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

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

design of finite length irregular protograph codes with low error floors over the binary input awgn channel using cyclic liftings we propose a technique to design finite length irregular low density parity check ldpc codes over the binary input additive white gaussian noise awgn channel with good performance in both the waterfall and the error floor region the design process starts from a protograph which embodies a desirable degree distribution this protograph is then lifted cyclically to a certain block length of interest the lift is designed carefully to maximize the components of the approximate cycle extrinsic message degree ace spectrum of the code s tanner graph in a greedy fashion as a consequence the designed code would perform well in the error floor region moreover the proposed construction results in quasi cyclic codes which are attractive in practice due to simple encoder and decoder implementation simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions

statistical inference with data dependent treatment allocation rules abstract in comparing two treatments with dichotomous responses the randomized play the winner rule wei and durham 1978 tends to assign more study subjects to the better treatment for ethical reasons this property is desirable for studies on human subjects the randomized play the winner rule which is a modification of zelen s play the winner rule zelen 1969 is not deterministic and is less vulnerable to experimental bias than other adaptive designs recently this design has been used in a trial to evaluate extracorporeal membrane oxygenation ecmo for treating newborns with respiratory failures at the university of michigan in this article exact conditional exact unconditional and approximate confidence intervals for the treatment difference are studied from a frequentist point of view with the randomized play the winner rule for small and moderate sized trials the exact unconditional procedures perform much better than the conditional ones because of the adaptive nature of the designs

immigrated urn models theoretical properties and applications urn models have been widely studied and applied in both scientific and social science disciplines in clinical studies the adoption of urn models in treatment allocation schemes has proved to be beneficial to researchers by providing more efficient clinical trials and to patients by increasing the likelihood of receiving the better treatment in this paper we propose a new and general class of immigrated urn imu models that incorporates the immigration mechanism into the urn process theoretical properties are developed and the advantages of the imu models are discussed in general the imu models have smaller variabilities than the classical urn models yielding more powerful statistical inferences in applications illustrative examples are presented to demonstrate the wide applicability of the imu models the proposed imu framework including many popular classical urn models not only offers a unify perspective for us to comprehend the urn process but also enables us to generate several novel urn models with desirable properties

on characterization of elementary trapping sets of variable regular ldpc codes in this paper we study the graphical structure of elementary trapping sets etss of variable regular low density parity check ldpc codes etss are known to be the main cause of error floor in ldpc coding schemes for the set of ldpc codes with a given variable node degree dl and girth g we identify all the nonisomorphic structures of an arbitrary class of a b etss where a is the number of variable nodes and b is the number of odd degree check nodes in the induced subgraph of the ets this paper leads to a simple characterization of dominant classes of etss those with relatively small values of a and b based on short cycles in the tanner graph of the code for such classes of etss we prove that any set s in the class is a layered superset lss of a short cycle where the term layered is used to indicate that there is a nested sequence of etss that starts from the cycle and grows one variable node at a time to generate s this characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the etss with lss property in a guaranteed fashion specific results on the structure of etss are presented for d l 3 4 5 6 g 6 8 and a b 10 in this paper the results of this paper can be used for the error floor analysis and for the design of ldpc codes with low error floors

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

high rate quantum low density parity check codes assisted by reliable qubits quantum error correction is an important building block for reliable quantum information processing a challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation a typical obstacle to constructing a variety of strong quantum error correcting codes is the complicated restrictions imposed on the structure of a code recently promising solutions to this problem have been proposed in quantum information science where in principle any binary linear code can be turned into a quantum error correcting code by assuming a small number of reliable quantum bits this paper studies how best to take advantage of these latest ideas to construct desirable quantum error correcting codes of very high information rate our methods exploit structured high rate low density parity check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths our approach to designing high rate quantum error correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes making it possible to deal with a situation where promising quantum analogues of low density parity check codes are difficult to find

asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials a general doubly adaptive biased coin design is proposed for the allocation of subjects to k treatments in a clinical trial this design follows the same spirit as efron s biased coin design and applies to the cases where the desired allocation proportions are unknown but estimated sequentially strong consistency a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions for two treatments a new family of designs is proposed and shown to be less variable than both the randomized play the winner rule and the adaptive randomized design also the proposed design tends toward a randomization scheme with a fixed target proportion as the size of the experiment increases

error correction capability of column weight three ldpc codes under the gallager a algorithm part ii the relation between the girth and the error correction capability of column weight three ldpc codes under the gallager a algorithm is investigated it is shown that a column weight three ldpc code with tanner graph of girth g 10 can correct all error patterns with up to g 2 1 errors in at most g 2 iterations of the gallager a algorithm for codes with tanner graphs of girth g 8 it is shown that girth alone cannot guarantee correction of all error patterns with up to g 2 1 errors under the gallager a algorithm sufficient conditions to correct g 2 1 errors are then established by studying trapping sets

a note on design when response has an exponential family distribution summary some simple design problems are considered when responses have an exponential family distribution with the difference between treatment groups in the canonical parameter being the objective of study it is shown that normal theory conclusions apply to a possibly surprising extent the loss from randomizing followed by subsequent stratification is the equivalent of one observation per stratum for binary data heterogeneity has to be quite large before failure to stratify induces serious loss whereas unnecessary use of a matched pair design is fully efficient asymptotically

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

necessary conditions for the existence of higher order extensions of univalent mappings from the disk to the ball if g cn 1 c is a holomorphic function such that g 0 0 and dg 0 0 and f is a normalized univalent mapping of the unit disk d c we consider the normalized extension of f to the euclidean unit ball b cn given by g f z f z1 g f z1 z f z1 z z b z z2 zn while for a given f g f will maintain certain geometric properties of f such as convexity or starlikeness if g is a polynomial of degree 2 of sufficiently small norm these properties may be lost whenever g contains a nonzero term of higher degree by establishing separate necessary and sufficient conditions for the extension of loewner chains from d to b through g we are able to completely classify those starlike and convex mappings f on d for which there exists a g with nonzero higher degree terms such that g f is a mapping of the same type on b

adaptive biased coin designs for skewing the allocation proportion in clinical trials with normal responses response adaptive designs are used in phase iii clinical trial for skewing the allocation pattern towards the better treatments we use optimum design theory to derive adaptive designs when the responses are normally distributed the performance of the designs is studied with respect to the loss and the proportion of allocation to different treatments the adaptive design does not affect inference

a unified family of covariate adjusted response adaptive designs based on efficiency and ethics response adaptive designs have recently attracted more and more attention in the literature because of its advantages in efficiency and medical ethics to develop personalized medicine covariate information plays an important role in both design and analysis of clinical trials a challenge is how to incorporate covariate information in response adaptive designs while considering issues of both efficiency and medical ethics to address this problem we propose a new and unified family of covariate adjusted response adaptive cara designs based on two general measurements of efficiency and ethics important properties including asymptotic properties of the proposed procedures are studied under categorical covariates this new family of designs not only introduces new desirable cara designs but also unifies several important designs in the literature we demonstrate the proposed procedures through examples simulations and a discussion of related earlier work

shift invariant target in allocation problems we provide a template for finding target allocation proportions in optimal allocation designs where the target will be invariant for both shifts in location and scale of the response distributions one possible application of such target allocation proportions is to carry out a response adaptive allocation while most of the existing designs are invariant for any change in scale of the underlying distributions they are not location invariant in most of the cases first we indicate this serious flaw in the existing literature and illustrate how this lack of location invariance makes the performance of the designs very poor in terms of allocation for any drastic change in location such as the changes from degrees centigrade to degrees fahrenheit we illustrate that unless a target allocation is location invariant it might lead to a completely irrelevant and useless target for allocation then we discuss how such location invariance can be achieved for general continuous responses we illustrate the proposed method using some real clinical trial data we also indicate the possible extension of the procedure for more than two treatments at hand and in the presence of covariates

extension operators and automorphisms abstract many of the commonly studied extension operators that take a normalized locally univalent function f defined in the open unit disk d c to a normalized locally biholomorphic mapping of the unit ball b c n n 2 are of the form f 0 f where 0 f z f z 1 f z 1 z z b z z 2 z n is the roper suffridge extension of f and is a first coordinate shear an automorphism of c n of the form w w 1 g w w where g c n 1 c is holomorphic and contains no terms of degree 0 or 1 in its taylor expansion at the origin other studies have considered the roper suffridge extension operator with the multiplier f z 1 replaced by a different expression recently it has been shown that the set of restrictions of automorphisms of c n to b satisfying some geometric property such as convexity or starlikeness is dense in the full respective family of geometric mappings when n 2 we study large families of biholomorphic mappings on b generated by automorphisms of c n abstract generalizations of the roper suffridge extension operator unitary rotations and families of locally univalent functions on d in many cases families arising from distinct automorphisms will be seen to be disjoint despite the fact that they contain the restrictions of the respective automorphisms to b and the previously noted density result of particular interest given our consideration of more general automorphisms related to extension operators we show that the compositions of the roper suffridge extension operator with a particular class of first coordinate overshears automorphisms of c n of the form w h w w 1 g w w for g h c n 1 c holomorphic and h never 0 extend certain bounded starlike mappings of d to starlike mappings of b we conclude with several open questions inspired by this work

cyclic and quasi cyclic ldpc codes on constrained parity check matrices and their trapping sets this paper is concerned with construction and structural analysis of both cyclic and quasi cyclic codes particularly low density parity check ldpc codes it consists of three parts the first part shows that a cyclic code given by a parity check matrix in circulant form can be decomposed into descendant cyclic and quasi cyclic codes of various lengths and rates some fundamental structural properties of these descendant codes are developed including the characterization of the roots of the generator polynomial of a cyclic descendant code the second part of the paper shows that cyclic and quasi cyclic descendant ldpc codes can be derived from cyclic finite geometry ldpc codes using the results developed in the first part of the paper this enlarges the repertoire of cyclic ldpc codes the third part of the paper analyzes the trapping set structure of regular ldpc codes whose parity check matrices satisfy a certain constraint on their rows and columns several classes of finite geometry and finite field cyclic and quasi cyclic ldpc codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances consequently their error floor performances are dominated by their minimum distances

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

finite alphabet iterative decoders part ii towards guaranteed error correction of ldpc codes via iterative decoder diversity recently we introduced a new class of finite alphabet iterative decoders faids for low density parity check ldpc codes these decoders are capable of surpassing belief propagation bp in the error floor region on the binary symmetric channel bsc with much lower complexity in this paper we introduce a novel scheme with the objective of guaranteeing the correction of a given and potentially large number of errors on column weight three ldpc codes the proposed scheme uses a plurality of faids which collectively correct more error patterns than a single faid on a given code the collection of faids utilized by the scheme is judiciously chosen to ensure that individual decoders have different decoding dynamics and correct different error patterns consequently they can collectively correct a diverse set of error patterns which is referred to as decoder diversity we provide a systematic method to generate the set of faids for decoder diversity on a given code based on the knowledge of the most harmful trapping sets present in the code using the well known column weight three 155 64 tanner code with dmin 20 as an example we describe the method in detail and show by means of exhaustive simulation that the guaranteed error correction capability on short length ldpc codes can be significantly increased with decoder diversity

ldpc codes on partial geometries construction trapping set structure and puncturing many known constructions of ldpc codes can be placed in a general framework using the notion of partial geometries based on this notion the structure of such ldpc codes can be analyzed using a geometric approach that illuminates important properties of their parity check matrices in this approach trapping sets are represented by subgeometries of the geometry used to construct the code based on the incidence relations between lines and points in this geometry the structure of trapping sets is investigated on the other hand it is shown that removing a subgeometry corresponding to a trapping set gives a punctured matrix which can be used as a parity check matrix of an ldpc code this relates trapping sets represented by subgeometries and punctured matrices represented by the residual geometries the null spaces of these punctured matrices are ldpc codes which inherit many of the good structural properties of the original code hence new ldpc codes with various lengths and rates can be obtained by puncturing an ldpc code constructed based on a partial geometry furthermore these punctured matrices and codes can be used in a two phase decoding scheme to correct combinations of errors and erasures

pseudocodeword performance analysis for ldpc convolutional codes message passing iterative decoders for low density parity check ldpc block codes are known to be subject to decoding failures due to so called pseudocodewords these failures can cause the large signal to noise ratio snr performance of message passing iterative decoding to be worse than that predicted by the maximum likelihood ml decoding union bound in this paper we address the pseudocodeword problem from the convolutional code perspective in particular we compare the performance of ldpc convolutional codes with that of their wrapped quasi cyclic block versions and we show that the minimum pseudoweight of an ldpc convolutional code is at least as large as the minimum pseudoweight of an underlying quasi cyclic code this result which parallels a well known relationship between the minimum hamming weight of convolutional codes and the minimum hamming weight of their quasi cyclic counterparts is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code s pseudocodeword this difference in the weight spectra leads to improved performance at low to moderate snrs for the convolutional code a conclusion supported by simulation results

controlling the error floor in ldpc decoding the error floor of ldpc is revisited as an effect of dynamic message behavior in the so called absorbing sets of the code it is shown that if the signal growth in the absorbing sets is properly balanced by the growth of set external messages the error floor can be lowered to essentially arbitrarily low levels importance sampling techniques are discussed and used to verify the analysis as well as to discuss the impact of iterations and message quantization on the code performance in the ultra low ber error floor regime

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

central limit theorems for doubly adaptive biased coin designs asymptotic normality of the difference between the number of subjects assigned to a treatment and the desired number to be assigned is established for allocation rules which use eisele s biased coin design subject responses are assumed to be independent random variables from standard exponential families in the proof it is shown that the difference may be magnified by appropriate constants so that the magnified difference is nearly a martingale an application to the behrens fisher problem in the normal case is described briefly

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

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

an o brien fleming sequential trial for comparing three treatments we consider a sequential procedure for comparing three treatments with the goal of ultimately selecting the best treatment this procedure starts with a sequential test to detect an overall treatment difference and eliminates the apparently inferior treatment if this test rejects the equality of the treatments it then proceeds with a sequential test of the remaining two treatments we base these sequential tests on the stopping boundaries popularized by o brien and fleming our procedure is similar in structure to that used by siegmund in conjunction with modified repeated significance tests we compare the performances of the two procedures via a simulation experiment we derive analytic approximations for an error probability the power and the expected sample size of our procedure which we compare to simulated values furthermore we propose a modification of the procedure for the comparison of a standard treatment with experimental treatments

optimum biased coin designs for sequential clinical trials with prognostic factors patients in a clinical trial arrive sequentially and are assigned to one of t treatments this assignment should maintain a balance between the numbers receiving each treatment yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating to balance these requirements efron 1971 introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically but with a bias towards the underrepresented treatment one disadvantage of efron s scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment biased coin schemes which do force balance over both treatments and prognostic factors are given by pocock simon 1975 and efron 1980 the properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials reviews of the literature and discussions of the practical implications of the designs are given by pocock 1979 and simon 1979 however the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance the procedures thus lack a firm theoretical framework an alternative approach in the presence of prognostic factors begg iglewicz 1980 uses optimum design theory to suggest a deterministic design criterion which is then modified for computational convenience in this paper i use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence or absence of prognostic factors this has the theoretical advantage of obviating dependence on a series of arbitrary functions the necessary optimum design theory is presented in 2 and in 3 applied to biased coin experiments the simplest case that of two treatments in the absence of prognostic factors is studied in 4 the extension to three or more treatments is in 5 followed in 6 by the allowance for prognostic factors although in the biased coin designs allocation of treatments is made probabilistically the results of the paper can be used for the sequential construction of da optimum designs in which randomization is not introduced by the experimenter all the designs mentioned so far are sequential but not adaptive in the useful distinction made by pericchi 1981 they are data dependent but not outcomedependent each allocation depends on the previous allocations and prognostic factors

frequentist performance of bayesian inference with response adaptive designs in controlled clinical trials where minimizing treatment failures is crucial response adaptive designs are attractive competitors to 1 1 randomized designs for comparing the success rates 1 and 2 of two treatments in these designs each new treatment assignment depends on previous outcomes through some predefined rule here play the winner pw randomized play the winner rpw drop the loser generalized drop the loser and doubly adaptive biased coin designs are considered for new treatment assignments as frequentist inference relies on complex sampling distributions in those designs we investigate how bayesian inference based on two independent beta prior distributions performs from a frequentist point of view performance is assessed through coverage probabilities of interval estimation procedures power and minimization of failure count it is shown that bayesian inference can be favorably compared to frequentist procedures where the latter are available the power of response adaptive designs is generally very close to the power of 1 1 randomized design however failure count savings are generally small except for the pw and doubly adaptive biased coin designs in particular ranges of the true success rates the rpw assignment rule has the worst performance while pw generalized drop the loser or doubly adaptive biased coin designs may outperform other designs depending on different particular ranges of the true success rates copyright 2010 john wiley sons ltd

bias and loss the two sides of a biased coin the paper assesses biased coin designs for sequential treatment allocation in clinical trials comparisons emphasise the importance of considering randomness as well as treatment balance which are calculated as bias and loss in the numerical examples the responses are assumed normally distributed perhaps after transformation and balance is required over a set of covariates the effect of covariate distribution on the properties of five allocation rules is investigated with an emphasis on methods of comparison which also apply to other forms of response the concept of admissibility shows that the widely used minimisation rule is outperformed by atkinson s rule derived from the theory of optimum experimental design we present a simplified form of this rule for this rule the ability to guess the next treatment allocation decreases with study size for the other rules it is constant

loewner chains and parametric representation in several complex variables abstract let b be the unit ball of c n with respect to an arbitrary norm we study certain properties of loewner chains and their transition mappings on the unit ball b we show that any loewner chain f z t and the transition mapping v z s t associated to f z t satisfy locally lipschitz conditions in t locally uniformly with respect to z b moreover we prove that a mapping f h b has parametric representation if and only if there exists a loewner chain f z t such that the family e tf z t t 0 is a normal family on b and f z f z 0 for z b also we show that univalent solutions f z t of the generalized loewner differential equation in higher dimensions are unique when e tf z t t 0 is a normal family on b finally we show that the set s0 b of mappings which have parametric representation on b is compact

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

hardness results on finding leafless elementary trapping sets and elementary absorbing sets of ldpc codes leafless elementary trapping sets letss are known to be the problematic structures in the error floor region of low density parity check ldpc codes over the additive white gaussian awgn channel under iterative decoding algorithms while problems involving the general category of trapping sets and the subcategory of elementary trapping sets etss have been shown to be np hard similar results for letss which are a subset of etss are not available in this paper we prove that for a general ldpc code finding a lets of a given size a with minimum number of odd degree check nodes b is np hard to approximate within any approximation factor we also prove that finding the minimum size a of a lets with a given b is np hard to approximate within any approximation factor similar results are proved for elementary absorbing sets a popular subcategory of letss

asymptotic properties of covariate adjusted response adaptive designs response adaptive designs have been extensively studied and used in clinical trials however there is a lack of a comprehensive study of response adaptive designs that include covariates despite their importance in clinical trials because the allocation scheme and the estimation of parameters are affected by both the responses and the covariates covariate adjusted response adaptive cara designs are very complex to formulate in this paper we overcome the technical hurdles and lay out a framework for general cara designs for the allocation of subjects to k 2 treatments the asymptotic properties are studied under certain widely satisfied conditions the proposed cara designs can be applied to generalized linear models two important special cases the linear model and the logistic regression model are considered in detail

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

lowering the error floor of ldpc codes using cyclic liftings cyclic liftings are proposed to lower the error floor of low density parity check ldpc codes the liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets we derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle of length l in the base graph such that the inverse image of in the lifted graph consists of only cycles of length strictly larger than l the proposed method is universal in the sense that it can be applied to any ldpc code over any channel and for any iterative decoding algorithm it also preserves important properties of the base code such as degree distributions and in some cases the code rate the constructed codes are quasi cyclic and thus attractive from a practical point of view the proposed method is applied to both structured and random codes over the binary symmetric channel bsc the error floor improves consistently by increasing the lifting degree and the results show significant improvements in the error floor compared to the base code a random code of the same degree distribution and block length and a random lifting of the same degree similar improvements are also observed when the codes designed for the bsc are applied to the additive white gaussian noise awgn channel

bayesian adaptive biased coin designs for clinical trials with normal responses des schemas adaptatifs sont utilises dans des essais cliniques de phase iii pour biaiser l attribution des traitements en faveur des meilleurs nous utilisons la theorie des schemas optimaux pour etablir une procedure bayesienne de tirage au sort biaise pour des schemas sequentiels avec reponses continues la performance de ces schemas adaptatifs est etudiee a la fois numeriquement et theoriquement des proprietes importantes sont la perte et la proportion de sujets alloues au meilleur traitement

multi objective optimal designs in comparative clinical trials with covariates the reinforced doubly adaptive biased coin design the present paper deals with the problem of allocating patients to two competing treatments in the presence of covariates or prognostic factors in order to achieve a good trade off among ethical concerns inferential precision and randomness in the treatment allocations in particular we suggest a multipurpose design methodology that combines efficiency and ethical gain when the linear homoscedastic model with both treatment covariate interactions and interactions among covariates is adopted the ensuing compound optimal allocations of the treatments depend on the covariates and their distribution on the population of interest as well as on the unknown parameters of the model therefore we introduce the reinforced doubly adaptive biased coin design namely a general class of covariate adjusted response adaptive procedures that includes both continuous and discontinuous randomization functions aimed to target any desired allocation proportion the properties of this proposal are described both theoretically and through simulations

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

error correction capability of column weight three ldpc codes in this paper the error correction capability of column weight three low density parity check ldpc codes when decoded using the gallager a algorithm is investigated it is proved that a necessary condition for a code to correct all error patterns with up to k ges 5 errors is to avoid cycles of length up to 2k in its tanner graph as a consequence of this result it is shown that given any alpha 0 exist n such that forall n n no code in the ensemble of column weight three codes can correct all alphan or fewer errors the results are extended to the bit flipping algorithms

properties of biased coin designs in sequential clinical trials martingale methods and the martingale invariance principle are used to derive central limit theorems and related results for biased coin designs of the kind previously studied by efron wei and many others the results are applied to the study of selection bias the method is developed for the simplest two treatment case and then extended first to the case of several treatments and secondly to the case of two treatments with prognostic factors

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

asymptotic properties of adaptive designs for clinical trials with delayed response for adaptive clinical trials using a generalized friedman s urn design we derive the limiting distribution of the urn composition under staggered entry and delayed response the stochastic delay mechanism is assumed to depend on both the treatment assigned and the patient s response a very general setup is employed with k treatments and l responses when l k 2 one example of a generalized friedman s urn design is the randomized playthe winner rule an application of this rule occurred in a clinical trial of depression which had staggered entry and delayed response we show that maximum likelihood estimators from such a trial have the usual asymptotic properties

sequential urn designs with elimination for comparing k 3 treatments a fully sequential procedure is proposed for comparing k or 3 treatments with immediate binary responses the procedure uses an adaptive urn design to randomize patients to the treatments and stopping rules are incorporated for eliminating less promising treatments simulation is used to assess the performance of the procedure for several adaptive urn designs in terms of expected numbers of treatment failures and allocation proportions and the effect on estimation at the end of the trial is also addressed it is concluded that the drop the loser rule is more effective than equal allocation and all of the other designs considered the practical benefits of the procedure are illustrated using the results of a three treatment lung cancer study it is then shown how the sequential elimination procedure may be used in dose finding studies and its performance is compared with a recently proposed method several possible extensions to the work are briefly indicated

new characterization and efficient exhaustive search algorithm for leafless elementary trapping sets of variable regular ldpc codes in this paper we propose a new characterization for leafless elementary trapping sets letss of variable regular low density parity check codes recently karimi and banihashemi proposed a characterization of letss which was based on viewing an lets as a layered superset lss of a short cycle in the code s tanner graph a notable advantage of lss characterization is that it corresponds to a simple lss based search algorithm expansion technique that starts from short cycles of the graph and finds the letss with lss structure efficiently compared with the lss based characterization of karimi and banihashemi which is based on a single lss expansion technique the new characterization involves two additional expansion techniques the introduction of the new techniques mitigates two problems that lss based characterization search suffers from 1 exhaustiveness not every lets structure is an lss of a cycle and 2 search efficiency lss based search algorithm often requires the enumeration of cycles with length much larger than the girth of the graph where the multiplicity of such cycles increases rapidly with their length we prove that using the three expansion techniques any lets structure can be obtained starting from a simple cycle no matter how large the size of the structure a or the number of its unsatisfied check nodes b are i e the characterization is exhaustive we also demonstrate that for the proposed characterization search to exhaustively cover all the lets structures within the a b classes with a leq a max and b leq b max for any value of a max and b max the length of the short cycles required to be enumerated is less than that of the lss based characterization search we in fact show that such a length for the proposed search algorithm is minimal we also prove that the three expansion techniques proposed here are the only expansions needed for characterization of lets structures starting from simple cycles in the graph if one requires each and every intermediate sub structure to be a lets as well extensive simulation results are provided to show that compared with lss based search significant improvement in search speed and memory requirements can be achieved

outcome adaptive randomization for a delayed outcome with a short term predictor imputation based designs delay in the outcome variable is challenging for outcome adaptive randomization as it creates a lag between the number of subjects accrued and the information known at the time of the analysis motivated by a real life pediatric ulcerative colitis trial we consider a case where a short term predictor is available for the delayed outcome when a short term predictor is not considered studies have shown that the asymptotic properties of many outcome adaptive randomization designs are little affected unless the lag is unreasonably large relative to the accrual process these theoretical results assumed independent identical delays however whereas delays in the presence of a short term predictor may only be conditionally homogeneous we consider delayed outcomes as missing and propose mitigating the delay effect by imputing them we apply this approach to the doubly adaptive biased coin design dbcd for motivating pediatric ulcerative colitis trial we provide theoretical results that if the delays although non homogeneous are reasonably short relative to the accrual process similarly as in the iid delay case the lag is also asymptotically ignorable in the sense that a standard dbcd that utilizes only observed outcomes attains target allocation ratios in the limit empirical studies however indicate that imputation based dbcds performed more reliably in finite samples with smaller root mean square errors the empirical studies assumed a common clinical setting where a delayed outcome is positively correlated with a short term predictor similarly between treatment arm groups we varied the strength of the correlation and considered fast and slow accrual settings

a class of loewner chain preserving extension operators abstract we consider operators that extend locally univalent mappings of the unit disk in c to locally biholomorphic mappings of the euclidean unit ball b of c n for such an operator we seek conditions under which e t e t f t t 0 is a loewner chain on b whenever f t t 0 is a loewner chain on we primarily study operators of the form g f z f z 1 g f z 1 z f z 1 z z z 2 z n where 0 1 2 and g c n 1 c is holomorphic finding that for g to preserve loewner chains the maximum degree of terms appearing in the expansion of g is a function of further applications involving bloch mappings and radius of starlikeness are given as are elementary results concerning extreme points and support points

on inferences from wei s biased coin design for clinical trials wei 1988 analyzed data from a clinical trial in which an urn sampling model was used to allocate patients to treatments the trial resulted in 11 patients being allocated to the experimental treatment all successes and with one patient allocated to the control treatment a failure wei analyzed these data using a randomization algorithm and concluded that the results were almost significant at p 0 051 he asserted that an analysis which ignored the design and presumed complete randomization leads to a p value of 0 001 in fact if a more conventional analysis is used in which both margins are fixed then this leads to p 0 083 fisher s exact test if complete randomization is assumed if the urn sampling allocation is taken into account much more conservative inferences are obtained an analysis which conditions on both margins leads to p 028 while one which conditions on the observed sequence of responses and the observed treatment totals leads to p 0 62 these serious discrepancies are discussed in addition to the inappropriateness of biased coin design and small sample sizes in important medical trials

on the construction of structured ldpc codes free of small trapping sets we present a method to construct low density parity check ldpc codes with low error floors on the binary symmetric channel codes are constructed so that their tanner graphs are free of certain small trapping sets these trapping sets are selected from the trapping set ontology for the gallager a b decoder they are selected based on their relative harmfulness for a given decoding algorithm we evaluate the relative harmfulness of different trapping sets for the sum product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets we apply this method to construct structured ldpc codes to facilitate the discussion we give a new description of structured ldpc codes whose parity check matrices are arrays of permutation matrices this description uses latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the tanner graph of a code to be free of four cycles

optimality variability power we provide a theoretical template for the comparison of response adaptive randomization procedures for clinical trials using a taylor expansion of the noncentrality parameter of the usual chi squared test for binary responses we show explicitly the relationship among the target allocation proportion the bias of the randomization procedure from that target and the variability induced by the randomization procedure we also generalize this relationship for more than two treatments under various multivariate alternatives this formulation allows us to directly evaluate and compare different response adaptive randomization procedures and different target allocations in terms of power and expected treatment failure rate without relying on simulation for k 2 treatments we compare four response adaptive randomization procedures and three target allocations based on multiple objective optimality criteria we conclude that the drop the loser rule and the doubly adaptive biased coin design are clearly super

conditioning in 2 2 tables two by two tables arise in a number of diverse settings in biomedical research including analysis of data from a clinical trial with a binary outcome and gating methods in flow cytometry to separate antigen specific immune responses from general immune responses these applications offer interesting challenges concerning what we should really be conditioning on the total number of events the number of events in the control condition etc we give several biostatistics examples to illustrate the complexities of analyzing what appear to be simple data

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

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

eliminating trapping sets in low density parity check codes by using tanner graph covers we discuss error floor asympotics and present a method for improving the performance of low density parity check ldpc codes in the high snr error floor region the method is based on tanner graph covers that do not have trapping sets from the original code the advantages of the method are that it is universal as it can be applied to any ldpc code channel decoding algorithm and it improves performance at the expense of increasing the code length without losing the code regularity without changing the decoding algorithm and under certain conditions without lowering the code rate the proposed method can be modified to construct convolutional ldpc codes also the method is illustrated by modifying tanner mackay and margulis codes to improve performance on the binary symmetric channel bsc under the gallager b decoding algorithm decoding results on awgn channel are also presented to illustrate that optimizing codes for one channel decoding algorithm can lead to performance improvement on other channels

adaptive designs for normal responses with prognostic factors summary adaptive designs are often used in clinical trials to force balance in sequential allocation of patients between two or more competitive treatments sometimes from ethical considerations the goal may be to allocate a larger number of patients to the better treatment in the course of the trial in the present paper we consider patients who are heterogeneous with respect to some prognostic factors and assume that the response from each patient is a continuous variable we consider a normal linear model and provide an allocation design with due attention to the prognostic factors the loss of efficiency incurred by using a balanced design is also indicated issues of inference based on adaptive allocation are also discussed finally a guideline is provided for the appropriate choice of the design parameters

exact two sample permutation tests based on the randomized play the winner rule in comparing two treatments in a clinical trial the randomized play the winner rule tends to assign more study subjects to the better treatment it is applicable when patients have delayed responses to treatments it is not deterministic and is less vulnerable to experimental bias than other adaptive designs in this paper an efficient algorithm is provided to construct exact permutation tests for testing the equality of the two treatments with the randomized play the winner rule the test procedure is illustrated with a real life example the example shows that the design used in the trial should not be ignored in the analysis