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

two double exponential gaps for automata with a limited pushdown abstract we shall consider nondeterministic and deterministic automata equipped with a limited pushdown constant height npda s and dpda s as well as their two way versions constant height 2 npda s and 2 dpda s we show two double exponential gaps for these devices namely i for complementing constant height one way npda s and ii for converting constant height 2 npda s or 2 dpda s into one way devices

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

design of serially concatenated systems depending on the block length based on extrinsic information transfer exit charts the convergence behavior of iterative decoding is studied for a number of serially concatenated systems such as a serially concatenated code coded data transmission over an intersymbol interference channel bit interleaved coded modulation or trellis coded modulation efficient optimization algorithms based on simplified exit chart construction are devised to find irregular codes improving the convergence of iterative decoding one optimization criterion is to find concatenated systems exhibiting thresholds of successful decoding convergence which are close to information theoretic limits however these thresholds are approached only for very long block lengths to overcome this problem the decoding convergence after a fixed finite number of iterations is optimized which yields systems performing very well for short block lengths too as an example optimal system configurations for communication over an additive white gaussian noise channel are presented

asymptotic behavior of solutions to elliptic and parabolic equations with unbounded coefficients of the second order in unbounded domains we study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain under the assumption that the solution belongs to l q with q in 2 infty we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients as a corollary we obtain the liouville type theorem in the case when the coefficients may grow at the spacial infinity we also study a corresponding parabolic problem in the n dimensional whole space and discuss the energy identity for solutions in l q as a corollary we show also the liouville type theorem for both forward and ancient solutions

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

bounds on information combining when the same data sequence is transmitted over two independent channels or when a data sequence is transmitted twice but independently over the same channel the independent observations can be combined at the receiver side from an information theory point of view the overall mutual information between the data sequence and the received sequences represents a combination of the mutual information of the two channels this concept is termed information combining a lower bound and an upper bound on the combined information is presented and it is proved that these bounds are tight furthermore this principle is extended to the computation of extrinsic information on single code bits for a repetition code and for a single parity check code of length three respectively for illustration of the concept and the bounds on information combining two applications are considered first bounds on the information processing characteristic ipc of a parallel concatenated code are derived from its extrinsic information transfer exit chart second bounds on the exit chart for an outer repetition code and for an outer single parity check code of a serially concatenated coding scheme are computed

density evolution for nonbinary ldpc codes under gaussian approximation this paper extends the work on density evolution for binary low density parity check ldpc codes with gaussian approximation to ldpc codes over gf q we first generalize the definition of channel symmetry for nonbinary inputs to include q ary phase shift keying psk modulated channels for prime q and binary modulated channels for q that is a power of 2 for the well defined q ary input symmetric output channel we prove that under the gaussian assumption the density distribution for messages undergoing decoding is fully characterized by q 1 quantities assuming uniform edge weights we further show that the density of messages computed by the check node decoder cnd is fully defined by a single number we then present the approximate density evolution for regular and irregular ldpc codes and show that the q 1 dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q 2p through application of approximate density evolution and linear programming we optimize the degree distribution of ldpc codes over gf 3 and gf 4 the optimized irregular ldpc codes demonstrate performance close to the shannon capacity for long codewords we also design gf q codes for high order modulation by using the idea of a channel adapter we find that codes designed in this fashion outperform those optimized specifically for the binary additive white gaussian noise awgn channel for a short codewords and a spectral efficiency of 2 bits per channel use b cu

analysis of the iterative decoding of ldpc and product codes using the gaussian approximation we propose a novel density evolution approach to analyze the iterative decoding algorithms of low density parity check ldpc codes and product codes based on gaussian densities namely for these classes of codes we derive a one dimensional 1d map whose iterates directly represent the error probability both for the additive white gaussian noise awgn and the rayleigh fading channel these simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm as an application we compute the decoding thresholds and show that they are consistent with the simulation results available in the literature

tree expectation propagation for ml decoding of ldpc codes over the bec we propose a decoding algorithm for ldpc codes that achieves the maximum likelihood ml solution over the binary erasure channel bec in this channel the tree structured expectation propagation tep decoder improves the peeling decoder pd by processing check nodes of degree one and two however it does not achieve the ml solution as the tree structure of the tep allows only for approximate inference in this paper we provide the procedure to construct the structure needed for exact inference this algorithm denoted as generalized tree structured expectation propagation gtep modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph the gtep decoder upon completion either provides the unique ml solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ml solution we also explain the algorithm as a gaussian elimination method relating the gtep to other ml solutions compared to previous approaches it presents an equivalent complexity it exhibits a simpler graphical message passing procedure and most interesting the algorithm can be generalized to other channels

liouville type theorems for the euler and the navier stokes equations abstract we prove liouville type theorems for weak solutions of the navier stokes and the euler equations in particular if the pressure satisfies p l 1 0 t l 1 r n with r n p x t d x 0 then the corresponding velocity should be trivial namely v 0 on r n 0 t in particular this is the case when p l 1 0 t h q r n where h q r n q 0 1 the hardy space on the other hand we have equipartition of energy over each component if p l 1 0 t l 1 r n with r n p x t d x 0 similar results hold also for the magnetohydrodynamic equations

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

extremes of information combining extreme densities for information combining are found for two important channel models the binary input symmetric parallel broadcast channel and the parity constrained input symmetric parallel channels following upper and lower mutual information thresholds are stated for per bit maximum a posteriori probability map decoding and low density parity check ldpc code ensembles

the size cost of boolean operations on constant height deterministic pushdown automata we study the size cost of boolean operations on constant height deterministic pushdown automata i e on the traditional pushdown automata with a built in constant limit on the height of the pushdown we design a simulation showing that a complement can be obtained with a polynomial tradeoff for intersection and union we show an exponential simulation and prove that the exponential blow up cannot be avoided

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

near capacity turbo coded soft decision aided dapsk star qam for amplify and forward based cooperative communications multilevel differential amplitude and phase shift keying dapsk schemes do not require any channel estimation which results in low complexity in this treatise we derive the soft output probability formulas required for a soft decision based demodulation of high order dapsk in order to facilitate iterative detection by exchanging extrinsic information with an outer turbo code tc furthermore when the tc block size is increased the system operates closer to the channel capacity compared to the identical throughput tc assisted 64 ary differential phase shift keying 64 dpsk scheme the 4 ring based tc assisted 64 ary dapsk arrangement has a power efficiency improvement of 2 3 db at a bit error rate ber of 10 5 furthermore when the tc block size is increased the system operates closer to the channel capacity more specifically when using a tc block length of 400 modulated symbols the 64 dapsk 4 16 scheme is 7 56 db away from its capacity curve while it had a reduced gap as low as 2 25 db when using a longer tc block length of 40 000 modulated symbols finally as a novel application example the soft decision m dapsk scheme was incorporated into an amplify and forward af based cooperative communication system which attains another 4 5 db snr improvement for a tc block length of 40 000 modulated symbols

language operations with regular expressions of polynomial size this work deals with questions regarding to what extent regularity preserving language operations affect the descriptional complexity of regular expressions some language operations are identified which are feasible for regular expressions in the sense that the result of the operation can be represented as a regular expression of size polynomial in that of the operands we prove that taking language quotients in particular the prefix and suffix closures of a regular set can incur at most a quadratic blow up on the required expression size the circular shift operation can cause only a cubic increase in size and at least a quadratic bloat can be necessary in the worst case

near capacity turbo trellis coded modulation design based on exit charts and union bounds transactions papers bandwidth efficient parallel concatenated turbo trellis coded modulation ttcm schemes were designed for communicating over uncorrelated rayleigh fading channels a symbol based union bound was derived for analysing the error floor of the proposed ttcm schemes a pair of in phase i and quadrature phase q interleavers were employed for interleaving the i and q components of the ttcm coded symbols in order to attain an increased diversity gain the decoding convergence of the iq ttcm schemes was analysed using symbol based extrinsic information transfer exit charts the best ttcm component codes were selected with the aid of both the symbolbased union bound and non binary exit charts for designing capacity approaching iq ttcm schemes in the context of 8 psk 16 qam 32 qam and 64 qam modulation schemes

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

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

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

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

ldpc codes for flat rayleigh fading channels with channel side information in this paper we design capacity approaching low density parity check ldpc codes in the low signal to noise ratio snr regime for flat rayleigh fading channels with channel side information at transmitter and receiver we use the structure advocated by caire et al which uses a single codebook with dynamic power allocation the extrinsic information transfer exit function method is used to design the ldpc codes which approach the channel capacities we also study the exit function properties of various demappers

on the design of fast convergent ldpc codes for the bec an optimization approach the complexity performance trade off is a fundamental aspect of the design of low density parity check ldpc codes in this paper we consider ldpc codes for the binary erasure channel bec use code rate for performance metric and number of decoding iterations to achieve a certain residual erasure probability for complexity metric we first propose a quite accurate approximation of the number of iterations for the bec moreover a simple but efficient utility function corresponding to the number of iterations is developed using the aforementioned approximation and the utility function two optimization problems w r t complexity are formulated to find the code degree distributions we show that both optimization problems are convex in particular the problem with the proposed approximation belongs to the class of semi infinite problems which are computationally challenging to be solved however the problem with the proposed utility function falls into the class of semi definite programming sdp and thus the global solution can be found efficiently using available sdp solvers numerical results reveal the superiority of the proposed code design compared to existing code designs from literature

on strong convergence to 3d steady vortex sheets in this paper we first establish a strong convergence criterion of approximate solutions for the 3d steady incompressible euler equations for axisymmetric flows under the assumption that the vorticity is of one sign and uniformly bounded in l 1 space we obtain a sufficient and necessary condition for the strong convergence in l 2 r 3 of approximate solutions furthermore for one sign and l 1 bounded vorticity it is shown that if a sequence of approximate solutions concentrates at an isolated point in r z plane then the concentration point can appear neither in the region near the axis including the symmetry axis itself nor in the region far away from the axis finally we present an example of approximates solutions which converge strongly in l 2 r 3 by using hill s spherical vortex

more concise representation of regular languages by automata and regular expressions we consider two formalisms for representing regular languages constant height pushdown automata and straight line programs for regular expressions we constructively prove that their sizes are polynomially related comparing them with the sizes of finite state automata and regular expressions we obtain optimal exponential and double exponential gaps i e a more concise representation of regular languages

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

an mse based transfer chart for analyzing iterative decoding schemes using a gaussian approximation an alternative to extrinsic information transfer exit charts called mean square error mse charts that use a measure related to the mse instead of mutual information is proposed using the relationship between mutual information and minimum mean square error mmse for the additive white gaussian noise awgn channel a relationship between the rate of any code and the area under a plot of mmse versus signal to noise ratio snr is obtained when the a priori log likelihood ratio llr is from a binary input gaussian channel using this result a justification is provided for designing concatenated codes by matching the exit curves of the inner and outer decoder when the llrs are assumed to be gaussian which is also the typical assumption used for code design using exit charts even though the gaussian assumption is almost never true the results presented in this paper represent a step toward the analysis of iterative decoding schemes using a single parameter finally for the special case of awgn channel it is shown that any capacity achieving code has an exit curve that is a step function

near capacity code design for entanglement assisted classical communication over quantum depolarizing channels we have conceived a near capacity code design for entanglement assisted classical communication over the quantum depolarizing channel the proposed system relies on efficient near capacity classical code designs for approaching the entanglement assisted classical capacity of a quantum depolarizing channel it incorporates an irregular convolutional code ircc a unity rate code urc and a soft decision aided superdense code sd which is hence referred to as an ircc urc sd arrangement furthermore the entanglement assisted classical capacity of an n qubit superdense code transmitted over a depolarizing channel is invoked for benchmarking it is demonstrated that the proposed system operates within 0 4 db of the achievable noise limit for both 2 qubit as well as 3 qubit sd schemes more specifically our design exhibits a deviation of only 0 062 and 0 031 classical bits per channel use from the corresponding 2 qubit and 3 qubit capacity limits respectively the proposed system is also benchmarked against the classical convolutional and turbo codes

efficient computation of exit functions for nonbinary iterative decoding the calculation of nonbinary extrinsic information transfer charts for the iterative decoding of concatenated index based codes is addressed we show that the extrinsic information at the output of a constituent a posteriori probability decoder can be calculated with very low complexity where expensive histogram measurements are not required any more an example for turbo trellis coded modulation demonstrates the capabilities of the proposed approach

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

the third support weight enumerators of the doubly even self dual 32 16 8 codes we present combinatorial methods for computing the third support weight enumerators of the five doubly even self dual 32 16 8 codes the methods exploit relationships that exist between support weight enumerators and complete coset weight enumerators of a self dual code

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

liouville theorems for the stationary navier stokes equation on a hyperbolic space abstract the problem for the stationary navier stokes equation in 3d under finite dirichlet norm is open in this paper we answer the analogous question on the 3d hyperbolic space we also address other dimensions and more general manifolds

unveiling turbo codes some results on parallel concatenated coding schemes a parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver the input bits to the first encoder are scrambled by the interleaver before entering the second encoder the codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders this construction can be generalized to any number of constituent codes parallel concatenated schemes employing two convolutional codes as constituent codes in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes have been previously shown to yield remarkable coding gains close to theoretical limits they have been named and are known as turbo codes we propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length the analytical bounding technique is then used to shed some light on some crucial questions which have been floating around in the communications community since the proposal of turbo codes