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

on interleaved differentially encoded convolutional codes we study a serially interleaved concatenated code construction where the outer code is a standard convolutional code and the inner code is a recursive convolutional code of rate 1 we focus on the ubiquitous inner differential encoder used in particular to resolve phase ambiguities double differential encoder used to resolve both phase and frequency ambiguities and another rate 1 recursive convolutional code of memory 2 we substantiate analytically the rather surprising result that the error probabilities corresponding to a maximum likelihood ml coherently detected antipodal modulation over the additive white gaussian noise awgn channel for this construction are advantageous as compared to the stand alone outer convolutional code this is in spite of the fact that the inner code is of rate 1 the analysis is based on the tangential sphere upper bound of an ml decoder incorporating the ensemble weight distribution wd of the concatenated code where the ensemble is generated by all random and uniform interleavers this surprising result is attributed to the wd thinning observed for the concatenated scheme which shapes the wd of the outer convolutional code to resemble more closely the binomial distribution typical of a fully random code of the same length and rate this gain is maintained regardless of a rather dramatic decrease as demonstrated here in the minimum distance of the concatenated scheme as compared to the minimum distance of the outer stand alone convolutional code the advantage of the examined serially interleaved concatenated code given in terms of bit and or block error probability which is decoded by a practical suboptimal decoder over the optimally decoded standard convolutional code is demonstrated by simulations and some insights into the performance of the iterative decoding algorithm are also discussed though we have investigated only specific constructions of constituent inner rate 1 and outer codes we trust hinging on the rational of the arguments here that these results extend to many other constituent convolutional outer codes and rate 1 inner recursive convolutional codes

arbitrarily varying channels with general alphabets and states previous results of csiszar and narayan 1988 1989 1991 on the capacity of discrete arbitrarily varying channels with input and state constraints are extended to the case of arbitrary alphabets and state sets for channels with scalar or vector inputs and additive interference consisting of a deterministic part and noise both arbitrarily varying subject to power constraints the capacity for deterministic codes is shown to be equal to the random coding capacity if the input power exceeds the power of the deterministic interference and zero otherwise explicit capacity formulas are also given

on the design of ldpc code ensembles for biawgn channels existing design methods for irregular low density parity check ldpc codes over the additive white gaussian noise channel are based on using asymptotic analysis tools such as density evolution in an optimization process such a process is computationally expensive particularly when a large number of constituent variable node degrees are involved in the design in this paper we propose a systematic approach for the design of irregular ldpc codes the proposed method which is based on a pre computed upper bound on the fraction of edges connected to variable nodes of degree 3 is considerably less complex than the conventional optimization approach through a number of examples we demonstrate that using our method ensembles with performance very close to those devised based on optimization can be designed in addition to having very good performance the number of constituent variable node degrees in the designed ensembles is only three or four this in some cases is much smaller than the corresponding number for optimization based designs with similar performance

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

identical capacity channel decomposition for design of universal ldpc codes design of low density parity check ldpc codes suitable for all channels which exhibit a given capacity c is investigated such codes are referred to as universal ldpc codes first based on numerous observations a conjecture is put forth that a code working on n equal capacity channels also works on any convex combination of these n channels as a supporting evidence we prove that a code satisfying the stability condition on n channels also satisfies the stability condition on the convex hull of these n channels then a channel decomposition method is suggested which spans any given channel with capacity c in terms of a number of identical capacity basis channels we expect codes that work on the basis channels to be suitable for any convex combination of the bases i e all channels with capacity c such codes are found over a wide range of rates an upper bound on the achievable rate of universal ldpc codes is suggested through examples it is shown that our codes achieve rates extremely close to this upper bound in comparison with existing ldpc codes designed for a given channel significant performance gain is reported when codes are used over various channels of equal capacity

bilayer low density parity check codes for decode and forward in relay channels this paper describes an efficient implementation of binning for decode and forward df in relay channels using low density parity check ldpc codes bilayer ldpc codes are devised to approach the theoretically promised rate of the df relaying strategy by incorporating relay generated parity bits in specially designed bilayer graphical code structures while conventional ldpc codes are sensitively tuned to operate efficiently at a certain channel parameter the proposed bilayer ldpc codes are capable of working at two different channel parameters and two different rates that at the relay and at the destination to analyze the performance of bilayer ldpc codes bilayer density evolution is devised as an extension of the standard density evolution algorithm based on bilayer density evolution a design methodology is developed for the bilayer codes in which the degree distribution is iteratively improved using linear programming further in order to approach to the theoretical df rate for a wide range of channel parameters this paper proposes two different forms of bilayer codes the bilayer expurgated and bilayer lengthened codes it is demonstrated that the rate of a properly designed bilayer ldpc code can closely approach the theoretical df limit finally it is shown that a generalized version of the proposed bilayer code construction is applicable to relay networks with multiple relays

multi edge type low density parity check codes for bandwidth efficient modulation a method of designing low density parity check codes for bandwidth efficient high order modulation is proposed a multi edge type ldpc code ensemble is used to improve the correspondence between modulation bit channel capacity and bit level protection in a bit interleaved coded modulation system a key innovation is the development of a multi dimensional extrinsic information transfer exit vector field technique for the analysis and design of multi edge type codes a condition sufficient for the decoder to converge is derived on the multi dimensional exit vector field and this condition is used as a constraint in code optimization code designs indicate that the proposed method produces codes matching the performance of codes designed using best available methods in ensemble threshold and is capable of achieving identical finite length error rates with shorter block lengths in addition to simplified design complexity the resulting codes allow for low complexity implementation and rate adaptivity and hence are well suited for adaptive modulation systems

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

design of irregular ldpc codes with optimized performance complexity tradeoff the optimal performance complexity tradeoff for error correcting codes at rates strictly below the shannon limit is a central question in coding theory this paper proposes a numerical approach for the minimization of decoding complexity for long block length irregular low density parity check ldpc codes the proposed design methodology is applicable to any binary input memoryless symmetric channel and any iterative message passing decoding algorithm with a parallel update schedule a key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence this paper shows that the proposed complexity measure can be accurately estimated from a density evolution and extrinsic information transfer chart analysis of the code a sufficient condition is presented for convexity of the complexity measure in the variable edge degree distribution when it is not satisfied numerical experiments nevertheless suggest that the local minimum is unique the results presented herein show that when the decoding complexity is constrained the complexity optimized codes significantly outperform threshold optimized codes at long block lengths within the ensemble of irregular codes

capacity and decoding rules for classes of arbitrarily varying channels the capacity of an arbitrarily varying channel avc is considered for deterministic codes with the average probability of error criterion and typically subject to at state constraint first sufficient conditions are provided that enable relatively simple decoding rules such as typicality maximum mutual information and minimum distance to attain capacity then the possibly noisy or channels and group adder channels are studied in detail for the former the capacity is explicitly determined and shown to be attainable by minimum distance decoding next for a large class of addictive avcs in addition to providing an intuitively suggestive simplification of the general avc capacity formula it is proven that capacity can be attained by a universal decoding rule finally the effect of random state selections on capacity is studied the merits and limitations of a previous mutual information game approach are also discussed

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

raptor codes on binary memoryless symmetric channels in this paper we will investigate the performance of raptor codes on arbitrary binary input memoryless symmetric channels bimscs in doing so we generalize some of the results that were proved before for the erasure channel we will generalize the stability condition to the class of raptor codes this generalization gives a lower bound on the fraction of output nodes of degree 2 of a raptor code if the error probability of the belief propagation decoder converges to zero using information theoretic arguments we will show that if a sequence of output degree distributions is to achieve the capacity of the underlying channel then the fraction of nodes of degree 2 in these degree distributions has to converge to a certain quantity depending on the channel for the class of erasure channels this quantity is independent of the erasure probability of the channel but for many other classes of bimscs this fraction depends on the particular channel chosen this result has implications on the universality of raptor codes for classes other than the class of erasure channels in a sense that will be made more precise in the paper we will also investigate the performance of specific raptor codes which are optimized using a more exact version of the gaussian approximation technique

properties of optimum binary message passing decoders we consider a class of message passing decoders for low density parity check ldpc codes whose messages are binary valued we prove that if the channel is symmetric and all codewords are equally likely to be transmitted an optimum decoding rule in the sense of minimizing message error rate should satisfy certain symmetry and isotropy conditions using this result we prove that gallager s algorithm b achieves the optimum decoding threshold among all binary message passing decoding algorithms for regular codes for irregular codes we argue that when the nodes of the message passing decoder do not exploit knowledge of their decoding neighborhood optimality of gallager s algorithm b is preserved we also consider the problem of designing irregular ldpc codes and find a bound on the achievable rates with gallager s algorithm b using this bound we study the case of low error rate channels and analytically find good degree distributions for them

on designing good ldpc codes for markov channels this paper presents a reduced complexity approximate density evolution de scheme for low density parity check ldpc codes in channels with memory in the form of a hidden markov chain this approximation is used to design degree sequences representing some of the best known ldpc code ensembles for the gilbert elliott channel and example optimizations are also given for other markov channels the problem of approximating the channel estimation is addressed by obtaining a specially constructed message passing schedule in which the channel messages all approach their stable densities it is shown that this new schedule is much easier to approximate than the standard schedule but has the same ultimate performance in the limits of long block length and many decoding iterations this result is extended to show that all message passing schedules that satisfy mild conditions will have the same threshold under density evolution

design of repeat accumulate codes for iterative detection and decoding extrinsic information transfer exit charts are used to design systematic and nonsystematic repeat accumulate ra codes for iterative detection and decoding the convergence problems of nonsystematic ra codes are solved by introducing a biregular or doped layer of check nodes as examples such nonsystematic codes are designed for multi input multi output mimo fading channels and are shown to operate close to capacity

efficient llr calculation for non binary modulations over fading channels log likelihood ratio llr computation for non binary modulations over fading channels is complicated a measure of llr accuracy on asymmetric binary channels is introduced to facilitate good llr approximations for non binary modulations considering piecewise linear llr approximations we prove convexity of optimizing the coefficients according to this measure for the optimized approximate llrs we report negligible performance loss compared to true llrs

nearest neighbor decoding for additive non gaussian noise channels we study the performance of a transmission scheme employing random gaussian codebooks and nearest neighbor decoding over a power limited additive non gaussian noise channel we show that the achievable rates depend on the noise distribution only via its power and thus coincide with the capacity region of a white gaussian noise channel with signal and noise power equal to those of the original channel the results are presented for single user channels as well as multiple access channels and are extended to fading channels with side information at the receiver

design of irregular weighted nonbinary repeat accumulate codes over gf q with q ary orthogonal modulation using a gaussian approximation using a gaussian approximation to the distribution of the message vectors under density evolution we design irregular weighted nonbinary repeat accumulate codes over gf q with q ary orthogonal modulation the resulting codes achieve a frame error rate of 10 1 within 0 56 to 0 91 db of channel capacity under the awgn channel

a new density evolution approximation for ldpc and multi edge type ldpc codes this paper considers density evolution for low density parity check ldpc and multi edge type ldpc met ldpc codes over the binary input additive white gaussian noise channel we first analyze three single parameter gaussian approximations for density evolution and discuss their accuracy under several conditions namely at low rates with punctured and degree one variable nodes we observe that the assumption of symmetric gaussian distribution for the density evolution messages is not accurate in the early decoding iterations particularly at low rates and with punctured variable nodes thus single parameter gaussian approximation methods produce very poor results in these cases based on these observations we then introduce a new density evolution approximation algorithm for ldpc and met ldpc codes our method is a combination of full density evolution and a single parameter gaussian approximation where we assume a symmetric gaussian distribution only after density evolution messages closely follow a symmetric gaussian distribution our method significantly improves the accuracy of the code threshold estimation additionally the proposed method significantly reduces the computational time of evaluating the code threshold compared with full density evolution thereby making it more suitable for code design

graph decomposition a new key to coding theorems a new and simple method is proposed for finding good encoders both for channels and for sources with side information this method relies on the continuous version of a graph decomposition result of lovasz the presently known best exponential error bounds for both problems follow in a unified manner with an improvement on the source coding bound the previous bounds for universal codes of the authors and marton are also improved

a more accurate one dimensional analysis and design of irregular ldpc codes we introduce a new one dimensional 1 d analysis of low density parity check ldpc codes on additive white gaussian noise channels which is significantly more accurate than similar 1 d methods our method assumes a gaussian distribution in message passing decoding only for messages from variable nodes to check nodes compared to existing work which makes a gaussian assumption both for messages from check nodes and from variable nodes our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence similar to previous work the problem of designing irregular ldpc codes reduces to a linear programming problem however our method allows irregular code design in a wider range of rates without any limit on the maximum variable node degree we use our method to design irregular ldpc codes with rates greater than 1 4 that perform within a few hundredths of a decibel from the shannon limit the designed codes perform almost as well as codes designed by density evolution

variations on the gallager bounds connections and applications there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles based on their distance spectrum we discuss many reported upper bounds on the maximum likelihood ml decoding error probability and demonstrate the underlying connections that exist between them in addressing the gallager bounds and their variations we focus on the duman and salehi see ieee trans commun vol 46 p 717 723 1998 variation which originates from the standard gallager bound a large class of efficient bounds or their chernoff versions is demonstrated to be a special case of the generalized second version of the duman and salehi bounds implications and applications of these observations are pointed out including the fully interleaved fading channel resorting to either matched or mismatched decoding the proposed approach can be generalized to geometrically uniform nonbinary codes finite state channels bit interleaved coded modulation systems and it can be also used for the derivation of upper bounds on the conditional decoding error probability

belief propagation algorithms on noisy hardware the wide recognition that emerging nano devices will be inherently unreliable motivates the evaluation of information processing algorithms running on noisy hardware as well as the design of robust schemes for reliable performance against hardware errors of varied characteristics in this paper we investigate the performance of a popular statistical inference algorithm belief propagation bp on probabilistic graphical models implemented on noisy hardware and we propose two robust implementations of the bp algorithm targeting different computation noise distributions we assume that the bp messages are subject to zero mean transient additive computation noise we focus on graphical models satisfying the contraction mapping condition that guarantees the convergence of the noise free bp we first upper bound the distances between the noisy bp messages and the fixed point of noise free bp as a function of the iteration number next we propose two implementations of bp namely censoring bp and averaging bp that are robust to computation noise censoring bp rejects incorrect computations to keep the algorithm on the right track to convergence while averaging bp takes the average of the messages in all iterations up to date to mitigate the effects of computation noise censoring bp works effectively when with high probability the computation noise is exactly zero and averaging bp although having a slightly larger overhead works effectively for general zero mean computation noise distributions sufficient conditions on the convergence of censoring bp and averaging bp are derived simulations on the ising model demonstrate that the two proposed implementations successfully converge to the fixed point achieved by noise free bp additionally we apply averaging bp to a bp based image denoising algorithm and as a bp decoder for ldpc codes in the image denoising application averaging bp successfully denoises an image even when nominal bp fails to do so in the presence of computation noise in the bp ldpc decoder application the power of averaging bp is manifested by the reduction in the residual error rates compared with the nominal bp decoder

mismatched decoding revisited general alphabets channels with memory and the wide band limit the mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given possibly suboptimal decoding rule this quantity has been studied extensively for single letter decoding rules over discrete memoryless channels dmcs here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non single letter decoding rules we also study the wide band limit and in particular the mismatch capacity per unit cost and the achievable rates on an additive noise spread spectrum system with single letter decoding and binary signaling

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

a converse coding theorem for mismatched decoding at the output of binary input memoryless channels an upper bound on the maximal transmission rate over binary input memoryless channels provided that the decoding decision rule is given is derived if the decision rule is equivalent to the maximum likelihood decoding matched decoding then the bound coincides with the channel capacity otherwise mismatched decoding it coincides with a known lower bound

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

performance of ldpc codes under faulty iterative decoding departing from traditional communication theory where decoding algorithms are assumed to perform without error a system where noise perturbs both computational devices and communication channels is considered here this paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low density parity check ldpc codes concentration of decoding performance around its average is shown to hold when noise is introduced into message passing and local computation density evolution equations for simple faulty iterative decoders are derived in one model computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases but arbitrarily small probability of error is not achievable probability of error may be driven to zero in another system model the decoding threshold again decreases smoothly with decoder noise as an application of the methods developed an achievability result for reliable memory systems constructed from unreliable components is provided

a counter example to the mismatched decoding converse for binary input discrete memoryless channels this paper studies the mismatched decoding problem for binary input discrete memoryless channels an example is provided for which an achievable rate based on superposition coding exceeds the csiszar korner hui rate thus providing a counter example to a previously reported converse result both numerical evaluations and theoretical results are used in establishing this claim

concatenated tree codes a low complexity high performance approach this paper is concerned with a family of concatenated tree ct codes ct codes are special low density parity check ldpc codes consisting of several trees with large spans they can also be regarded as special turbo codes with hybrid recursive nonrecursive parts and multiple constituent codes ct codes are decodable by the belief propagation algorithm they combine many advantages of ldpc and turbo codes such as low decoding cost fast convergence speed and good performance

does the performance of ldpc codes depend on the channel in this letter we discuss the performance of low density parity check ldpc codes on memoryless channels using a recently proposed analysis technique based on extrinsic information transfer exit charts we present an interpretation of the known fact that the bit error rate ber performance of an ensemble of ldpc codes shows little dependence on the specific memoryless channel this result has been partially observed in the literature for symmetric channels and is here extended to asymmetric channels we conjecture and demonstrate that the performance of an ensemble of ldpc codes depends primarily and solely on the mutual information mi between the input and the output of the channel as a validation of this conjecture we compare the performance of a few ldpc codes with various rates for five representative memoryless both symmetric and asymmetric channels obtaining results in excellent agreement with the exit chart based prediction

design of low density parity check codes for modulation and detection a coding and modulation technique is studied where the coded bits of an irregular low density parity check ldpc code are passed directly to a modulator at the receiver the variable nodes of the ldpc decoder graph are connected to detector nodes and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder the code is optimized by performing a curve fitting on extrinsic information transfer charts design examples are given for additive white gaussian noise channels as well as multiple input multiple output mimo fading channels where the receiver but not the transmitter knows the channel for the mimo channels the technique operates within 1 25 db of capacity for various antenna configurations and thereby outperforms a scheme employing a parallel concatenated turbo code by wide margins when there are more transmit than receive antennas

an analytic framework for identifying finite time coherent sets in time dependent dynamical systems abstract the study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems barriers to transport which mitigate mixing are currently the subject of intense study in the autonomous setting the use of transfer operators perron frobenius operators to identify invariant and almost invariant sets has been particularly successful in the nonautonomous time dependent setting coherent sets a time parameterised family of minimally dispersive sets are a natural extension of almost invariant sets the present work introduces a new analytic transfer operator construction that enables the calculation of finite time coherent sets sets are that minimally dispersive over a finite time interval this new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude the construction can also be applied to general markov processes on continuous state space

a constrained coding alternative to mppm multipulse pulse position modulation mppm has been widely proposed to improve data rate over traditional pulse position modulation ppm in free space optical communication systems encoders for mppm are typically lookup table based thus restricting the practical size of mppm codebooks power of two sized mppm codebooks typically do not have an efficient soft in soft out decoder making them poorly suited for concatenation with an outer code under iterative decoding in this paper a new coding technique based on constrained coding is introduced that allows construction of codes which have an efficient encoding algorithm more importantly these new codes are suitable for iterative soft decision decoding in concatenation with an outer error correcting code simulation results for both the gaussian and poisson channels show that a serial concatenation with an outer low density parity check code ldpc can achieve between 2 to 3 db coding gain over comparable reed solomon and ldpc coded mppm systems

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