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

correction of mismatched l values in bicm receivers in this work we analyze the problem of correction of the reliability metrics l values in bit interleaved coded modulation bicm receivers first we propose a method for finding the linear correction factors that minimize the probability of error of a maximum likelihood decoder that uses the corrected l values to this end we use the efficient approximation of the pairwise error probability in the domain of the cumulant generating functions cgf of the l values and conclude that the optimal correction factors are equal to the twice of value of the saddlepoint of the cgf next a simple extension of the proposed method to the non linear correction is presented we provide numerical examples demonstrating improvements attainable with the proposed method comparing it to the competitive solutions

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

imprimitive ninth degree number fields with small discriminants we present tables of ninth degree nonprimitive i e containing a cubic subfield number fields each table corresponds to one signature except for fields with signature 3 3 for which we give two different tables depending on the signature of the cubic subfield details related to the computation of the tables are given as well as information about the cpu time used the number of polynomials that we deal with etc for each field in the tables we give its discriminant the discriminant of its cubic subfields the relative polynomial generating the field over one of its cubic subfields the corresponding irreducible polynomial over q and the galois group of the galois closure fields having interesting properties are studied in more detail especially those associated with sextic number fields having a class group divisible by 3

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

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

turbo spc codes this letter is concerned with a family of modified turbo type codes referred to as turbo spc single parity check codes a technique based on the spc code is introduced to replace puncturing for rate adjustment a noticeable feature of the proposed scheme is the significantly reduced decoding complexity compared with the standard punctured turbo code the cost reduction factor becomes more noticeable as rate increases over ten times for rate spl rarr 1 the error rate floor problem known to turbo codes can also be improved using the proposed scheme for moderate to high rates performances of about 0 5 db from the theoretical limits at ber 10 sup 5 have been observed for turbo spc codes

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

mmse decision feedback equalizers and coding ii coding results for pt i see ibid vol 43 no 10 p 2582 1995 the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure in systems that combine equalization with coded modulation with a given symbol rate 1 t and transmit spectrum the output signal to noise ratio snr sub mmse dfe u of a mmse dfe with an unbiased decision rule is a single parameter that characterizes the channel for coding purposes indeed the transmit spectrum that maximizes snr sub mmse dfe u is the capacity achieving water pouring spectrum and the capacity c t in bits per two dimensions is given by c t log sub 2 1 snr sub mmse dfe u regardless of the channel characteristics the performance of a coded system with a mmse dfe equalization structure may be accurately estimated using the gain of the coding scheme at a given pr e this performance is shown to be approximately the same as that of a multicarrier system using the same transmit spectrum and similar coding such systems are known to be able to approach capacity arbitrarily closely the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels simulation results indicate that performance of the mmse dfe is surprisingly insensitive to transmit spectral shaping as long as the transmit spectrum exceeds the capacity achieving band but that there is an optimal symbol rate that should approximately be used

on the real cubic fields in this paper the authors announce a table of the 4753 totally real nonconjugate nonabelian cubic fields with discriminant less than 100000 each field is given by its discriminant the coefficients of a generating polynomial and the index of this polynomial over the field a basis of the integers of the field is also given some differences with other tables are pointed out godwin and samet 3 have described the construction of a table of real cubic fields with discriminants d 20000 using similar methods angell 1 extends these results up to d 100000 using a different method the authors 5 have constructed a table of the 4753 real nonabelian cubic fields with discriminants d 100000 here and in the sequel triplets of conjugate fields are counted once only the method developed in 5 generalizes the one used in 4 and the table constructed there gives for each cubic field k its discriminant an irreducible polynomial f x which defines k the index of f x over k and a basis for the integers of k in this work we present some consequences derived from 5 in particular we have discovered that ten fields are missing from angell s table in table 1 we give each of these ten fields and its corresponding class numbers h the field k with discriminant d is generated by a root 0 of the polynomial f x x3 ax b s is the index of 0 and 1 0 a is a basis of the integers of k where 02 to t 2 a a

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

optimal puncturing ratios and energy allocation for multiple parallel concatenated codes we propose a systematic design framework for optimal low complexity punctured multiple parallel concatenated codes mpccs based on minimizing the convergence threshold using extrinsic information transfer exit charts as the convergence threshold is related to the area between the two exit curves the corresponding optimization problem is equivalent to a curve fitting problem the exit curves are determined by the respective exit functions of the constituents which can be conveniently shaped through the use of random puncturing and unequal energy allocations across parallel coding streams the design task is therefore to find the optimal combination of constituents puncturing ratios and energy allocation for matching the exit curves a search over all rate one convolutional codes of memory length four or less is performed identifying 98 classes of codes with unique exit functions out of a total of 310 codes low complexity mpccs with up to four constituents are found where the convergence thresholds are observed to be within 0 1 db or less of the fundamental minimum signal to noise ratio snr corresponding to the binary phase shift keying bpsk capacity for code rates 1 3 les r les 7 8 further allowing for unequal energy allocation the convergence thresholds for lower code rates are similarly improved

optimization of signal sets for partial response channels i numerical techniques given a linear time invariant discrete time channel the problem of constructing n input signals of finite length k that maximize minimum l sub 2 distance between pairs of outputs is considered two constraints on the input signals are considered a power constraint on each of the n inputs hard constraint and an average power constraint over the entire set of inputs soft constraint the hard constraint problem is equivalent to packing n points in an ellipsoid in min k n 1 dimensions to maximize the minimum euclidean distance between pairs of points gradient based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves shannon capacity assuming additive white gaussian noise

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

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

classification of unique mappings for 8psk based on bit wise distance spectra the performance of bit interleaved coded modulation bicm with or without iterative decoding id is significantly influenced by the mapping of bits to the symbol constellation our main objective in this paper is to develop a systematic design approach for bicm id schemes ensuring the best possible performance with iterative decoding although useful mappings for bicm id have been found based on various search strategies no attempt has been made to systematically enumerate and classify all unique mappers for a given constellation as the basis for a systematic enumeration and classification we define the average bit wise distance spectrum for a mapping from bits to symbols different bit wise distance spectra are derived assuming no prior information or full prior information respectively the bit wise distance spectra determine corresponding bit wise error probability and bit wise mutual information the latter allows us to use the classification of mappings with unique bit wise distance spectra to also classify mappings with unique extremal points in the corresponding extrinsic information transfer exit curves as an example of our approach we classify 8psk mappings into 86 classes of unique mappings according to bit wise distance spectra the classification can be used to significantly reduce the complexity of the search for suitable mappers for bicm id for 8psk and a given encoder only 86 different mappings need to be investigated as examples of the systematic design approach the best 8psk mappings for minimizing the convergence threshold are found for concatenation with the rate 1 2 5 7 8 and 133 171 8 convolutional codes and the rate 1 2 umts turbo code with identical constituent convolutional codes 15 13 8

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

coset codes for partial response channels or coset codes with spectral nulls known coset codes are adapted for use on partial response channels or to generate signals with spectral nulls by using coset precoding and running digital sum feedback any desired tradeoff can be achieved between the power and spectra of the relevant sequences up to the optimum tradeoff possible a fundamental theorem specifying this optimum tradeoff is given a maximum likelihood sequence estimation mlse decoder for the original code may be used for the adapted code and such a decoder then attains the minimum squared distance of the original code these methods sometimes generate codes with greater minimum squared distance than that of the original code this distance can be attained by augmented decoders although such decoders inherently require long decoding delays and may be subjected to quasi catastrophic error propagation the authors conclude that at least for sequences supporting large numbers of bits per symbol coset codes can be adapted to achieve effectively the same performance and complexity on partial response channels or for sequences with spectral nulls as they do in the ordinary memoryless case

multilevel codes theoretical concepts and practical design rules this paper deals with 2 sup l ary transmission using multilevel coding mlc and multistage decoding msd the known result that mlc and msd suffice to approach capacity if the rates at each level are appropriately chosen is reviewed using multiuser information theory it is shown that there is a large space of rate combinations such that mlc and full maximum likelihood decoding mld can approach capacity it is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of mld the capacity rule the balanced distances rules and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies moreover topics relevant in practical applications such as signal set labeling dimensionality of the constituent constellation and hard decision decoding are emphasized bit interleaved coded modulation proposed by caire et al see ibid vol 44 p 927 46 1998 is reviewed in the context of mlc finally the combination of signal shaping and coding is discussed significant shaping gains are achievable in practice only if these design rules are taken into account

salem numbers in dynamics on kahler threefolds and complex tori let x be a compact kahler manifold of dimension k le 4 and f x rightarrow x a pseudo automorphism if the first dynamical degree lambda 1 f is a salem number we show that either lambda 1 f lambda k 1 f or lambda 1 f 2 lambda k 2 f in particular if dim x 3 then lambda 1 f lambda 2 f we use this to show that if x is a complex 3 torus and f is an automorphism of x with lambda 1 f 1 then f has a non trivial equivariant holomorphic fibration if and only if lambda 1 f is a salem number if x is a complex 3 torus having an automorphism f with lambda 1 f lambda 2 f 1 but is not a salem number then the picard number of x must be 0 3 or 9 and all these cases can be realized

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

punctured turbo codes for bpsk qpsk channels we discuss a procedure for designing high rate turbo codes via puncturing with applications to bpsk qpsk channels rates in the form k k 1 2 spl les k spl les 16 are considered for constituent encoders with memory sizes m 3 and 4 the algorithm includes the selection of constituent encoder generator polynomials puncture patterns and interleavers with the goal of maximizing the minimum codeword weight for weight two and weight three inputs the performance of the proposed codes are found via computer simulation and are observed in each case to be less than 0 9 db m 3 and 0 75 db m 4 from their respective channel capacity limits at a bit error rate of 10 sup 5 we consider two applications of punctured turbo codes

exit chart based power allocation for iterative frequency domain mimo detector transmission power allocation in single carrier multiple input multiple output mimo systems with iterative frequency domain fd soft cancellation sc minimum mean squared error mmse equalization is considered a novel framework for transmission power minimization subject to equalizer convergence constraints referred as convergence constrained power allocation ccpa method is proposed based on extrinsic information transfer exit chart analysis the proposed method decouples the spatial interference between the streams using singular value decomposition svd and minimizes the transmission power while achieving the target mutual information for each stream after iterations at the receiver we show that the transmission power optimization can be formulated as a convex optimization problem three ccpa methods one approximately optimal and other two heuristic methods inspired by the lagrange duality are derived the numerical results demonstrate that the proposed scheme outperforms the existing linear precoding schemes moreover the proposed heuristic schemes can achieve performance close with that of the approximately optimal method in terms of the equalizer convergence properties as well as transmission power

a vertex centered and positivity preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids abstract we suggest a new positivity preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids the scheme has vertex centered edge midpoint and cell centered unknowns the vertex centered unknowns are primary and have finite volume equations associated with them the edge midpoint and cell centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns which makes the final scheme a pure vertex centered one unlike most existing positivity preserving schemes the construction of the scheme is based on a special nonlinear two point flux approximation that has a fixed stencil and does not require the convex decomposition of the co normal in order to solve efficiently the nonlinear systems resulting from the nonlinear scheme picard method and its anderson acceleration are discussed numerical experiments demonstrate the second order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the anderson acceleration is also shown on reduction of the number of nonlinear iterations moreover the proposed scheme does not have the so called numerical heat barrier issue suffered by most existing cell centered and hybrid schemes however further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

the effect of a precoder on serially concatenated coding systems with an isi channel the performance of a serially concatenated system which includes a channel with memory preceded by a precoder as a rate 1 inner coder is presented the effect of different precoders on the maximum likelihood bit error performance is analyzed the precoder weight gain which explains the good bit error rate ber performance is identified through a union bound analysis precoders are divided into two groups based on an analysis of the euclidean distance and its multiplicity and each precoder group shows a distinct ber curve behavior it is shown that the ber curves for two precoder groups cross over each other convolutional codes are considered as outer codes in simulations on various intersymbol interference channels several important design considerations for the choice of precoders are derived based on the analysis and these are confirmed through simulations with an iterative decoding algorithm

a modified trellis coding technique for partial response channels the problem of trellis coding for multilevel baseband transmission over partial response channels with transfer polynomials of the form 1 or d sup n is addressed the novel method presented here accounts for the channel memory by using multidimensional signal sets and partitioning the signal set present at the noiseless channel output it is shown that this coding technique can be viewed as a generalization of a well known procedure for binary signaling the concatenation of convolutional codes and inner block codes that are tuned to the channel polynomial it results in high coding gains with moderate complexity if some bandwidth expansion is accepted

general formulas for solving solvable sextic equations abstract let g be a transitive solvable subgroup of s6 we show that there is a common formula for finding the roots of all irreducible sextic polynomials f x q x with gal f g moreover once the roots ri are calculated there is an explicit procedure for numbering them so that the galois group acts via ri r i for g s6 we also demonstrate new criteria for determining the galois group of an irreducible sextic polynomial f x q x the first two results generalize results of d s dummit 1991 math comp 57 387 401 for quintic polynomials

irregular trellis for the near capacity unary error correction coding of symbol values from an infinite set irregular joint source and channel coding jscc scheme is proposed which we refer to as the irregular unary error correction iruec code this code operates on the basis of a single irregular trellis instead of employing a set of separate regular trellises as in previous irregular trellis based codes our irregular trellis is designed with consideration of the uec free distance which we characterize for the first time in this paper we conceive the serial concatenation of the proposed iruec code with an irregular unity rate code irurc code and propose a new extrinsic information transfer exit chart matching algorithm for parametrizing these codes this facilitates the creation of a narrow exit tunnel at a low e text b n 0 value and provides near capacity operation owing to this our scheme is found to offer a low symbol error ratio ser which is within 0 4 db of the discrete input continuous output memoryless channel dcmc capacity bound in a particular practical scenario where gray mapped quaternary phase shift keying qpsk modulation is employed for transmission over an uncorrelated narrowband rayleigh fading channel with an effective throughput of 0 508 text bit text s 1 text hz 1 furthermore the proposed iruec irurc scheme offers a ser performance gain of 0 8 db compared to the best of several regular and irregular separate source and channel coding sscc benchmarkers which is achieved without any increase in transmission energy bandwidth transmit duration or decoding complexity

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

synchronizable codes for the optical oppm channel random overlapping pulse position modulation oppm sequences result in an unrecoverable error floor on both the probability of erroneous synchronization and the probability of symbol error when only chip synchronization is present it is known however that for a given sequence length m a subset of the set of all possible sequences is synchronizable in the sense that in the absence of noise the receiver can correctly symbol synchronize by observing m or more symbol intervals the authors design finite state machines and codes over a j ary alphabet which produce sequences with the property that every subsequence of length l is synchronizable some of the codes in addition to being synchronizable produce a coding gain for an optical poisson channel the authors introduce joint synchronization and detection algorithms that utilize the memory in the encoded sequences to produce joint estimates of timing and sequences their performance is analyzed through simulations and analytical results

a decoupled and positivity preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes abstract we suggest a new positivity preserving discrete duality finite volume ddfv scheme for anisotropic diffusion problems on general polygonal meshes like existing ddfv schemes this scheme is built on the primary and dual meshes and has finite volume fv equations for both vertex and cell centered unknowns the transpose of the coefficient matrix for the fv equations of the cell centered unknowns is an m matrix while that for the vertex unknowns is not an m matrix but a symmetric and positive definite matrix by employing a certain truncation technique for the vertex unknowns the positivity preserving property for both categories of unknowns is guaranteed local conservation is strictly maintained for the cell centered unknowns and conditionally maintained for the vertex unknowns since the fv equations of the vertex unknowns can be solved independently the two sets of fv equations are decoupled in contrast to existing nonlinear positivity preserving schemes the new scheme requires no nonlinear iterations for linear problems for nonlinear problems the positivity preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted moreover the positivity of the discrete solution is proved and the well posedness is analyzed rigorously for linear problems in numerical experiments the new scheme is examined extensively and compared with two positivity preserving cell centered schemes and a nonlinear ddfv scheme numerical results show that the present ddfv scheme achieves second order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations

coherent and differential space time shift keying a dispersion matrix approach motivated by the recent concept of spatial modulation sm we propose a novel space time shift keying stsk modulation scheme for multiple input multiple output mimo communication systems where the concept of sm is extended to include both the space and time dimensions in order to provide a general shift keying framework more specifically in the proposed stsk scheme one out of q dispersion matrices is activated during each transmitted block which enables us to strike a flexible diversity and multiplexing tradeoff this is achieved by optimizing both the space time block duration as well as the number of the dispersion matrices in addition to the number of transmit and receive antennas we will demonstrate that the resultant equivalent system model does not impose any inter channel interference ici and hence the employment of single stream maximum likelihood ml detection becomes realistic at a low complexity furthermore we propose a differential stsk dstsk scheme assisted by the cayley unitary transform which does not require any channel state information csi at the receiver here the usual error doubling caused by the differential decoding gives rise to 3 db performance penalty in comparison to coherent stsk cstsk additionally we introduce an enhanced cstsk scheme which avoids the requirement of inter antenna synchronization ias between the rf chains associated with the transmit antenna elements aes by imposing a certain constraint on the dispersion matrix design where each column of the dispersion matrices includes only a single non zero component moreover according to the turbo coding principle the proposed cstsk and dstsk schemes are combined with multiple serially concatenated codes and an iterative bit to symbol soft demapper more specifically the associated stsk parameters are optimized with the aid of extrinsic information transfer exit charts for the sake of achieving a near capacity performance

concatenated codes for class iv partial response channels we present the design and analysis of parallel and serial concatenated codes with iterative decoding for class iv partial response pr4 signaling code rates of the form k sub 0 k sub 0 1 for k sub 0 4 8 16 32 and 64 are considered we show that the 1 1 spl oplus d sup 2 tomlinson precoder first proposed by wolf and ungerboeck 1991 for trellis coded pr4 channels is effective for both parallel and serial concatenations but the error rate floor is relatively high for the serial case this leads to new interleaver and precoder designs for serial concatenations the latter involving an examination of the conditional distance enumerator of each precoded pr4 channel the new precoders substantially reduce the floors of the serial concatenated codes but degrade the performance of the parallel concatenated codes for a bit error rate of 10 sup 5 the best parallel and serial concatenated codes come within 1 7 db of a lower bound on the capacity limit for pr4 signaling on an additive white gaussian noise channel

coded sampling bound how much training is needed for iterative semi blind channel estimation coherent detection is commonly facilitated by means of pilot aided channel estimation while this approach offers a high estimation accuracy the spectral and power efficiencies are reduced due to the pilot overhead in the case that redundancy due to channel coding is not used for the purpose of channel estimation the maximum pilot spacing for which a time varying channel can be reconstructed without distortion is given by the nyquist shannon sampling theorem in this paper it is shown that if channel coding is taken into account for channel estimation in an iterative fashion the maximum spacing can be significantly extended which results in an increased spectral efficiency towards this goal a so called coded sampling bound is derived this semi analytical bound is compared with the nyquist shannon sampling theorem the presented results indicate that the maximum pilot spacing can be arbitrarily large given a suitable channel code and code rate

trellis coding for partial response channels we consider trellis coding techniques for improving the reliability of digital transmission over noisy partial response channels such channels are commonly encountered in digital communication systems and also play a role in devices for data recording concentrating on the channels with characteristics 1 mp d we study methods to obtain codes which increase free euclidean distance between permitted sequences of channel outputs and avoid the occurrence of unlimited runs of identical outputs at the expense of some loss in data rate one technique employs the concept of set partitioning the other is based on using convolutional codes designed for maximum free hamming distance in conjunction with a precoder both methods lead to essentially equivalent codes

new ternary line codes based on trellis structure the trellis coding technique is applied to line coded baseband digital transmission systems for r n n 1 n 1 2 3 coding rates a new codeword assignment model is proposed to accomplish basic requirements for line coding in which each length n binary data sequence is encoded into a length n 1 ternary 0 line codeword chosen among the code alphabet with 2 sup n 2 elements assuming viterbi decoding the system error performance is improved by increasing the free euclidean distance between coded sequences a new algorithm is given for the calculation of the free distance between line coded sequences so obtained for r 1 2 and r 3 4 rates the analytical error performance upper bounds are derived the power spectral densities of the new line codes are also calculated and compared with those of known line codes

on invariant polynomials and their application in field theory certain polynomials invariant under a permutation group g so called g polynomials play an important role in several computational methods of galois theory since their practical value depends on the degree it is important to know g polynomials of smallest possible degree a reasonable technique to find such g polynomials is presented and for certain classes of groups an explicit description is obtained the list of g polynomials given by stauduhar in vol 27 of this journal is thereby enlarged and improved as an application of g polynomials three important resolvents of quintic and sextic algebraic equations are computed and a parametric family of sextic equations with given galois group is exhibited introduction let k be a field and x1 x indeterminates let sm be the group of all permutations of 1 m which acts on the polynomial ring k xl xmi by s o xi xs i s e sm i 1 m let g be a subgroup of sm several problems in field theory e g the determination of the galois group or the computation of a galois resolvent of a given algebraic equation can be solved explicitly if one can do the following see 7 p 276 16 11 2 3 and references in the last paper i find a polynomial p belonging to g also called a g polynomial i e a polynomial p in k x1 xm whose stabilizer s e sm i p p is equal to g ii let z be an additional indeterminate and a m am the elementary symmetric functions defined by z x1 zxm zm a1zm l am let r e k x xm z be the polynomial of z degree sm g whose zeros in k x1 xm are the elements of s o p i s e sm find the unique polynomial r e k x xm z such that r r a am z suppose that problems i and ii have been solved for g 5 sm if f zm a1zm l am e k z is a polynomial with coefficients in k and splitting field k f then r a1 am z is a certain resolvent of f i e its splitting field is a subfield of k f the prime factor decomposition of r a1 am z in k z for example gives insight into the relation between g and the galois group g f gal k f k of f by this method one can determine the group g f loc cit up to now actual solutions of problem ii exist only in very few cases if g is the alternating group am take r z2 d where d is the discriminant of zm xizm xm for subgroups g of sm m 4 see 10 p 72 ff if m 5 and received september 16 1982 1980 mathematics subject classification primary 12 04 12f10 20b99 1987 american mathematical society 0025 5718 87 1 00 25 per page

enumeration of quartic fields of small discriminant with the mixed type case now completed all algebraic number fields of degree 4 with absolute discriminant 106 have been enumerated methods from the totally real and totally complex cases were used without major mod ification isomorphism of fields was determined by a method similar to one of lenstra the t2 criterion of pohst was applied to reduce the number of redundant examples

reduced state representations for trellis codes using constellation symmetry this paper presents a symmetry based technique for trellis code state diagram reduction that has more general applicability than the quasi regularity technique of rouanne et al and zehavi et al for trellis codes using standard constellations and labelings for a 2 sup spl nu x state trellis code the new technique reduces the 2 sup 2 spl nu x state diagram to 2 sup spl nu x spl nu q state diagram where 0 spl les spl nu sub q spl les spl nu sub x the particular value of spl nu sub q depends on the constellation labeling and the convolutional encoder for standard rate k k 1 set partitioned trellis codes spl nu sub q 0 and the overall number of states is the same with the new technique as with quasi regularity for codes that are not quasi regular and thus not amenable to the quasi regularity technique the new technique often provides some improvement when spl nu sub q spl nu sub x for 8 phase shift keying trellis codes the new technique always yields spl nu sub q 0

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

a targeted martinet search constructing number fields with prescribed ramification is an important problem in computational number theory in this paper we consider the problem of computing all imprimitive number fields of a given degree which are unramified outside of a given finite set of primes s by combining the techniques of targeted hunter searches with martinet s relative version of hunter s theorem we then carry out this algorithm to generate complete tables of imprimitive number fields for degrees 4 through 10 and certain sets s of small primes

design of protograph ldpc codes for partial response channels we investigate the performance of the protograph low density parity check ldpc codes which have been shown to possess simple structures and outstanding error performance over additive white gaussian noise awgn channels over partial response pr channels using the finite length extrinsic information transfer exit algorithm due to the intersymbol interference isi caused by the pr channels we observe that the conventional protograph ldpc codes do not perform well in terms of error rates we further propose a new design scheme and construct three new types of protograph ldpc codes unlike conventional protograph ldpc codes in which the highest degree variable nodes are punctured the new protograph ldpc codes have their lowest degree variable nodes punctured moreover some edge re connections are made in one of the proposed codes the exit chart analysis the convergence analysis and the bit error rate simulation have shown that all three new codes outperform the conventional protograph ldpc codes moreover two of the proposed codes are superior to the regular column weight 3 ldpc code and thus they are good alternatives compared to other error correction codes for use in data storage systems

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

a table of quintic number fields all algebraic number fields f of degree 5 and absolute discriminant less than 2 x 107 totally real fields respectively 5 x 106 other signatures are determined we describe the methods which we applied and list significant data

symmetric functions m sets and galois groups given the elementary symmetric functions in ri i 1 n we describe algorithms to compute the elementary symmetric functions in the products ri ri2 rimi 1 il im n and in the sums ri ri2 rim 1 il im n the computation is performed over the coefficient ring generated by the elementary symmetric functions we apply fft multiplication of series to reduce the complexity of the algorithm for sums an application to computing galois groups is given

a new class of multiple rate codes based on block markov superposition transmission the hadamard transform ht over the binary field provides a natural way to implement multiple rate codes referred to as ht coset codes where the code length n 2 p is fixed but the code dimension k can be varied from 1 to n 1 by adjusting the set of frozen bits the ht coset codes including reed muller rm codes and polar codes as typical examples can share the same fundamental encoder decoder architecture with the implementation complexity of order o n log n however to guarantee that all codes with designated rates perform well ht coset coding usually requires a sufficiently large code length which in turn causes difficulties in the determination of which bits are better for being frozen in this paper we propose to transmit short ht coset codes in the so called block markov superposition transmission bmst manner at the transmitter signals are spatially coupled via superposition resulting in long codes at the receiver these coupled signals are recovered by a sliding window iterative soft successive cancellation decoding algorithm most importantly the performance around or below the bit error rate ber of 10 5 can be predicted by a simple genie aided lower bound both these bounds and simulation results show that the bmst of short ht coset codes performs well within one db away from the corresponding shannon limits in a wide range of code rates

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

optimization of a lattice based constellation for signaling over a partial response channel this article discusses the problem of optimizing a lattice based signal constellation for signaling over a partial response channel the objective is to minimize the probability of error which is determined by the combined effects of the additive gaussian noise and channel memory

numerical artifacts in the generalized porous medium equation why harmonic averaging itself is not to blame abstract the degenerate parabolic generalized porous medium equation gpme poses numerical challenges due to self sharpening and its sharp corner solutions for these problems we show results for two subclasses of the gpme with differentiable k p with respect to p namely the porous medium equation pme and the superslow diffusion equation spurious temporal oscillations and nonphysical locking and lagging have been reported in the literature these issues have been attributed to harmonic averaging of the coefficient k p for small p and arithmetic averaging has been suggested as an alternative we show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues here we investigate the causes of these numerical artifacts using modified equation analysis the modified equation framework can be used for any type of discretization we show results for the second order finite volume method the observed problems with harmonic averaging can be traced to two leading error terms in its modified equation this is also illustrated numerically through a modified harmonic method mhm that can locally modify the critical terms to remove the aforementioned numerical artifacts

the mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient numerical schemes for nonlinear parabolic equations based on the harmonic averaging of cell centered diffusion coefficients break down when some of these coefficients go to zero or their ratio grows to tackle this problem we propose new mimetic finite difference schemes that use a staggered discretization of the diffusion coefficient the primary mimetic operator approximates div k the derived dual mimetic operator approximates the new mimetic schemes preserve symmetry and positive definiteness of the continuum problem which allows us to use algebraic solvers with optimal complexity we perform detailed numerical analysis of the new schemes for linear elliptic problems and a specially designed linear parabolic problem that has solution dynamics typical for nonlinear problems we show that the new schemes are competitive with the state of the art schemes for steady state problems but provide much more accurate solution dynamics for the transient problem

on codes satisfying mth order running digital sum constraints multi level sequences with a spectral null of order m at frequency f meaning that the power spectral density and its first 2m 1 derivatives vanish at f are characterized by finite state transition diagrams fstds whose edge labels satisfy bounds on the variation of the mth order running digital sum rds necessary and sufficient conditions for fstds with higher order null constraints at dc and at an arbitrary submultiple of the symbol frequency are derived analytical results are given concerning the performance of codes satisfying an mth order rds constraint on partial response channels specific code designs for quaternary channel inputs are presented the euclidean distance properties of this new class of codes aside from their spectral shaping properties are demonstrated

near capacity three stage turbo detection of irregular convolutional coded joint sphere packing modulation and space time coding conventional two stage turbo detected schemes typically suffer from a bit error rate ber floor preventing them from achieving infinitesimally low ber values especially when the inner coding stage is of non recursive nature we circumvent this deficiency by proposing a three stage turbo detected sphere packing sp aided space time block coding stbc stbc sp scheme where a rate 1 recursive inner precoder is employed to avoid having a ber floor the convergence behaviour of this serially concatenated scheme is investigated with the aid of 3d extrinsic information transfer exit charts furthermore the capacity of the stbc sp scheme is determined and an algorithm is proposed for calculating a tighter upper bound on the maximum achievable bandwidth efficiency based on the exit charts of the stbc sp demapper the proposed three stage turbo detected scheme operates within about 1 0 db of the capacity and within 0 5 db of the maximum achievable bandwidth efficiency limit

baseband trellis codes with a spectral null at zero a method is described for modifying classical n dimensional trellis codes to provide baseband codes that combine a spectral null at dc with significant coding gain the information rate of the classical code is decreased by one bit and this extra redundancy is used to keep the running digital sum bounded equivalently if the rate is held constant then twice as many signal points are needed causing a power penalty of 6 n db baseband trellis codes are presented for several information rates together with complete spectral plots and performance comparisons a method of constructing baseband codes with multiple spectral nulls is also described

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

binary convolutional codes with application to magnetic recording calderbank heegard and ozarow 1 have suggested a method of designing codes for channels with intersymbol interference such as the magnetic recording channel these codes are designed to exploit intersymbol interference the standard method is to minimize intersymbol interference by constraining the input to the channel using run length limited sequences calderbank heegard and ozarow considered an idealized model of an intersymbol interference channel that leads to the problem of designing codes for a partial response channel with transfer function 1 d n 2 where the channel inputs are constrained to be pm 1 this problem is considered here channel inputs are generated using a nontrivial coset of a binary convolutional code the coset is chosen to limit the zero run length of the output of the channel and so maintain clock synchronization the minimum squared euclidean distance between outputs corresponding to distinct inputs is bounded below by the free distance of a second convolutional code which we call the magnitude code an interesting feature of the analysis is that magnitude codes that are catastrophic may perform better than those that are noncatastrophic

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

discrete multiple tone modulation with coset coding for the spectrally shaped channel a discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference isi and additive gaussian noise multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete fourier transform dft code vectors and high gain coset or trellis codes symbol blocks from an inverse dft idft are cyclically extended to generate isi free channel output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels asymptotic performance of this system is derived and examples of asymptotic and finite block length coding gain performance for several channels are evaluated at different values of bits per sample this discrete multiple tone technique is linear in both the modulation and the demodulation and is free from the effects of error propagation that often afflict systems employing bandwidth optimized decision feedback plus coset codes

reduced complexity approx log map and max log map soft psk qam detection algorithms in this paper we propose to reduce the complexity of both the approx log map algorithm as well as of the max log map algorithm conceived for generalized psk qam detection where only a reduced size subset of the psk qam constellation points is taken into account for producing a single soft bit output although the detectors of gray labelled low order psk qam schemes generally produce near horizontal exit curves our proposed detectors exploit the a priori llrs gleaned from a channel decoder in order to retain the optimum detection capability for all psk qam constellations furthermore we demonstrate in this paper that the widely applied mimo schemes including v blast and stbc which invoke the proposed soft psk qam detectors may also benefit from our reduced complexity design our simulation results confirm that a near capacity performance may be achieved by the proposed detectors at a substantially reduced detection complexity

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

trellis based scalar vector quantizer for memoryless sources the paper describes a structured vector quantization approach for stationary memoryless sources that combines the scalar vector quantizer svq ideas laroia and farvardin 1993 with trellis coded quantization marcellin and fischer 1990 the resulting quantizer is called the trellis based scalar vector quantizer tb svq the svq structure allows the tb svq to realize a large boundary gain while the underlying trellis code enables it to achieve a significant portion of the total granular gain for large block lengths and powerful possibly complex trellis codes the tb svq can in principle achieve the rate distortion bound as indicated by the results obtained even for reasonable block lengths and relatively simple trellis codes the tb svq outperforms all other fixed rate quantizers at reasonable complexity

on achieving an asymptotically error free fixed point of iterative decoding for perfect a priori information in this paper we provide necessary and sufficient conditions for constituent codes in multiple concatenated and graph based coding schemes to achieve an asymptotically error free iterative decoding fixed point if the maximum possible a priori information is available at least one constituent code in an iterative decoding scheme must satisfy these conditions in order to ensure an asymptotically vanishing bit error probability at the convergence point of the decoder our results are proved for arbitrary binary input symmetric memoryless channels bismcs and thus can be universally applied to many transmission scenarios specifically using a factor graph framework it is shown that non inner codes in a serial concatenation or check nodes in generalized ldpc codes achieve perfect extrinsic information if and only if the minimum hamming distance between codewords is two or greater for inner codes in a serial concatenation constituent codes in a parallel concatenation or variable nodes in doubly generalized ldpc codes the corresponding encoder condition for acquiring perfect extrinsic information is an infinite codeword weight for a weight one input sequence for this case we provide a general proof which holds for all linear encoders and bismcs we also show that these results can improve the performance of concatenated coding schemes

information rates and power spectra of digital codes the encoding of independent data symbols as a sequence of discrete amplitude real variables with given power spectrum is considered the maximum rate of such an encoding is determined by the achievable entropy of the discrete sequence with the given constraints an upper bound to this entropy is expressed in terms of the rate distortion function for a memoryless finite alphabet source and mean square error distortion measure a class of simple dc free power spectra is considered in detail and a method for constructing markov sources with such spectra is derived it is found that these sequences have greater entropies than most codes with similar spectra that have been suggested earlier and that they often come close to the upper bound when the constraint on the power spectrum is replaced by a constraint on the variance of the sum of the encoded symbols a stronger upper bound to the rate of dc free codes is obtained finally the optimality of the binary biphase code and of the ternary bipolar code is decided

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

trellis coded multidimensional phase modulation a 2l dimensional multiple phase shift keyed l mpsk signal set is obtained by forming the cartesian product of l two dimensional mpsk signal sets a systematic approach to partitioning l mpsk signal sets that is based on block coding is used an encoder system approach is developed it incorporates the design of a differential precoder a systematic convolutional encoder and a signal set mapper trellis coded l 4psk l 8psk and l 16psk modulation schemes are found for 1

two layer coded spatial modulation with block markov superposition transmission this paper is concerned with the spatial modulation sm a multiple input multiple output mimo transmission technique that maps information bits not only into the conventional two dimensional signal points but also into the indices of active transmit antennas we present a two layer coded sm scheme in which the spatial bits carried by the antenna indices and the signal bits carried by the conventional signals are protected separately by two error correction codes for the ease of decoding process the code rates are allocated according to the chain rule of the mutual information we choose block markov superposition transmission bmst codes for each layer since they are easily designed for any given code rate with a predictable performance lower bound an iterative sliding window decoding algorithm is also presented by exchanging messages iteratively between the two bmst decoders and the soft in soft out siso demapper of the sm to reduce the computational complexity we propose to implement the siso demapping algorithm by employing only partial soft inputs numerical results show that the bmst sm system performs well over uncorrelated rayleigh fading channels

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

on the computation of totally real quartic fields of small discriminant all totally real quartic fields of discriminant less than 106 are computed the method used to generate the fields is derived from delone and faddeev with corrections and improvements a new method for deciding field i8omorphism is used to eliminate redundant examples integral bases and galois groups are given for each field 1 introduction in 3 pp 184 200 delone and faddeev give a number geometric method for computing generating equations for all totally real quartic fields of discriminant not exceeding a fixed upper bound l e r 0 this method is well suited for implementation on a computer unfortunately the original paper contains many mistakes and the proofs are sometimes hard to understand in the present paper we give a corrected version of the delone faddeev algorithm we describe its implementation and we give a table of all totally real quartic fields of discriminant less than 1 000 000 including their integral bases and the galois groups of their normal closures a different method for computing totally real quartic fields was given by god win 6 and for general algorithms for computing fields of small discriminants see martinet 8 and pohst 9 10

reduced state sequence estimation with set partitioning and decision feedback a reduced state sequence estimator for linear intersymbol interference channels is described the estimator uses a conventional viterbi algorithm with decision feedback to search a reduced state subset trellis that is constructed using set partitioning principles the complexity of maximum likelihood sequence estimation mlse due to the length of the channel memory and the size of the signal set is systematically reduced an error probability analysis shows that a good performance complexity tradeoff can be obtained in particular the results indicate that the required complexity to achieve the performance of mlse is independent of the size of the signal set for large enough signal sets simulation results are provided for two partial response systems a simple technique for quadrature partial response signaling qprs is described that eliminates the quasicatastrophic nature of the ml trellis

a table of totally real quintic number fields we give a table of the 1077 totally real number fields of degree five having a discriminant less than 2 000 000 there are two nonisomorphic fields of discriminant 1 810 969 and two nonisomorphic fields of discriminant 1 891 377 all the other number fields in the table are characterized by their discriminant among these fields three are cyclic and four have a galois closure whose galois group is the dihedral group d5 the galois closure for all the other fields in the table has a galois group isomorphic to s5

tables of octic fields with a quartic subfield we describe the computation of extended tables of degree 8 fields with a quartic subfield using class field theory in particular we find the minimum discriminants for all signatures and for all the possible galois groups we also discuss some phenomena and statistics discovered while making the tables such as the occurrence of 11 non isomorphic number fields having the same discriminant or several pairs of non isomorphic number fields having the same dedekind zeta function