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

a note on the nonexistence of solutions for prescribed mean curvature equations on a ball we prove the nonexistence of solutions for a prescribed mean curvature equation div u1 u 2 u p 1u x br rn u 0 x br when p 1 and the positive parameter is small the result extends theorems of narukawa and suzuki and finn from the case of n 2 p 1 to all n 2 p 1 moreover our proof is very simple and the result is not limited to positive and negative solutions we also show that a similar result for positive solutions is still true if u p 1u is replaced by the exponential nonlinearity eu 1

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

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

exact multiplicity and bifurcation diagrams of positive solutions of a one dimensional multiparameter prescribed mean curvature problem abstract we study the exact multiplicity and bifurcation diagrams of positive solutions u c 2 l l c l l of the one dimensional multiparameter prescribed mean curvature problem u x 1 u x 2 u p u q l x l u l u l 0 where 0 is a bifurcation parameter l 0 the radius of the one dimensional ball l l is an evolution parameter and 0 p q are two constants we prove that the problem has at most two positive solutions for any 0 p q and l 0 in addition if 0 p q q p p 1 2 p 1 we give a classification of totally three qualitatively different bifurcation diagrams on the u plane for any l 0 for any fixed p 0 and q q p p 2 2 2 p 3 we prove that there exist positive l l such that the bifurcation diagrams on the u plane are individually qualitatively different for the cases i 0 l l or l l ii l l iii l l l

positive solutions of the dirichlet problem for the prescribed mean curvature equation abstract we discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem div u 1 u 2 f x u in u 0 on in a general bounded domain r n depending on the behavior at zero or at infinity of f x s or of its potential f x s 0 s f x t d t our main effort here is to describe in a way as exhaustive as possible all configurations of the limits of f x s s 2 at zero and of f x s s at infinity which yield the existence of one two three or infinitely many positive solutions either strong or weak or bounded variation solutions are considered our approach is variational and combines critical point theory the lower and upper solutions method and elliptic regularization

global bifurcation diagrams and exact multiplicity of positive solutions for a one dimensional prescribed mean curvature problem arising in mems abstract we study global bifurcation diagrams and exact multiplicity of positive solutions for the one dimensional prescribed mean curvature problem arising in mems u x 1 u x 2 1 u p u 1 l x l u l u l 0 where 0 is a bifurcation parameter and p l 0 are two evolution parameters we determine the exact number of positive solutions by the values of p l and moreover for p 1 the bifurcation diagram undergoes fold and splitting bifurcations while for 0 p 1 the bifurcation diagram undergoes fold splitting and segment shrinking bifurcations our results extend and improve those of brubaker and pelesko n d brubaker j a pelesko analysis of a one dimensional prescribed mean curvature equation with singular nonlinearity nonlinear anal 75 2012 5086 5102 and pan and xing h pan r xing exact multiplicity results for a one dimensional prescribed mean curvature problem related to a mems model nonlinear anal rwa 13 2012 2432 2445 by generalizing the nonlinearity 1 u 2 to 1 u p with general p 1 we also answer an open question raised by brubaker and pelesko on the extension of global bifurcation diagram results to general p 0 concerning this open question we find and prove that global bifurcation diagrams for 0 p 1 are different to and more complicated than those for p 1

simple capacity achieving ensembles of rateless erasure correcting codes this paper is concerned with a simple binary erasure recovery coding scheme that falls into the family of so called semi random low density parity check sr ldpc codes based on a constrained random scrambling technique the proposed coding scheme is systematic rateless and capacity achieving we provide simulation examples comparing the new scheme with the well known luby transform lt and raptor codes it is shown that the new scheme has advantages in complexity and performance over its counterparts especially in channels with a relatively low erasure rate

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

exact number of solutions of a prescribed mean curvature equation abstract we study the exact number of positive solutions of the dirichlet problem for the one dimensional prescribed mean curvature equation u 1 u 2 u p u q u 0 u 1 0 where 0 is a parameter and p q satisfy either 1 p q or 0 p q 1

classical and non classical solutions of a prescribed curvature equation we discuss existence and multiplicity of positive solutions of the one dimensional prescribed curvature problem u root 1 u 2 lambda f t u u 0 0 u 1 0 depending on the behaviour at the origin and at infinity of the potential integral u 0 f t s ds besides solutions in w 2 w 1 0 1 we also consider solutions in w loc 2 1 0 1 which are possibly discontinuous at the endpoints of 0 1 our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem c 2007 elsevier inc all rights reserved

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

radial solutions for a prescribed mean curvature equation with exponential nonlinearity abstract we establish an existence result for radial solutions for a prescribed mean curvature equation with exponential nonlinearity our methods are based on degree theory combined with a time map analysis we also obtain two nonexistence results for positive solutions for more general f one of them is not limited to radial solutions

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

low density graph codes that are optimal for binning and coding with side information in this paper we describe and analyze the source and channel coding properties of a class of sparse graphical codes based on compounding a low density generator matrix ldgm code with a low density parity check ldpc code our first pair of theorems establishes that there exist codes from this ensemble with all degrees remaining bounded independently of block length that are simultaneously optimal for both channel coding and source coding with binary data when encoding and decoding are performed optimally more precisely in the context of lossy compression we prove that finite degree constructions can achieve any pair r d on the rate distortion curve of the binary symmetric source in the context of channel coding we prove that the same finite degree codes can achieve any pair c p on the capacity noise curve of the binary symmetric channel bsc next we show that our compound construction has a nested structure that can be exploited to achieve the wyner ziv bound for source coding with side information scsi as well as the gelfand pinsker bound for channel coding with side information ccsi although the results described here are based on optimal encoding and decoding the proposed graphical codes have sparse structure and high girth that renders them well suited to message passing and other efficient decoding procedures

exact number of solutions of a one dimensional prescribed mean curvature equation with concave convex nonlinearities abstract in this paper an exact number of solutions is obtained for the one dimensional prescribed mean curvature equation with concave convex nonlinearities in the form of u 1 u 2 u p u q u x 0 0 x 1 u 0 u 1 0 where 0 is a parameter and p q satisfy 0 p 1 q the arguments are based upon a time map method

a linear encoding approach to index assignment in lossy source channel coding we present a general scheme for the lossy transmission of a source with arbitrary statistics through a noisy channel under the mean square error fidelity criterion our approach is based on transform coding scalar quantization of the transform coefficients and linear encoding of the quantization indices entropy coding and channel coding are merged into a single linear encoding function such that the catastrophic behavior of conventional entropy coding is avoided and the full power of modern coding techniques and iterative belief propagation decoding can be exploited we show that this approach is asymptotically optimal in the limit of large block length for arbitrary source statistics and binary input output symmetric channel in the practical regime of finite block length and low decoding complexity we show through the explicit construction of codes for the lossy transmission of digital images over a binary symmetric channel that our approach yields significant improvements with respect to previously proposed channel optimized quantization schemes and also with respect to the conventional concatenation of state of the art image coding with state of the art channel coding although our constructive example focuses on a special case the approach is general and can be applied to other classes of sources of practical interest

one dimensional prescribed mean curvature equation with exponential nonlinearity abstract we establish the existence and multiplicity results for the one dimensional prescribed mean curvature equation u 1 u 2 e u u 0 0 u l 0 in particular we characterize the precise dependence of the multiplicity of solutions with respect to both the length l of the interval and the value of the parameter

exact multiplicity for semilinear elliptic dirichlet problems involving concave and convex nonlinearities let b be the unit ball in r n n 3 let 0 p q n 2 n 2 in 1994 ambrosetti et al found that the semilinear elliptic dirichlet problem admits at least two solutions for small 0 and no solution for large in this paper we prove that there is a critical number 0 such that this problem has exactly two solutions for 0 exactly one solution for and no solution for

fountain coding via multiplicatively repeated non binary ldpc codes we study fountain codes transmitted over the binary input symmetric output channel for channels with small capacity receivers in fountain coding systems needs to collects many channel outputs to recover information bits since a collected channel output yields a check node in the decoding tanner graph the channel with small capacity leads to large decoding complexity in this paper we introduce a novel fountain coding scheme with non binary ldpc codes the decoding complexity of the proposed fountain code does not depend on the channel numerical experiments show that the proposed codes exhibit better performance than conventional fountain codes especially for moderate number of information bits

optimization design and asymptotic analysis of systematic luby transform codes over biawgn channels in this paper we study systematic luby transform slt codes over binary input additive white gaussian noise biawgn channels to improve the bit error rate ber performance of slt codes on the biawgn channel this paper presents a novel optimization design of degree distributions first we apply the gaussian approximation ga to analyze the asymptotic ber performance of slt codes and calculate overhead thresholds for successful decoding second we derive an approximate closed form expression for lower bound on ber by applying the ga and further simplify the expression in low and high signal to noise ratio regimes respectively third we adopt the conventional linear programming clp constrained by the ga to optimize the degree distribution of slt codes the objective of clp is to minimize the average degree of the degree distribution that may result in bad error floor to improve the error floor caused by clp we put forth a novel optimization model constrained by the lower bound on ber and using the model we can design the degree distribution more flexibly with any desired ber simulation results show that the proposed degree distribution can provide better ber performance of slt codes over the biawgn channel

a physical layer rateless code for wireless channels in this paper we propose a physical layer rateless code for wireless channels a novel rateless encoding scheme is developed to overcome the high error floor problem caused by the low density generator matrix ldgm like encoding scheme in conventional rateless codes this is achieved by providing each symbol with approximately equal protection in the encoding process an extrinsic information transfer exit chart based optimization approach is proposed to obtain a robust check node degree distribution which can achieve near capacity performances for a wide range of signal to noise ratios snr simulation results show that under the same channel conditions and transmission overheads the bit error rate ber performance of the proposed scheme considerably outperforms the existing rateless codes in additive white gaussian noise awgn channels particularly at low ber regions

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

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

the prescribed mean curvature equation for nonparametric surfaces we study the prescribed mean curvature equation for nonparametric surfaces obtaining existence and uniqueness results in the sobolev space w2 p we also prove that under appropriate conditions the set of surfaces of mean curvature h is a connected subset of w2 p moreover we obtain existence results for a boundary value problem which generalizes the one dimensional periodic problem for the mean curvature equation

message wise unequal error protection based on low density parity check codes we propose a practical message wise unequal error protection uep scheme using low density parity check ldpc codes where one or more special messages are more protected than other ordinary messages in contrast to the information theoretic cavity coding scheme which discards the codewords of ordinary messages near those of special messages the proposed coding scheme performs codeword flipping to separate the codewords of special and ordinary messages without discarding any codewords to better distinguish the original and flipped codewords the ldpc codes with all odd degree check nodes are employed the decoder performs message type detection and codeword flipping detection based on the unsatisfied check nodes in iterative decoding we provide performance analysis for both the message type detection and the codeword flipping detection moreover we provide an asymptotic analysis on the detection error exponent to reveal the relationship between the proposed practical coding scheme and the information theoretically optimal cavity coding simulation results are provided to show that the proposed practical message wise uep schemes offer capacity approaching protections to both types of messages as if only one type is transmitted

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

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

raptor codes in the low snr regime in this paper we revisit the design of raptor codes for binary input additive white gaussian noise channels where we are interested in very low signal to noise ratios snrs a linear programming degree distribution optimization problem is defined for raptor codes in the low snr regime through several approximations we also provide an exact expression for the polynomial representation of the degree distribution with infinite maximum output node degree in the low snr regime which enables us to calculate the exact value of the fractions of output nodes of small degrees a more practical degree distribution design is also proposed for raptor codes in the low snr regime where we include the rate efficiency and the decoding complexity in the optimization problem and an upper bound on the maximum rate efficiency is derived for given design parameters simulation results show that the raptor code with the designed degree distributions can approach rate efficiencies larger than 0 95 in the low snr regime

response surface designs for quantitative and qualitative variables when some variables are quantitative and some qualitative in a response surface context standard designs may not be suitable the reasons for this are illuminated some alternative designs are discussed every n point design provides n linearly independent estimation contrasts some of these p say are needed to estimate the p parameters of the postulated model the remaining n p linearly independent estimation contrasts are available to estimate pure error if used and to test for lack of fit either overall or in particular ways the key to choosing a good design is to use the available degrees of freedom well given certain assumptions about the model to be fitted when there is also uncertainty about the model assumptions dogmatic design advice is not possible sound guidelines are available however and these are presented and illustrated

time maps and exact multiplicity results for one dimensional prescribed mean curvature equations ii abstract we investigate various properties of time maps for one dimensional prescribed mean curvature equations using these properties we obtain some exact multiplicity results of positive solutions and sign changing solutions as it turned out these quasilinear problems show many different phenomena from semilinear problems our methods are based on a detailed analysis of time maps

joint source channel coding for deep space image transmission using rateless codes a new coding scheme for image transmission over noisy channel is proposed similar to standard image compression the scheme includes a linear transform followed by successive refinement scalar quantization unlike conventional schemes though in the proposed system the quantized transform coefficients are linearly mapped into channel symbols using systematic linear encoders this fixed to fixed length linear index coding approach avoids the use of an explicit entropy coding stage e g arithmetic or huffman coding which is typically fragile to channel post decoding residual errors we use linear codes over gf 4 which are particularly suited for this application since they are matched to the dead zone quantizer symbol alphabet and to the qpsk modulation used on the deep space communication channel we optimize the proposed system where the linear codes are systematic raptor codes over gf 4 the rateless property of raptor encoders allows to implement a continuum of coding rates in order to accurately match the channel coding rate to the transmission channel capacity and to the quantized source entropy rate for each transform subband and refinement level comparisons are provided with respect to the concatenation of state of the art image coding and channel coding schemes used by jet propulsion laboratories jpl for the mars exploration rover mer mission

linear llr approximation for iterative decoding on wireless channels on a fading channel with no channel state information at the receiver true log likelihood ratios llr are complicated functions of the channel output it is assumed in the literature that the power of the additive noise is known and the expected value of the fading gain is used in a linear function of the channel output to find approximate llrs this approach however is not optimal in the sense of bit error rate performance in this paper we introduce a measure of accuracy for the approximate llrs based on their probability density function and we show that this measure provides a very convenient tool for finding good approximate llrs assuming that the power of the additive noise is known and using the proposed measure we find a linear llr approximation whose performance is extremely close to that of the true llr calculation on an uncorrelated rayleigh fading channel these results are then extended to the case that the noise power is also unknown and a performance almost identical to the previous case is obtained

ripple design of lt codes for biawgn channels this paper presents a novel framework which enables a design of rateless codes for binary input additive white gaussian noise biawgn channels using the ripple based approach known from the works for the binary erasure channel bec we reveal that several aspects of the analytical results from the bec also hold in biawgn channels the presented framework is applied in a code design example which shows promising results compared to existing work in particular it shows a great robustness towards variations in the signal to noise power ratio snr contrary to existing codes

protograph based raptor like ldpc codes this paper proposes protograph based raptor like pbrl codes as a class of rate compatible low density parity check codes for binary input awgn channels as with the raptor codes exclusive or operations on precoded bits produce additional parity bits providing extensive rate compatibility unlike raptor codes each additional parity bit in the protograph is explicitly designed to optimize the density evolution threshold during the lifting process approximate cycle extrinsic message degree ace and circulant progressive edge growth cpeg constraints are used to avoid undesirable graphical structures some density evolution performance is sacrificed to obtain lower error floors particularly at short blocklengths simulation results are shown for information block sizes of k 1032 hbox and 16 384 for a target frame error rate of 10 5 at each rate the k 1032 hbox and 16 384 code families perform within 1 db and 0 4 db of both the gallager bound and the normal approximation respectively the 16 384 code family outperforms the best known standardized code family namely the ar4ja codes the pbrl codes also outperform dvb s2 codes that have the advantages of longer blocklengths and outer bch codes performance is similar to rc code families designed by nguyen et al that do not constrain codes to have the pbrl structure and involve simulation in the optimization process at each rate

rateless coding for gaussian channels a rateless code i e a rate compatible family of codes has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones a perfect family of such codes is one in which each of the codes in the family is capacity achieving we show by construction that perfect rateless codes with low complexity decoding algorithms exist for additive white gaussian noise channels our construction involves the use of layered encoding and successive decoding together with repetition using time varying layer weights as an illustration of our framework we design a practical three rate code family we further construct rich sets of near perfect rateless codes within our architecture that require either significantly fewer layers or lower complexity than their perfect counterparts variations of the basic construction are also developed including one for time varying channels in which there is no a priori stochastic model

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