maximum likelihood erasure decoding of ldpc codes pivoting algorithms and code design this paper investigates efficient maximum likelihood ml decoding of low density parity check ldpc codes over erasure channels a set of algorithms referred to as pivoting algorithms is developed the aim is to limit the average number of pivots or reference variables from which all the other erased symbols are recovered iteratively the suggested algorithms exhibit different trade offs between complexity of the pivoting phase and average number of pivots moreover a systematic procedure to design ldpc code ensembles for efficient ml decoding is proposed numerical results illustrate that the designed ldpc codes achieve a near optimum performance very close to the singleton bound at least down to a codeword error rate level 10 8 with an affordable decoding complexity for one of the presented codes and algorithms a software implementation has been developed which is capable to provide data rates above 1 5 gbps on a commercial computing platform

analysis of the block error probability of concatenated polar code ensembles in this paper we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check crc codes separated by an interleaver in the short and moderate block length regimes the analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding the analysis is carried out by introducing the concatenated polar code cpc ensembles whose distance properties can be analyzed for sufficiently short block lengths by means of the uniform interleaver approach at moderate block lengths we resort to the monte carlo simulations results show that if the inner polar code possesses a low minimum distance and the outer crc code has a sufficiently large amount of redundancy then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting cpc

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

construction of near optimum burst erasure correcting low density parity check codes in this paper a simple and effective tool for the design of low density parity check ldpc codes for iterative correction of bursts of erasures is presented the design method consists of starting from the parity check matrix of an ldpc code and developing an optimized parity check matrix with the same performance over the memoryless erasure channel and suitable also for the iterative correction of single erasure bursts the parity check matrix optimization is performed by an algorithm called pivot searching and swapping pss algorithm it executes permutations of carefully chosen columns of the parity check matrix after a local analysis of particular variable nodes called stopping set pivots this algorithm can be in principle applied to any ldpc code if the input parity check matrix is designed to achieve a good performance over the memoryless erasure channel then the code obtained after the application of the algorithm provides a good joint correction of independent erasures and single erasure bursts numerical results are provided in order to show the algorithm effectiveness when applied to different categories of ldpc codes

practical codes for photon communication in a recent paper pierce studied the problems of communicating at optical frequencies using photon counting techniques and concluded that at low temperatures we encounter insuperable problems of encoding long before we approach channel capacity in this paper it is shown that even assuming a noiseless model for photon communication for which capacity measured in nats photon is infinite it is unlikely that a signaling efficiency of even 10 nats photon could be achieved practically on the positive side it is shown that pulse position modulation plus reed solomon coding yields practical results in the range of 2 to 3 nats photon

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

parallel concatenation of non binary linear random fountain codes with maximum distance separable codes the performance and the decoding complexity of a novel coding scheme based on the concatenation of maximum distance separable mds codes and linear random fountain codes are investigated differently from raptor codes which are based on a serial concatenation of a high rate outer block code and an inner luby transform code the proposed coding scheme can be seen as a parallel concatenation of a mds code and a linear random fountain code both operating on the same finite field upper and lower bounds on the decoding failure probability under maximum likelihood ml decoding are developed it is shown how for example the concatenation of a 15 10 reed solomon rs code and a linear random fountain code over a finite field of order 16 f 16 brings to a decoding failure probability 4 orders of magnitude lower than the one of a linear random fountain code for the same receiver overhead in a channel with a erasure probability of e 5 cdot10 2 it is illustrated how the performance of the novel scheme approaches that of an idealized fountain code for higher order fields and moderate erasure probabilities an efficient decoding algorithm is developed for the case of a generalized rs code

uniqueness and approximation of solutions of first order non linear equations guy metivier d6partement de math6matiques universit6 rennes i f 35042 rennes c6dex france this paper is devoted to non linear first order analytic partial differential equa tions and contains the proof of the following results a uniqueness of c 2 complex valued solutions to the non characteristic cauchy problem b any c 2 solution is the limit of a sequence of analytic solutions of the same equation for linear equations of any order a is of course well known 7 and b which is also well known in the constant coefficients case 8 has also been proved for wide classes of equations see baouendi treves 3 4 baouendi rothschild 23 for non linear equations the situation is quite different and so far only the first order case has been investigated indeed in that case baouendi goulaouic and treves 1 have already proved a and b in the semilinear case and a either in dimension 2 or when the cauchy data is analytic so the aim of this paper is to extend their result to a general first order non linear equation leaving still completely open the problem for higher order equations as in 1 the proofs given in this paper rely on the analysis of the hamil tonian field of the equation the new point is that we are able to produce directly a formula expressing any smooth enough solution in terms of its cauchy data uniqueness follows at once in the linear case this formula is exactly the one given in baouendi treves 3 and gives in the same time the approximation property here we still derive this property from the mentioned formula but because of the nonlinearity the arguments are just a little more complicated this paper was completed while the author was visiting at rutgers university

on the cubic lattice green functions wheatstone physics laboratory king s college university of london strand london wc2r 2ls u k it is proved that k k 4 eta frac 1 2 1 eta frac 1 2 k k where eta is a complex variable which lies in a certain region scr r 2 of the eta plane and k k pm are complete elliptic integrals of the first kind with moduli k pm which are given by k pm 2 equiv k pm 2 eta frac 1 2 pm frac 1 4 eta 4 eta frac 1 2 frac 1 4 2 eta 1 eta frac 1 2 this basic result is then used to express the face centred cubic and simple cubic lattice green functions at the origin in terms of the square of a complete elliptic integral of the first kind several new identities involving the heun function f a b alpha beta gamma delta eta are also derived next it is shown that the three cubic lattice green functions all have parametric representations which involve the green function for the two dimensional honeycomb lattice finally the results are applied to a variety of problems in lattice statistics in particular a new simplified formula for the generating function of staircase polygons on a four dimensional hypercubic lattice is derived

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

results on the improved decoding algorithm for low density parity check codes over the binary erasure channel in this correspondence we first investigate some analytical aspects of the recently proposed improved decoding algorithm for low density parity check ldpc codes over the binary erasure channel bec we derive a necessary and sufficient condition for the improved decoding algorithm to successfully complete decoding when the decoder is initialized to guess a predetermined number of guesses after the standard message passing terminates at a stopping set furthermore we present improved bounds on the number of bits to be guessed for successful completion of the decoding process when a stopping set is encountered under suitable conditions we derive a lower bound on the number of iterations to be performed for complete decoding of the stopping set we then present a superior novel improved decoding algorithm for ldpc codes over the binary erasure channel bec the proposed algorithm combines the observation that a considerable fraction of unsatisfied check nodes in the neighborhood of a stopping set are of degree two and the concept of guessing bits to perform simple and intuitive graph theoretic manipulations on the tanner graph the proposed decoding algorithm has a complexity similar to previous improved decoding algorithms finally we present simulation results of short length codes over bec that demonstrate the superiority of our algorithm over previous improved decoding algorithms for a wide range of bit error rates

structural analysis of array based non binary ldpc codes structural properties of array based non binary low density parity check nbldpc codes are studied in this paper first we characterize graphical substructures induced by codewords of symbol weight six in array based nbldpc codes defined by parity check matrices with column weight three we also reveal necessary conditions for these graphical substructures to incur weight 6 codewords such conditions can be used to select nonzero elements for avoiding weight 6 codewords or reducing the multiplicity of weight 6 codewords second we show that there exist weight 7 codewords in array based nbldpc codes defined by parity check matrices with column weight three as a byproduct we find that the graphical substructure induced by a weight 7 codeword takes the graphical substructure induced by the related weight 6 codewords as a subgraph third we show that there may exist codewords with symbol weight four six and seven in array based nbldpc codes defined by parity check matrices with column weight two these results enrich the structural analysis of array based ldpc codes in addition simulation results show the performance advantage of array based nbldpc codes

on decoding of low density parity check codes over the binary erasure channel this paper investigates decoding of low density parity check ldpc codes over the binary erasure channel bec we study the iterative and maximum likelihood ml decoding of ldpc codes on this channel we derive bounds on the ml decoding of ldpc codes on the bec we then present an improved decoding algorithm the proposed algorithm has almost the same complexity as the standard iterative decoding however it has better performance simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm we also provide some graph theoretic properties of different decoding algorithms of ldpc codes over the bec which we think are useful to better understand the ldpc decoding methods in particular for finite length codes

short non binary low density parity check codes for phase noise channels this paper considers the design of short non binary low density parity check ldpc codes over finite fields of order m for channels with phase noise in particular m ary differential phase shift keying dpsk modulated code symbols are transmitted over an additive white gaussian noise awgn channel with the wiener phase noise at the receiver side non coherent detection takes place with the help of a multi symbol detection algorithm followed by a non binary decoding step both the detector and the decoder operate on a joint factor graph as a benchmark finite length bounds and information rate expressions are computed and compared with the codeword error rate cer performance as well as the iterative threshold of the obtained codes as a result performance within 1 2 db from finite length bounds is obtained down to a cer of 10 3

the decoding error probability of linear codes over the erasure channel in this paper we study the decoding error probability of linear codes over the erasure channel under the list decoding the notion of the q ell incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding by calculating the q ell incorrigible set distributions the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions for the ensemble of all n k linear codes the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined furthermore the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all n nr linear codes

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

efficient maximum likelihood decoding of ldpc codes over the binary erasure channel we propose an efficient maximum likelihood ml decoding algorithm for decoding low density parity check ldpc codes over the binary erasure channel bec we also analyze the computational complexity of the proposed algorithm

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

performance bounds for nonbinary linear block codes over memoryless symmetric channels the performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum likelihood ml decoding the transmission of these codes is assumed to take place over a memoryless and symmetric channel the new bounds which are based on the gallager bounds and their variations are applied to the gallager ensembles of nonbinary and regular low density parity check ldpc codes these upper bounds are also compared with sphere packing lower bounds this study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

non binary ldpc erasure codes with separated low degree variable nodes the code design of non binary low density parity check codes for the erasure channel under maximum a posteriori decoding is addressed in particular a partially structured ensemble of codes characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees is proposed the identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade off between waterfall performance error floor and decoding complexity as an example the performance curve of a short 256 128 code on the memoryless 16 ary erasure channel tightly approaches the singleton bound at least down to a codeword error rate of 10 9 at low decoding complexity

remarks on the integrability of first order complex pde abstract the classical method of characteristics for the integration of real first order pde is extended to complex equations via a symplectic structure on the complex cotangent bundle that is holomorphic along the fibres integrability is shown to be equivalent to a variant of darboux s theorem i e the existence of symplectic frames that incorporate the equations applications of the linear theory to the associated hamiltonian system yield integrability results for the nonlinear pde

coding in the block erasure channel in this correspondence we study an m ary block erasure channel with b blocks where with probability epsi a block of l coded symbols is erased the behavior of the error probability of coded systems over such channels is studied and we show that if the code is diversity wise maximum distance separable its word error probability is equal to the outage probability which admits a very simple expression this correspondence is intended to complement the error probability analysis in previous work by lapidoth and shed some light on the design of coding schemes for nonergodic channels

exact evaluation of the baxter bazhanov green function the analytic properties of the lattice green function where t lies in a complex plane which is cut along the real axis from 2 to 2 are investigated in particular it is proved that tg t can be written in the product form where and denotes a hypergeometric function this result and the analytic continuation formulae for the function are then used to obtain various exact closed form expressions for the related functions where it is also shown that g t is a solution of a third order fuchsian differential equation

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

capacity achieving ensembles for the binary erasure channel with bounded complexity we present two sequences of ensembles of nonsystematic irregular repeat accumulate ira codes which asymptotically as their block length tends to infinity achieve capacity on the binary erasure channel bec with bounded complexity per information bit this is in contrast to all previous constructions of capacity achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap in rate to capacity the new bounded complexity result is achieved by puncturing bits and allowing in this way a sufficient number of state nodes in the tanner graph representing the codes we derive an information theoretic lower bound on the decoding complexity of randomly punctured codes on graphs the bound holds for every memoryless binary input output symmetric mbios channel and is refined for the binary erasure channel

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

an integral arising from the chiral sl n potts model we show that the integral can be expressed in terms of 5f4 hypergeometric functions the integral arises in the solution by baxter and bazhanov of the free energy of the sl n potts model which includes the term j 2 our result immediately gives the logarithmic mahler measure of the laurent polynomial in terms of the same hypergeometric functions