also be applied to nonlinear codes

asymptotic analysis and spatial coupling of counter braids a counter braid cb is a novel counter architecture introduced by lu et al in 2007 for per flow measurements on high speed links which can be decoded with low complexity using message passing mp cbs achieve an asymptotic compression rate under optimal decoding that matches the entropy lower bound of the flow size distribution in this paper we apply the concept of spatial coupling to cbs to improve the performance of the original cbs and analyze the performance of the resulting spatially coupled cbs sc cbs we introduce an equivalent bipartite graph representation of cbs with identical iteration by iteration finite length and asymptotic performance based on this equivalent representation we then analyze the asymptotic performance of single layer cbs and sc cbs under the mp decoding algorithm proposed by lu et al in particular we derive the potential threshold of the uncoupled system and show that it is equal to the area threshold we also derive the maxwell decoder for cbs and prove that the potential threshold is an upper bound on the maxwell decoding threshold which in turn is a lower bound on the maximum a posteriori map decoding threshold we then show that the area under the extended mp extrinsic information transfer curve defined for the equivalent graph computed for the expected residual cb graph when a peeling decoder equivalent to the mp decoder stops is equal to zero precisely at the area threshold this combined with the analysis of the maxwell decoder and simulation results leads us to the conjecture that the potential threshold is in fact equal to the maxwell decoding threshold and hence a lower bound on the map decoding threshold interestingly sc cbs do not show the well known phenomenon of threshold saturation of the mp decoding threshold to the potential threshold characteristic of spatially coupled low density parity check codes and other coupled systems however sc cbs yield better mp decoding thresholds than their uncoupled counterparts finally we also consider sc cbs as a compressed sensing scheme and show that low undersampling factors can be achieved

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

optimized design of finite length separable circulant based spatially coupled codes an absorbing set based analysis in this paper we characterize the finite length performance of separable circulant based spatially coupled scb sc ldpc codes for transmission over the additive white gaussian noise channel for a general class of finite length graph based codes it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime we first present the mathematical conditions for the existence of absorbing sets in binary scb sc codes this analysis enables us to find the exact number of absorbing sets as a function of the design parameters in particular our results show that the choice of the cutting vector affects the number of absorbing sets and therefore the error floor performance of the code for a fixed column weight we find provably optimal cutting vectors that result in the least number of absorbing sets furthermore we extend our analysis to nonbinary scb sc codes where we show that the choice of the cutting vector is not as critical as in the binary case we provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary scb sc codes by informed selection of edge labels our simulation results show the superior error floor performance of our designed binary and nonbinary scb sc codes compared with binary unstructured and nonbinary quasi cyclic sc codes available in the open literature

moment modification multipeakons and nonisospectral generalizations abstract firstly a formal correspondence is established between the camassa holm ch equation and a two component modified ch or called sqq equation according to the method of moment modification for multipeakon formulae secondly based on the generalized nonisospectral ch equation in chang et al 2014 14 and the interlacing multipeakons of the two component modified ch equation in chang et al 2016 15 we propose a new generalized two component modified ch equation with two parameters which possesses a nonisospectral lax pair the proposed equation still admits multipeakon solutions of explicit and closed form sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented last of all as a by product a generalized nonisospectral modified ch equation is deduced together with its lax pair

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

searching for voltage graph based ldpc tailbiting codes with large girth the relation between parity check matrices of quasi cyclic qc low density parity check ldpc codes and biadjacency matrices of bipartite graphs supports searching for powerful ldpc block codes using the principle of tailbiting compact representations of bipartite graphs based on convolutional codes can be found bounds on the girth and the minimum distance of ldpc block codes constructed in such a way are discussed algorithms for searching iteratively for ldpc block codes with large girth and for determining their minimum distance are presented constructions based on all one matrices steiner triple systems and qc block codes are introduced finally new qc regular ldpc block codes with girth up to 24 are given

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

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

bilayer ldpc convolutional codes for decode and forward relaying in this paper we present bilayer ldpc convolutional codes for half duplex relay channels two types of codes bilayer expurgated ldpc convolutional codes and bilayer lengthened ldpc convolutional codes are proposed for decode and forward df relaying in the case of the binary erasure relay channel we prove analytically that both code constructions achieve the capacities of the source relay link and the source destination link simultaneously provided that the channel conditions are known when designing the codes meanwhile both codes enable the highest transmission rate possible with df relaying for a wide range of channel parameters in addition the regular degree distributions can easily be computed from the channel parameters which significantly simplifies the code optimization the code construction and performance analysis are extended to the general binary memoryless symmetric channel where a capacity achieving performance is conjectured numerical results are provided for both types of codes with finite node degrees over binary erasure channels and binary input additive white gaussian noise channels which verify the aforementioned theoretical analysis

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

pade approximants and complex high order toda lattices tridiagonal operators with complex coefficients are considered the correspondence between dynamics of high order toda equations for the coefficients of the operator and its resolvent function is established it gives a method to solve an inverse problem integration of high order toda equations with complex initial data based on pade approximants and continued fractions for the resolvent function

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

a new class of multiple rate codes based on block markov superposition transmission the hadamard transform ht over the binary field provides a natural way to implement multiple rate codes referred to as ht coset codes where the code length n 2 p is fixed but the code dimension k can be varied from 1 to n 1 by adjusting the set of frozen bits the ht coset codes including reed muller rm codes and polar codes as typical examples can share the same fundamental encoder decoder architecture with the implementation complexity of order o n log n however to guarantee that all codes with designated rates perform well ht coset coding usually requires a sufficiently large code length which in turn causes difficulties in the determination of which bits are better for being frozen in this paper we propose to transmit short ht coset codes in the so called block markov superposition transmission bmst manner at the transmitter signals are spatially coupled via superposition resulting in long codes at the receiver these coupled signals are recovered by a sliding window iterative soft successive cancellation decoding algorithm most importantly the performance around or below the bit error rate ber of 10 5 can be predicted by a simple genie aided lower bound both these bounds and simulation results show that the bmst of short ht coset codes performs well within one db away from the corresponding shannon limits in a wide range of code rates

on differential equations for orthogonal polynomials on the unit circle in this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding caratheodory function satisfies a riccati differential equation with polynomial coefficients in terms of second order matrix differential equations in the semi classical case a characterization in terms of second order linear differential equations with polynomial coefficients is deduced

szegő difference equations transfer matrices and orthogonal polynomials on the unit circle we develop the theory of orthogonal polynomials on the unit circle based on the szeg recurrence relations written in matrix form the orthogonality measure and c function arise in exactly the same way as weyl s function in the weyl approach to second order linear differential equations on the half line the main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one dimensional schrodinger operators discrete and continuous and the notion of subordinacy from the gilbert pearson theory we study the relations between transfer matrices and the structure of orthogonality measures the theory is illustrated by the szeg equations with reflection coefficients having bounded variation

on the full kostant toda system and the discrete korteweg de vries equations abstract the relation between the solutions of the full kostant toda lattice and the discrete korteweg de vries equation is analyzed a method for constructing solutions of these systems is given as a consequence of the matricial interpretation of this method the transform of darboux is extended for general hessenberg banded matrices

matrix toda and volterra lattices abstract we consider matrix toda and volterra lattice equations and their relation with matrix biorthogonal polynomials from that relation we give a method for constructing a new solution of these systems from another given one an illustrative example is presented

intermittency stochastic growth and phase transition in a simple deterministic partial differential equation with a singular term we show by numerical integration that the discretized version of simple deterministic partial differential equations with singular terms proposed by zhang 1992 exhibit rich spatio temporal behaviour representing a mixture of stochastic and deterministic regimes varying the relative weight b of the singular term we have been able to detect transitions in the global behaviour of the solutions by determining their total width w t in particular we have found intermittent solutions as well as a power law dependence of w w t to infinity on b

the cauchy problem for ut δu u q with q a positive real number the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces in that context it is known as the generalized deterministic kpz equation the paper is concerned with the initial value problem for the equation under the assumption that the initial data function is bounded and continuous results on the existence uniqueness and regularity of solutions are obtained

robust rate compatible punctured ldpc convolutional codes a family of robust rate compatible rc punctured low density parity check convolutional codes ldpc ccs is derived from a time invariant ldpc cc mother code by periodically puncturing encoded bits variable nodes with respect to several criteria 1 ensuring the recoverability of punctured variable nodes 2 minimizing the number of completely punctured cycle trapping sets cpctss and 3 minimizing the number of punctured variable nodes involved in short cycles the influence of 1 and 3 on iterative decoding performance is felt most strongly in the waterfall region of the bit error rate ber curve while 2 has a larger effect in the error floor or high signal to noise ratio snr region we show that the length of the puncturing period is an important parameter when designing high rate punctured codes and moreover that extending the puncturing period can improve the decoding performance and extend the range of compatible rates as examples we obtain families of rc ldpc ccs from several time invariant ldpc cc mother codes with monomial and binomial entries in their polynomial syndrome former matrices

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

distributional equation for laguerre hahn functionals on the unit circle let u be a hermitian linear functional defined in the linear space of laurent polynomials and f its corresponding caratheodory function we establish the equivalence between a riccati differential equation with polynomial coefficients for f zaf bf 2 cf d and a distributional equation for u d au b u 2 c u h l where l is the lebesgue functional and the polynomials b c h are defined in terms of the polynomials a b c d

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

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

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

special solutions of the toda chain and combinatorial numbers we classify all solutions of the restricted toda chain from the ansatz that all moments are expressed in terms of polynomials of some variable we show that each such solution corresponds to the sheffer class orthogonal polynomials it is shown that corresponding hankel determinants are related to some well known combinatorial numbers

rate compatible ldpc convolutional codes achieving the capacity of the bec in this paper we propose a new family of rate compatible regular low density parity check ldpc convolutional codes the construction is based on graph extension i e the codes of lower rates are generated by successively extending the graph of the base code with the highest rate theoretically the proposed rate compatible family can cover all the rational rates from 0 to 1 in addition the regularity of degree distributions simplifies the code optimization we prove analytically that all the ldpc convolutional codes of different rates in the family are capable of achieving the capacity of the binary erasure channel bec the analysis is extended to the general binary memoryless symmetric channel for which a capacity approaching performance can be achieved analytical thresholds and simulation results for finite check and variable node degrees are provided for both becs and binary input additive white gaussian noise channels the results confirm that the decoding thresholds of the rate compatible codes approach the corresponding shannon limits over both channels

bounds on the error probability of block codes over the q ary erasure channel in this paper tight bounds on the block error probability of linear block codes over order q finite fields for the q ary erasure channel under maximum likelihood ml decoding are developed upper bounds are obtained for uniform parity check ensembles sparse parity check ensembles general parity check ensembles e g gallager regular nonbinary low density parity check ensembles and for any given linear code with known distance spectrum the tightness of the upper bounds is confirmed both by the comparison with simple lower bounds and for gallager low density parity check ensembles by extensive monte carlo simulations exploiting the derived bounds it is shown how already for short blocks and small q 2 sparse ensembles attain block error probabilities close to those of idealized maximum distance separable mds codes down to low error probabilities whereas in the same regime binary codes show visible losses with respect to the singleton bound thanks to the accurate performance estimates the developed bounds can support the design of near optimum erasure correcting codes with short and moderate lengths

elliptic solutions of the restricted toda chain lame polynomials and generalization of the elliptic stieltjes polynomials we construct new families of elliptic solutions of the restricted toda chain the main tool is a special so called stieltjes ansatz for the moments of corresponding orthogonal polynomials we show that the moments thus obtained are related to three types of lam polynomials the corresponding orthogonal polynomials can be considered as a generalization of the stieltjes carlitz elliptic polynomials

complex spatiotemporal patterns in two lattice models with instability abstract in this paper we consider two lattice models i the discretized version of a simple nonlinear partial differential equation pde with a singular term and ii a coupled map lattice cml with unstable coupling both systems are defined by simple iteration rules involving the nearest neighbour grid points of the corresponding one dimensional lattice we show numerically that these models result in an apparently stochastic spatiotemporal behaviour and exhibit interesting regimes as the function of the parameters of the models in particular we find that by varying the relative weight b of the singular term in the pde the total width of the solution scales with b the cml leads to an intermittent behaviour between structured and turbulent regimes

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

on polynomials associated with an uvarov modification of a quartic potential freud like weight in this contribution we consider sequences of monic polynomials orthogonal with respect to the standard freud like inner product involving a quartic potential p q r p x q x e x 4 2 t x 2 d x m p 0 q 0 we analyze some properties of these polynomials such as the ladder operators and the holonomic equation that they satisfy and as an application we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field it is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation finally an equation of motion for their zeros in terms of their dependence on t is given

deterministic and stochastic surface growth with generalized nonlinearity the scaling behavior of the interface width for the generalized kardar parisi zhang kpz equation ensuremath partial h ensuremath partial t ensuremath nu mathrm ensuremath nabla 2 h ensuremath lambda ensuremath vert ensuremath nabla h mathrm ensuremath vert mathrm ensuremath mu ensuremath eta x t is studied in two dimensions as a function of ensuremath mu with and without the additive noise term ensuremath eta in the case of additive noise the scaling of the surface width is found for all ensuremath mu to be the same as for the ordinary kpz equation ensuremath mu 2 in contrast to a previous conjecture this appears to be due to the combined action of the nonlinearity ensuremath vert ensuremath nabla h mathrm ensuremath vert mathrm ensuremath mu and the noise under renormalization which together induce a ensuremath vert ensuremath nabla h mathrm ensuremath vert 2 term for the deterministic case corresponding to the smoothing of an initially rough interface with roughness exponent ensuremath alpha good agreement with the scaling relation z min 2 ensuremath alpha 1 ensuremath mu ensuremath mu is obtained for ensuremath mu ensuremath ge 1 however for ensuremath mu 1 an instability is observed which leads to a fluctuating grooved surface for an asymptotically large system the roughness exponent is 1 and the growth exponent is approximately equal to 1 2 the evolution of two slightly different surfaces is studied in order to determine a lyapunov exponent characterizing this instability

perturbations on the subdiagonals of toeplitz matrices let l be an hermitian linear functional defined on the linear space of laurent polynomials it is very well known that the gram matrix of the associated bilinear functional in the linear space of polynomials is a toeplitz matrix in this contribution we analyze some linear spectral transforms of l such that the corresponding toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial toeplitz matrix we focus our attention in the analysis of the quasi definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one