metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

optimal information bit decoding of linear block codes we consider the problem of decoding a linear block code used over a binary symmetric channel with crossover probability p when the goal of the decoding is to minimize the probability of an information bit error in general there will be several different decoding strategies each corresponding to the optimal rule over some range p spl isin p sub i 1 p sub i each of these strategies can be implemented by standard array with the appropriate choice of coset leaders that are not in general of minimum weight we give optimal rules for selecting coset leaders in the first region 0 p sub 1 and the region p sub max 1 2 for any linear mapping between information bits and codewords the determination of the optimal rule has been an open problem for small p the rule for p p sub max is important because for many codes when p is fixed the optimal rule for any coset corresponds to either the optimal rule for small p or the optimal rule for p near 1 2 the optimal decoding strategy varies with the generator matrix used in general but it is demonstrated that the optimal generator matrix is separable when d sup spl perp 2 the results are shown to extend in a natural way to the case where the goal is to minimize the probability that arbitrary collections of information bits are in error we examine some specific codes and compare the performance under optimal decoding to the performance of previously suggested strategies

linear time nearest point algorithms for coxeter lattices the coxeter lattices are a family of lattices containing many of the important lattices in low dimensions this includes an e 7 e 8 and their duals an e 7 and e 8 we consider the problem of finding a nearest point in a coxeter lattice we describe two new algorithms one with worst case arithmetic complexity o nlogn and the other with worst case complexity o n where n is the dimension of the lattice we show that for the particular lattices an and an the algorithms are equivalent to nearest point algorithms that already exist in the literature

asymptotic performance of single parity check product codes this article investigates the asymptotic performance of single parity check spc product codes pcs from a decoding point of view specifically the probability of bit error is bounded before and after the decoding of each dimension similar to the analysis of iterated codes by elias 1954 it is shown that the asymptotic probability of bit error can be driven to zero as the number of dimensions and hence the block length increases at signal to noise ratios snrs within 2 db of capacity over the additive white gaussian noise awgn channel

nonlinear codes the product construction the standard product construction is discussed with respect to nonlinear codes thus so called nonlinear product codes are obtained that are better than linear product codes of similar length and code rate and at the same time amenable for encoding decoding on the other hand it is shown that certain notorious nonlinear codes have an augmented product construction namely they can be constructed by taking the union of a product code and certain of its cosets the binary hamming codes are shown to have similar construction a simple two stage decoder is proposed for nonlinear product nlp codes the decoder is shown to be a bounded distance bd information decoder that is the nonlinear equivalent of the bd decoder employed for linear codes a list based maximum likelihood decoder is also discussed

rule chaining in fuzzy expert systems a fuzzy expert system must do rule chaining differently than a nonfuzzy expert system in particular any rule that can fire with a particular linguistic variable in its consequent must fire before any rule whose antecedent conditions depend upon the resultant fuzzy set value of the consequent linguistic variable is allowed to fire the dependent rules would be considered in a chain with the fuzzy rules which generate or assert the needed fuzzy linguistic variable a recent paper by j pan et al 1998 points out that a version of the fuzzyclips expert system shell does not operate with chained fuzzy rules as one would expect they introduce fuzzyshell which is described as the only known shell to have the expected fuzzy rule chaining performance we show several approaches to obtaining the desired behavior in fuzzyclips further a potential pitfall with the fuzzyshell approach to dealing with chaining is pointed out

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

an efficient algorithm for soft decision decoding of the 24 12 extended golay code an efficient algorithm for soft decision decoding of the 24 12 extended golay code is described results obtained for white gaussian noise are presented which show that performance is only a few tenths of a decibel away from that of an ideal correlator

the stretched golay and other codes for high snr finite delay quantization of the gaussian source at 1 2 bit per sample error correcting codes are considered as codebooks for high performance vector quantization vq of the iid gaussian source at fractional bit rate a family of good rate one half codes is introduced the stretched golay codes the performance of these codes is compared to other good block codes trellis coded quantization and other techniques the stretched golay codes are shown to outperform previously published results for block lengths 32 40 56 and 64 the good performance together with fast decoding make these techniques attractive for applications such as low bit rate coding of speech

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

corrections to sub riemannian geometry an error in the proof of corollary 6 2 of 1 has been pointed out by gerard ben arous the computation of m x in the case o 0 on p 243 is incorrect because m x 0 when 0 0 and jg x 0 for all k there is also a factor of missing in the formula as stated for 0 0 but this is not significant thus when applying the pontryagin theorem theorem 6 1 it is necessary to consider one additional case o 0 and m x9 0 along the curve the conditions of the theorem in this case lead to the equations

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

reliability based code search algorithms for maximum likelihood decoding of block codes efficient code search maximum likelihood decoding algorithms based on reliability information are presented for binary linear block codes the codewords examined are obtained via encoding the information set utilized for encoding comprises the positions of those columns of a generator matrix g of the code which for a given received sequence constitute the most reliable basis for the column space of g substantially reduced computational complexity of decoding is achieved by exploiting the ordering of the positions within this information set the search procedures do not require memory the codeword to be examined is constructed from the previously examined codeword according to a fixed rule consequently the search algorithms are applicable to codes of relatively large size they are also conveniently modifiable to achieve efficient nearly optimum decoding of particularly large codes

tables of sphere packings and spherical codes the theta function of a sphere packing gives the number of centers at each distance from the origin the theta functions of a number of important packings a n d n e n the leech lattice and others and tables of the first fifty or so of their coefficients are given in this paper

decoding the golay codes we introduce exceptionally simple decoding algorithms for the two extended golay codes the algorithms are based on recent methods of conway and curtis of finding the unique blocks containing five points in either the 5 8 24 steiner system or the 5 6 12 steiner system these decoding methods are simple enough to enable decoding extended golay codes by hand both of the methods involve relations between the extended golay codes and other self dual codes proofs are given explaining these relationships and why the decoding methods work the decoding algorithms are explained and illustrated with many examples 3 chap 12 has facts about the mathieu group and some details about decoding the golay codes

bounded distance decoding algorithms decision regions and pseudo nearest neighbors for a code c bounded distance decoding algorithms perform as optimal algorithms within the balls b c centered at the codewords c spl isin c with radius equal to half the minimum euclidean distance of the code thus distinct bounded distance algorithms vary in performance due to their different behavior outside the balls b c we investigate this issue by analyzing the decision regions of some known e g gmd and some new bounded distance algorithms presented in this work in particular we show that there are three distinct types of nearest neighbors and classify them according to their influence on the decision region simulation results and computer generated images of the decision regions are provided to illustrate the analytical results for block and lattice codes on additive white gaussian noise awgn channels

sub riemannian geometry of the coefficients of univalent functions abstract we consider coefficient bodies m n for univalent functions based on the lowner kufarev parametric representation we get a partially integrable hamiltonian system in which the first integrals are kirillov s operators for a representation of the virasoro algebra then m n are defined as sub riemannian manifolds given a lie poisson bracket they form a grading of subspaces with the first subspace as a bracket generating distribution of complex dimension two with this sub riemannian structure we construct a new hamiltonian system to calculate regular geodesics which turn to be horizontal lagrangian formulation is also given in the particular case m 3

generalization of chase algorithms for soft decision decoding of binary linear codes soft decision decoding of binary linear block codes transmitted over the additive white gaussian channel awgn using antipodal signaling is considered a set of decoding algorithms called generalized chase algorithms is proposed in contrast to chase algorithms which require a lfloor d 1 2 rfloor binary error correcting decoder for decoding a binary linear block code of minimum distance d the generalized chase algorithms can use a binary decoder that can correct less than lfloor d 1 2 rfloor hard errors the chase algorithms are particular cases of the generalized chase algorithms the performance of all proposed algorithms is asymptotically optimum for high signal to noise ratio snr simulation results for the 47 23 quadratic residue code indicate that even for low snr the performance level of a maximum likelihood decoder can be approached by a relatively simple decoding procedure

complementary sets generalized reed muller codes and power control for ofdm the use of error correcting codes for tight control of the peak to mean envelope power ratio pmepr in orthogonal frequency division multiplexing ofdm transmission is considered in this correspondence by generalizing a result by paterson it is shown that each q phase q is even sequence of length 2m lies in a complementary set of size 2k 1 where k is a nonnegative integer that can be easily determined from the generalized boolean function associated with the sequence for small k this result provides a reasonably tight bound for the pmepr of q phase sequences of length 2 m a new 2h ary generalization of the classical reed muller code is then used together with the result on complementary sets to derive flexible ofdm coding schemes with low pmepr these codes include the codes developed by davis and jedwab as a special case in certain situations the codes in the present correspondence are similar to paterson s code constructions and often outperform them

error trellises for convolutional codes ii decoding methods for pt i see ibid vol 46 p 1592 1601 1998 soft decision maximum likelihood decoding of convolutional codes over gf q can be accomplished via searching through an error trellis for the least weighing error sequence the error trellis is obtained by a syndrome based construction its structure lends itself particularly well to the application of expedited search procedures the method to carry out such error trellis based decoding is formulated by four algorithms three of these algorithms are aimed at reducing the worst case computational complexity whereas by applying the fourth algorithm the average computational complexity is reduced under low to moderate channel wise level the syndrome decoder achieves substantial worst case and average computational gains in comparison with the conventional maximum likelihood decoder namely the viterbi decoder which searches for the most likely codeword directly within the code

differential trellis decoding of convolutional codes this paper investigates the principle of metric differences for trellis decoding of convolutional codes based on this differential method a new algorithm referred to as differential trellis decoding dtd is proposed dtd offers an alternative to the conventional add compare select acs method for implementing the viterbi algorithm

reed muller codes projections onto gf 4 and multilevel construction a projection of binary reed muller codes r r m onto gf 4 sup m 2 is presented for an r r m code this operation yields a linear quaternary code with the same length dimension and minimum distance as the reed muller r r 1 m 2 code based upon this projection multilevel construction is given for r r m where the constituent codes applied to the different levels are themselves the reed muller codes r r 2 m 2 and r r m 2 as well as the aforementioned quaternary code this construction of reed muller codes is readily applicable for their efficient decoding

performance analysis of fiber optic bppm cdma systems with single parity check product codes the application of a family of single parity check spc product codes in the fiber optic code division multiple access cdma systems with binary pulse position modulation bppm is considered in this paper we evaluate the bit error probabilities of the coded systems taking into account the photodetector noise thermal noise and the multiple access interference in this paper the performance of the employed coding schemes over a binary input output symmetric channel is analyzed based on the density evolution approach the validity of the analytically obtained bit error rates bers over binary symmetric channels bscs for each of the decoding iterations is verified by the system simulation it is then shown that the application of these coding schemes in the optical bppm cdma communication system which can be modeled as a bsc significantly improves the system performance particularly for a certain ber less than 10 7 and bandwidth the coded systems not only permit a higher number more than twice of active users compared with the uncoded systems but also can operate at higher channel bit rate up to four times with more than 10 energy saving furthermore by increasing the number of code dimensions less power less than 40 is required to achieve the desired performance

minimal permutation sets for decoding the binary golay codes corresp for permutation decoding of an e error correcting linear code a set of permutations which move all error vectors of weight leq e out of the information places is needed a method of finding minimal decoding sets is given along with minimal sets obtained with this method for the binary golay codes

efficient bounded distance decoding of the hexacode and associated decoders for the leech lattice and the golay code two soft decision decoding algorithms for the 6 3 4 quaternary code hexacode are presented both algorithms realize half the minimum euclidean distance of the code the proposed algorithms are most practical in using them bounded distance decoding of the golay code and the leech lattice are performed with at most 187 and 519 real number operations respectively compare this to 651 respectively 3595 operations required by the best known maximum likelihood decoders vardy and be ery 1991 1993 and 431 respectively 1007 operations required by the bounded distance decoders amrani et al 1994 we present some simulation results for the proposed leech lattice decoders revealing near optimal performance a comparison to known trellis codes is also provided

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 z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

the nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

univalent functions and integrable systems we study one parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters these evolution parameters in some cases admit hamiltonian formulation and lead to integrable systems one example of such parameters is complex moments for the laplacian growth that form a whitham toda integrable hierarchy another example we deal with is related to expanding coefficient bodies for conformal maps given by lowner subordination chains the coefficients bodies are proved to form a liouville partially integrable hamiltonian system for each fixed index and the first integrals are obtained we also discuss the contact structure of this system

on the voronoi neighbor ratio for binary linear block codes soft decision decoding of block codes is regarded as the geometrical problem of identifying the voronoi region within which a given input vector lies a measure called the neighbor ratio is proposed to characterize how many facets a voronoi region has theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes an asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1 2 or not for rates below this threshold all pairs of codewords tend to share a voronoi facet for higher rates a relatively small fraction of them do

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

second generation fuzzy expert system abstract we describe our approach to designing a fuzzy expert systems shell which is decision theoretic in nature we wish to determine the interval operating characteristics in order to maximize the information in relevant working memory our solution is a shell which generalizes and contains our present shell flops major additions are 1 1 upper and lower confidence values 2 2 allowing rules to fire on either or both confidence values and 3 3 a memory update algorithm designed to maximize information

a new linear programming approach to decoding linear block codes in this paper we propose a new linear programming formulation for the decoding of general linear block codes different from the original formulation given by feldman the number of total variables to characterize a parity check constraint in our formulation is less than twice the degree of the corresponding check node the equivalence between our new formulation and the original formulation is proven the new formulation facilitates to characterize the structure of linear block codes and leads to new decoding algorithms in particular we show that any fundamental polytope is simply the intersection of a group of the so called minimum polytopes and this simplified formulation allows us to formulate the problem of calculating the minimum hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables we then propose a branch and bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem in addition we prove that for the family of single parity check spc product codes the fractional distance and the pseudodistance are both equal to the minimum distance finally we propose an efficient algorithm for decoding spc product codes with low complexity and maximum likelihood ml decoding performance

even more efficient bounded distance decoding of the hexacode the golay code and the leech lattice we present a new bounded distance decoding algorithm for the hexacode which requires at most 34 real operations in the worst case as compared to 57 such operations in the best previously known decoder the new algorithm is then employed for bounded distance decoding of the leech lattice and the golay code the error correction radius of the resulting decoders is equal to that of a maximum likelihood decoder the resulting decoding complexity is at most 331 real operations for the leech lattice and at most 121 operations for the golay code for all the three codes the hexacode the golay code and the leech lattice the proposed decoders are considerably more efficient than any decoder presently known

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

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

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

on the structure of vector quantizers vector quantization is intrinsically superior to predictive coding transform coding and other suboptimal and em ad hoc procedures since it achieves optimal rate distortion performance subject only to a constraint on memory or block length of the observable signal segment being encoded the key limitation of existing techniques is the very large randomly generated code books which must be stored and the computational complexity of the associated encoding procedures the quantization operation is decomposed into its rudimentary structural components this leads to a simple and elegant approach to derive analytical properties of optimal quantizers some useful properties of quantizers and algorithmic approaches are given which are relevant to the complexity of both storage and processing in the encoding operation highly disordered quantizers which have been designed using a clustering algorithm are considered finally lattice quantizers are examined which circumvent the need for a code book by using a highly structured code based on lattices the code vectors are algorithmically generated in a simple manner rather than stored in a code book and fast algorithms perform the encoding algorithm with negligible complexity

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

encoding and decoding binary product lattices a binary product lattice is generated from two binary component lattices of lower dimensions by employing the kronecker product this work focuses on codes carved from binary product lattices defined as such an intriguing problem is that of effectively mapping independent data sequences onto a selected subset of lattice points a novel approach is disclosed yielding an explicit connection between source bits and lattice points decoding methods typically used for binary product codes do not apply for product lattices several alternative decoding approaches are discussed in particular a provably bounded distance decoder is presented it relies on the fact that a product lattice code point may be regarded as a two dimensional array whose rows and columns are points in the component lattices the obtained results are compared with classical lattices known in the art

on the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes it was shown earlier that for a punctured reed muller rm code or a primitive bch code which contains a punctured rm code of the same minimum distance as a large subcode the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions the problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized hamming weight hierarchy of a code and it is shown that for rm codes the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to v k wei 1991 the state complexity of the trellis diagram for the extended and permuted 64 24 bch code is discussed

complex rotary codes revisited a low complexity high performance decoding approach in this paper based on a belief propagation decoding strategy a class of generalized parity check codes called complex rotary codes is investigated it is shown that by using iterative sum product decoding the complex rotary codes have a much lower decoding complexity than turbo codes but have almost the same performance for the high code rate and short frame case frame length 500 bits it is also shown that the prime block size of complex rotary codes is essential to achieve better performance because of its uniform checking characteristic

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

linear precoders for bit interleaved coded modulation on awgn channels analysis and design criteria this paper investigates linear precoding for bit interleaved coded modulation bicm on additive white gaussian noise awgn channels concatenating a linear precoder as an inner encoder with an outer convolutional encoder produces a powerful code with a limited decoding complexity linear precoders are examined and optimized for two scenarios using i a noniterative decoding strategy and ii an iterative decoding strategy under a perfect feedback assumption the precoder design is based on an information theoretical approach on the one hand and a pair wise error probability pep analysis on the other hand both approaches render convenient precoder design rules for a binary phase shift keying bpsk signal set the optimal precoders that result from these rules are also derived numerical results confirm the analytical findings and simulations illustrate the effectiveness of the approach

single parity check product codes this paper considers the performance of iteratively decoded single parity check spc multidimensional product codes in an additive white gaussian noise channel asymptotic performance bounds are compared to simulation results a new code structure based on spc product codes is introduced this structure involves interleaving between the encoding of each dimension in the product code an analysis of the weight distribution is used to explain the good performance results for these randomly interleaved spc product codes

augmented product codes and lattices reed muller codes and barnes wall lattices this paper concerns the construction of the so called augmented product codes and augmented product lattices these are obtained by augmenting product codes or product lattices from certain classes thus obtaining higher dimensional codes or lattices from the same class respectively certain properties of the augmented product construction are derived and specific construction examples are given in particular it is shown that the reed muller codes the golay code the barnes wall lattices as well as the leech lattice all have various augmented product constructions

complex and integral laminated lattices in an earlier paper we studied real laminated lattices or z modules a where a is the lattice of even integers and a is obtained by stacking layers of a suitable n l dimensional lattice a as densely as possible keeping the same minimal norm the present paper considers two generalizations a complex and quaternionic lattices obtained by replacing z module by module where j may be the eisenstein gaussian or hurwitzian integers etc and b integral laminated lattices in which a is required to be an integral lattice with the prescribed minimal norm this enables us to give a partial answer to a question of j g thompson on integral lattices and to explain some of the computer generated results of plesken and pohst on this problem also a number of familiar lattices now arise in a canonical way for example the coxeter todd lattice is the 6 dimensional integral laminated lattice over z to of minimal norm 2 the paper includes tables of the best real integral lattices in up to 24 dimensions

practical encoders and decoders for euclidean codes from barnes wall lattices we address the application of barnes wall bw lattice codes for communication over additive white gaussian noise awgn channels we introduce construction a prime of complex bw lattices that makes new connection between linear codes over polynomial rings and lattices we show that construction a prime of bw lattices is equivalent to the multilevel construction from reed muller codes proposed by forney to decode the bw lattice code we adapt the low complexity sequential bw lattice decoder sbwd proposed by micciancio and nicolosi first we study the error performance of sbwd for decoding the infinite lattice and demonstrate that it is powerful in making correct decisions well beyond the packing radius subsequently we use the sbwd to decode lattice codes through a novel noise trimming technique where the received vector is appropriately scaled before applying the sbwd we show that the noise trimming technique is most effective for decoding bw lattice codes in smaller dimensions while the gain diminishes for decoding codes in larger dimensions

on the problem of finding zero concurring codewords zero concurring codewords disclose a certain structure of the code that may be used for efficient soft decision decoding and for designing dc free codes methods for constructing sets of zero concurring codewords are presented for several families of codes for the general case an algorithm solution of the problem is offered a table of results obtained using the proposed techniques is supplied for all the primitive narrow sense binary bch codes of length up to 127

fuzzy classification using pattern discovery rule based classifiers allow rationalization of classifications made this in turn improves understanding which is essential for effective decision support as a rule based classifier the pattern discovery pd algorithm functions well in discrete nominal and continuous data domains a drawback when using pd as a classifier for decision support is that it has an unbounded decision space that confounds the understanding of the degree of support for a decision incorporating pd into a fuzzy inference system fis allows the the degree of support for a decision to be expressed with intuitively understandable terms in addition using discrete algorithms in continuous domains can result in reduced accuracy due to quantization fuzzification reduces this ldquocost of quantizationrdquo and improves classification performance in this work the pd algorithm was used as a source of rules for a series of fiss implemented using different rule weighting and defuzzification schemes each providing a linguistic basis for rule description and a bounded space for expression of decision support the output of each fis consists of a suggested outcome a strong confidence metric describing suggestions within this space and a linguistic expression of the rules this constitutes a stronger basis for decision making than that provided by pd alone a variety of synthetic continuous class distributions with varying degrees of separation was used to evaluate the performance of fuzzy pd back propagation and bayesian classifiers overall the accuracy of the fuzzy system was found to be similar but slightly below that of the inherently continuous valued classifiers and was somewhat improved with respect to the pd classifiers for the difficult spiral class distributions studied the fuzzy classifiers were able to make more classifications than the pd classifiers the correct classification rates for the fuzzy classifiers were similar across the various rule weighting and defuzzification schemes demonstrating the strength of the statistical method for rule generation analysis of several real world data sets shows that a pd based fis has comparable performance to a neuro fuzzy system the use of a pd based fis however provides insight into the structure of the data analyzed not available through the other approaches

on the pless construction and ml decoding of the 48 24 12 quadratic residue code we present a method for maximum likelihood decoding of the 48 24 12 quadratic residue code this method is based on projecting the code onto a subcode with an acyclic tanner graph and representing the set of coset leaders by a trellis diagram this results in a two level coset decoding which can be considered a systematic generalization of the wagner rule we show that unlike the 24 12 8 golay code the 48 24 12 code does not have a pless construction which has been an open question in the literature it is determined that the highest minimum distance of a 48 24 binary code having a pless 1986 construction is 10 and up to equivalence there are three such codes

reduced lists of error patterns for maximum likelihood soft decoding a method whereby a substantially reduced family of error patterns called survivors may be created for maximum likelihood soft decoding is introduced the family of survivor depends on the received word a decoder based on this approach first forms the survivors then scores them rather than obtaining the survivors by online elimination of error patterns the use of predetermined lists that represent all families of survivors is proposed

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

twenty three constructions for the leech lattice in the previous chapter we classified the points at maximal distance from the leech lattice the deep holes in that lattice and showed that there are 23 classes of such holes the classes being in one to one correspondence with the 23 niemeier lattices in 24 dimensions we now present 23 constructions for the leech lattice one for each class of hole or niemeier lattice two of these are the usual constructions of the leech lattice from the golay codes over f 2 and f 3