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

an efficient semiparametric estimator for binary response models this paper proposes an estimator for discrete choice models that makes no assumption concerning the functional form of the choice probability function where this function can be characterized by an index the estimator is shown to be consistent asymptotically normally distributed and to achieve the semiparametric efficiency bound monte carlo evidence indicates that there may be only modest efficiency losses relative to maximum likelihood estimation when the distribution of the disturbances is known and that the small sample behavior of the estimator in other cases is good

a transform theory for a class of group invariant codes a binary cyclic code of length of n can be defined by a set of parity check equations that is invariant under the additive cyclic group generated by a i to i 1 modulo n a class of quasicyclic codes that have parity check equations invariant under the group generated by a subgroup of a and a subgroup of the multiplicative group m i to 2 sup k i modulo n is studied in detail the fourier transform transforms the action of the additive group and a linearized polynomial transform transforms the action of the multiplicative group the form of the transformed code equations permits a bch like bound on the minimum distance of the code the techniques are illustrated by the construction of several codes that are decodable by the berlekamp massey algorithm and that have parameters that approach or surpass those of the best codes known

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

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

semidefinite extreme points of the unit ball in a polynomial space abstract let be the triangle in r 2 bounded by the lines x 0 y 0 x y 1 the standard simplex in r 2 denote by 2 the set of all real bivariate algebraic polynomials of total degree at most two let b be the unit ball of the space 2 endowed with the supremum norm on we give a full description of the semidefinite extreme points of b the present paper completes the description of the set e of all extreme points of b started in milev and naidenov 2008 13 and milev and naidenov 2011 14 we study as an application the path connectedness of e the conclusion is that e 1 consists of two path connected components

breaking the curse of dimensionality in conditional moment inequalities for discrete choice models abstract this paper studies inference of preference parameters in semiparametric discrete choice models when these parameters are not point identified and the identified set is characterized by a class of conditional moment inequalities exploring the semiparametric modeling restrictions we show that the identified set can be equivalently formulated by moment inequalities conditional on only two continuous indexing variables such formulation holds regardless of the covariate dimension thereby breaking the curse of dimensionality for nonparametric inference based on the underlying conditional moment inequalities we further apply this dimension reducing characterization approach to the monotone single index model and to a variety of semiparametric models under which the sign of conditional expectation of a certain transformation of the outcome is the same as that of the indexing variable

a smoothed maximum score estimator for the binary response model this paper describes a semiparametric estimator for binary response models in which there may be arbitrary heteroskedasticity of unknown form the estimator is obtained by maximizing a smoothed version of the objective function of c manski s maximum score estimator the smoothing procedure is similar to that used in kernel nonparametric density estimation the resulting estimator s rate of convergence in probability is the fastest possible under the assumptions that are made the centered normalized estimator is asymptotically normally distributed methods are given for consistently estimating the parameters of the limiting distribution and for selecting the bandwidth required by the smoothing procedure copyright 1992 by the econometric society

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

distance bounds for periodically time varying and tail biting ldpc convolutional codes existence type lower bounds on the free distance of periodically time varying ldpc convolutional codes and on the minimum distance of tail biting ldpc convolutional codes are derived it is demonstrated that the bound on free distance of periodically time varying ldpc convolutional codes approaches the bound on free distance of general nonperiodic time varying ldpc convolutional codes as the period increases the proof of the bound is based on lower bounding the minimum distance of corresponding tail biting ldpc convolutional codes which is of interest in its own right

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

rank estimators for monotonic index models we present a new class of rank estimators of scaled coefficients in semiparametric monotonic linear index models the estimators require no subjective bandwidth choices and have attractive computational properties we establish n consistency and asymptotic normality and provide the general form and consistent estimators of the asymptotic covariance matrix we also provide a generalization covering single equation multiple indices models satisfying certain monotonicity constraints an analogue of consistency when all explanatory variables are categorical is established and an application is presented

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

geometry of three homogeneous polynomials on real hilbert spaces let h be a two dimensional real hilbert space we give a characterisation of the extreme and the smooth points of the unit ball of the space of three homogeneous polynomials on h in terms of the coefficients of the polynomial we also determine the smooth points of the unit ball of the predual of the space of three homogeneous polynomials

distribution free maximum likelihood estimator of the binary choice model1 is a given function of the exogenous variables z and unknown parameters 9 representing the systematic component of the utility difference and f is the distribution function of the random component of the utility difference this paper describes a method of estimating the parameters 9 without assuming any functional form for the distribution function f and proves that this estimator is consistent f is also consistently estimated the method uses maximum likelihood estimation in which the likelihood is maximized not only over the parameter 9 but also over a space which contains all distribution functions

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

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

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

non parametric analysis of a generalized regression model the maximum rank correlation estimator the paper considers estimation of a model yi d f x i 0 ui where the composite transformation d f is only specified that d r r is non degenerate monotonic and f r2 r is strictly monotonic in each of its variables the paper thus generalizes standard data analysis which assumes that the functional form of d f is known and additive the estimator which it proposes is the maximum rank correlation estimator which is non parametric in the functional form of d f and non parametric in the distribution of the error terms ui the estimator is shown to be strongly consistent for the parameters 0 up to a scale coefficient

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

identification of binary response models abstract this article studies identification of the threshold crossing model of binary response most research on binary response has considered specific estimators and tests the study of identification exposes the foundations of binary response analysis by making explicit the assumptions needed to justify different methods it also clarifies the connections between reduced form and structural analyses of binary response data assume that the binary outcome z is determined by an observable random vector x and by an unobservable scalar u through a model z 1 x u 0 also assume that fu x the probability distribution of u conditional on x is continuous and strictly increasing given these maintained assumptions we investigate the identifiability of given the following restrictions on the distributions fu x x x mean independence quantile independence index sufficiency statistical independence and statistical independence with the distribution known we find that mean independence has n

binary choice models with discrete regressors identification and misspecification this paper explores the inferential question in semiparametric binary response models when the continuous support condition is not satisfied and all regressors have discrete support i focus mainly on the models under the conditional median restriction as in manski 1985 i find sharp bounds on the components of the parameter of interest and outline several applications the formulas for bounds obtained using a recursive procedure help analyze cases where one regressor s support becomes increasingly dense furthermore i investigate asymptotic properties of estimators of the identification set i describe a relation between the maximum score estimation and support vector machines and propose several approaches to address the problem of empty identification sets when the model is misspecified

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

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

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

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

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

bounds on parameters in panel dynamic discrete choice models identification of dynamic nonlinear panel data models is an important and delicate problem in econometrics in this paper we provide insights that shed light on the identification of parameters of some commonly used models using these insights we are able to show through simple calculations that point identification often fails in these models on the other hand these calculations also suggest that the model restricts the parameter to lie in a region that is very small in many cases and the failure of point identification may therefore be of little practical importance in those cases although the emphasis is on identification our techniques are constructive in that they can easily form the basis for consistent estimates of the identified sets

operational characteristics of maximum score estimation abstract this paper reports on the operational characteristics of maximum score estimation of a linear model from binary response data a series of previous articles have shown that in theory the maximum score method makes possible binary response analysis under very weak distributional assumptions here we present evidence on the properties of maximum score estimation in practice after reviewing the known asymptotic theory of maximum score estimation the paper describes an algorithm for maximum score estimation and characterizes its performance then findings from a monte carlo study comparing maximum score and logit maximum likelihood estimation are reported finally the accuracy of bootstrap estimation of maximum score root mean square errors is evaluated

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

the need for structure in quantum ldpc codes the existence of quantum ldpc codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information despite years of research good quantum ldpc codes are not known to exist but at the very least it is known they cannot be defined on very regular topologies like low dimensional grids in this work we establish a complementary result showing that good quantum css codes which are sparsely generated require structure in the local terms that constrain the code space so as not to be too random in a well defined sense to show this we prove a weak converse to a theorem of krasikov and litsyn on weight distributions of classical codes due to which may be of independent interest subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight tantamount to having a non negligible approximate minimal distance while they may not have a large minimal non zero weight they still have very few words of low hamming weight

the unit ball of the complex mathcal p 3h let h be a two dimensional complex hilbert space and mathcal p 3h the space of 3 homogeneous polynomials on h we give a characterization of the extreme points of its unit ball mathsf b mathcal p 3h from which we deduce that the unit sphere of mathcal p 3h is the disjoint union of the sets of its extreme and smooth points we also show that an extreme point of mathsf b mathcal p 3h remains extreme as considered as an element of mathsf b mathcal l 3h finally we make a few remarks about the geometry of the unit ball of the predual of mathcal p 3h and give a characterization of its smooth points

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

estimation of best predictors of binary response abstract this paper investigates analog estimation of predictors of binary response given semiparametric restrictions on the response process results are presented for the isotonic response model and for the weaker single crossing model the isotonic model assumes that the response probability is monotone in a linear function of the conditioning variables in the single crossing model the response probability is above a given value when a linear function of the conditioning variables is positive and below that value otherwise it is found that analog estimators computed under the isotonicity restriction are generally consistent estimation under the single crossing restriction is consistent only for special loss functions

monge ampere measures for convex bodies and bernstein markov type inequalities we use geometric methods to calculate a formula for the complex monge ampere measure dd c v k n for k n c n a convex body and v k its siciak zaharjuta extremal function bedford and taylor had computed this for symmetric convex bodies k we apply this to show that two methods for deriving bernstein markov type inequalities i e pointwise estimates of gradients of polynomials yield the same results for all convex bodies a key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area ellipses determining the complex monge ampere solution v k

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

confidence sets for partially identified parameters that satisfy a finite number of moment inequalities this paper proposes a computationally simple way to construct confidence sets for a parameter of interest in models comprised of moment inequalities building on results from the literature on multivariate one sided tests i show how to test the hypothesis that any particular parameter value is logically consistent with the maintained moment inequalities the associated test statistic has an asymptotic chi bar square distribution and can be inverted to construct an asymptotic confidence set for the parameter of interest even if that parameter is only partially identified critical values for the test are easily computed and a monte carlo study demonstrates implementation and finite sample performance

geometry of homogeneous polynomials on two dimensional real hilbert spaces we describe a method which characterises extreme points of the unit ball of the space of homogeneous polynomials of degrees 3 and 4 on two dimensional hilbert spaces and provides classes of examples of extreme and smooth polynomials of an arbitrary degree n

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

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

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

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

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

inference on regressions with interval data on a regressor or outcome this paper examines inference on regressions when interval data are available on one variable the other variables being measured precisely let a population be characterized by a distribution p y x v v 0 v 1 where y r 1 x r k and the real variables v v 0 v 1 satisfy v 0 v v 1 let a random sample be drawn from p and the realizations of y x v 0 v 1 be observed but not those of v the problem of interest may be to infer e y x v or e v x this analysis maintains interval i monotonicity m and mean independence mi assumptions i p v 0 v v 1 1 m e y x v is monotone in v mi e y x v v 0 v 1 e y x v no restrictions are imposed on the distribution of the unobserved values of v within the observed intervals v 0 v 1 it is found that the immi assumptions alone imply simple nonparametric bounds on e y x v and e v x these assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score mms method the immi assumptions combined with a parametric model for e y x v or e v x yield an identification region that may be estimated consistently by a modified minimum distance mmd method monte carlo methods are used to characterize the finite sample performance of these estimators empirical case studies are performed using interval wealth data in the health and retirement study and interval income data in the current population survey

confidence intervals for partially identified parameters recently a growing body of research has studied inference in settings where parameters of interest are partially identified in many cases the parameter is real valued and the identification region is an interval whose lower and upper bounds may be estimated from sample data for this case confidence intervals cis have been proposed that cover the entire identification region with fixed probability here we introduce a conceptually different type of confidence interval rather than cover the entire identification region with fixed probability we propose cis that asymptotically cover the true value of the parameter with this probability however the exact coverage probabilities of the simplest version of our new cis do not converge to their nominal values uniformly across different values for the width of the identification region to avoid the problems associated with this we modify the proposed ci to ensure that its exact coverage probabilities do converge uniformly to their nominal values we motivate this modified ci through exact results for the gaussian case

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

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

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

non linear regression with discrete explanatory variables with an application to the earnings function abstract in this paper it will be shown that regression models with discrete explanatory variables take the form of a polynominal of a linear function of the regressors a two stage estimation procedure and various model specification tests will be developed in close harmony with an empirical application to the earnings function

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point

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