enumerating fuzzy switching functions and free kleene algebras abstract we investigate the free kleene algebra on n free generators fk n this algebra is essentially the same as the set of n variable fuzzy switching functions the cardinality of fk n has been previously determined only for n 1 2 and 3 we present a general method for enumeration problems of this kind we show fk 4 has 160 297 985 276 elements the final section contains new upper and lower bounds for the cardinality of fk n we also consider the number of regular functions and show that there are 11 197 129 615 and 430 904 428 717 such functions in 1 2 3 and 4 variables

a unified method for evaluating the error correction radius of reliability based soft decision algorithms for linear block codes this paper presents a unified method for evaluating the error correction radii of many reliability based soft decision decoding algorithms for binary linear block codes based on this unified method these decoding algorithms as well as their potential improvements can be compared directly the error correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft decision decoder to the transmitted signal sequence it is shown that this problem can be formulated as a constrained optimization problem although no general closed form expression is possible a simple algorithm that always converges to the optimum solution of this optimization problem is presented based on these results the error correction radii of some well known reliability based soft decision decoding algorithms are revisited and compared

tailbiting codes bounds and search results tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied the notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower bounded the asymptotic behavior of the derived bound is investigated furthermore for regular tailbiting codes the product state complexity is lower bounded tables of new tailbiting trellis representations of linear block codes of rates 1 2 1 3 and 1 4 are presented almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time varying trellises which are optimal five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters four of them are also superior to any previously known nonlinear code with the same parameters also more than 40 other quasi cyclic codes have been found that improve the parameter set of previously known quasi cyclic codes

a bounded distance decoding algorithm for the leech lattice with generalizations an algorithm is given that decodes the leech lattice with not much more than twice the complexity of soft decision decoding of the golay code the algorithm has the same effective minimum distance as maximum likelihood decoding and increases the effective error coefficient by less than a factor or two the algorithm can be recognized as a member of the class of multistage algorithms that are applicable to hierarchical constructions it is readily generalized to lattices that can be expressed in terms of binary code formulas and in particular to construction b lattices

fast generalized minimum distance decoding of algebraic geometry and reed solomon codes generalized minimum distance gmd decoding is a standard soft decoding method for block codes we derive an efficient general gmd decoding scheme for linear block codes in the framework of error correcting pairs special attention is paid to reed solomon rs codes and one point algebraic geometry ag codes for rs codes of length n and minimum hamming distance d the gmd decoding complexity turns out to be in the order o nd where the complexity is counted as the number of multiplications in the field of concern for ag codes the gmd decoding complexity is highly dependent on the curve in consideration it is shown that we can find all relevant error erasure locating functions with complexity o o sub 1 nd where o sub 1 is the size of the first nongap in the function space associated with the code a full gmd decoding procedure for a one point ag code can be performed with complexity o dn sup 2

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

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

on random rotations diversity and minimum mse decoding of lattices we establish a simple relation between high diversity multidimensional rotations obtained from totally complex cyclotomic fields and the discrete fourier transform the diversity distribution of an hadamard like random rotation is derived analytically it is shown that a random multidimensional rotation exhibits an excellent diversity distribution and can be combined to quadrature amplitude modulation qam constellations to combat channel fading we also describe a mean square error mse universal lattice decoder suitable for large dimensions up to 1024 the mse criterion treats the lattice structure as intersymbol interference the universal decoder is applied to both gaussian and rayleigh fading channels to decode dense lattice sphere packings and rotated cubic constellations respectively

the leech lattice and the golay code bounded distance decoding and multilevel constructions multilevel constructions of the binary golay code and the leech lattice are described both constructions are based upon the projection of the golay code and the leech lattice onto the 6 3 4 hexacode over gf 4 however unlike the previously reported constructions the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates based upon the multilevel structure of the golay code and the leech lattice efficient bounded distance decoding algorithms are devised the bounded distance decoder for the binary golay code requires at most 431 operations as compared to 651 operations for the best known maximum likelihood decoder efficient bounded distance decoding of the leech lattice is achieved by means of partitioning it into four cosets of q sub 24 beyond the conventional partition into two h sub 24 cosets the complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case as compared to about 3600 operations for the best known in maximum likelihood decoder it is shown that the proposed algorithms decode correctly at least up to the guaranteed error correction radius of the maximum likelihood decoder thus the loss in coding gain is due primarily to an increase in the effective error coefficient which is calculated exactly for both algorithms furthermore the performance of the leech lattice decoder on the awgn channel is evaluated experimentally by means of a comprehensive computer simulation the results show a loss in coding gain of less than 0 1 db relative to the maximum likelihood decoder for ber ranging from 10 sup 1 to 10 sup 7

maximum likelihood soft decoding of binary block codes and decoders for the golay codes maximum likelihood soft decision decoding of linear block codes is addressed a binary multiple check generalization of the wagner rule is presented and two methods for its implementation one of which resembles the suboptimal forney chase algorithms are described besides efficient soft decoding of small codes the generalized rule enables utilization of subspaces of a wide variety thereby yielding maximum likelihood decoders with substantially reduced computational complexity for some larger binary codes more sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the 24 12 golay code yielding a computational gain factor of about 2 with respect to previous methods a ternary single check version of the wagner rule is applied for efficient soft decoding of the 12 6 ternary golay code

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

geometrical and performance analysis of gmd and chase decoding algorithms the overall number of nearest neighbors in bounded distance decoding bdd algorithms is given by n sub 0 eff n sub 0 n sub bdd where nbdd denotes the number of additional non codeword neighbors that are generated during the suboptimal decoding process we identify and enumerate the nearest neighbors associated with the original generalized minimum distance gmd and chase 1972 decoding algorithms after careful examination of the decision regions of these algorithms we derive an approximated probability ratio between the error contribution of a noncodeword neighbor one of n sub bdd points and a codeword nearest neighbor for chase algorithm 1 it is shown that the contribution to the error probability of a noncodeword nearest neighbor is a factor of 2 sup d 1 less than the contribution of a codeword while for chase algorithm 2 the factor is 2 sup d 2 1 d being the minimum hamming distance of the code for chase algorithm 3 and gmd a recursive procedure for calculating this ratio which turns out to be nonexponential in d is presented this procedure can also be used for specifically identifying the error patterns associated with chase algorithm 3 and gmd utilizing the probability ratio we propose an improved approximated upper bound on the probability of error based on the union bound approach simulation results are given to demonstrate and support the analytical derivations

general fuzzy acceptors for syntactic pattern recognition abstract in this paper the behavior of general fuzzy acceptor over a commutative cl semigroup l is considered and the properties of equivalence relation are investigated from these results we can apply for pattern recognition of deformed images or for enhancement of distorted patterns

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

computing generators of free modules over orders in group algebras abstract let e be a number field and g be a finite group let a be any o e order of full rank in the group algebra e g and x be a left a lattice we give a necessary and sufficient condition for x to be free of given rank d over a in the case that the wedderburn decomposition e g m is explicitly computable and each m is in fact a matrix ring over a field this leads to an algorithm that either gives elements 1 d x such that x a 1 a d or determines that no such elements exist let l k be a finite galois extension of number fields with galois group g such that e is a subfield of k and put d k e the algorithm can be applied to certain galois modules that arise naturally in this situation for example one can take x to be o l the ring of algebraic integers of l and a to be the associated order a e g o l e g the application of the algorithm to this special situation is implemented in magma under certain extra hypotheses when k e q

a new reed solomon code decoding algorithm based on newton s interpolation a reed solomon code decoding algorithm based on newton s interpolation is presented this algorithm has as main application fast generalized minimum distance decoding of reed solomon codes it uses a modified berlekamp massey algorithm to perform all necessary generalized minimum distance decoding steps in only one run with a time domain form of the new decoder the overall asymptotic generalized minimum distance decoding complexity becomes o dn with n the length and d the distance of the code including the calculation of all error locations and values this asymptotic complexity is optimal other applications are the possibility of fast decoding of reed solomon codes with adaptive redundancy and a general parallel decoding algorithm with zero delay

sphere bound achieving coset codes and multilevel coset codes a simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array l and its error probability on an additive white gaussian noise awgn channel it is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding including certain binary lattices these codes have structure of the kind that has been found to be useful in practice capacity curves and design guidance for practical codes are given exponential error bounds for coset codes are developed generalizing poltyrev s 1994 bounds for lattices these results are based on the channel coding theorems of information theory rather than the minkowski hlawka theorem of lattice theory

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

error trellises for convolutional codes i construction an error trellis is a directed graph that represents all the sequences belonging to the coset which contains the symbol by symbol detected version of a given received sequence a modular construction of error trellises for an n k convolutional code over gf q is presented the trellis is designed on the basis of partitioning the scalar check matrix of the code into submatrices of l rows accompanied with a corresponding segmentation of the syndrome the value of the design parameter l is an essentially unconstrained multiple of n k for all the cosets of the code the sections of the error trellis are drawn from a collection of only q sup l modules the module for each section is determined by the value of the associated syndrome segment in case the construction is based on a basic polynomial check matrix either canonical or noncanonical then the error trellis is minimal in the sense that spl sigma spl les spl mu where spl sigma is the dimension of the state space of the trellis and spl mu is the constraint length of a canonical generator matrix for the code for basic check matrices with delay free columns the inequality reduces to spl sigma spl mu

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

asymptotic optimality of the gmd and chase decoding algorithms the generalized minimum distance gmd and chase 1972 decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white gaussian noise awgn channel we compute the limitation of the ratio between the probability of decoding error for the gmd or any one of the chase decoding algorithms and that of the maximum likelihood ml decoding when the signal to noise ratio snr approaches infinity if the minimum hamming distance of the code is greater than 2 the limitation is shown to be equal to 1 and thus the gmd and chase decoding algorithms are asymptotically optimum

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

computing class fields via the artin map based on an explicit representation of the artin map for kummer extensions we present a method to compute arbitrary class fields as in the proofs of the existence theorem the problem is first reduced to the case where the field contains sufficiently many roots of unity using kummer theory and an explicit version of the artin reciprocity law we show how to compute class fields in this case we conclude with several examples

an improvement to generalized minimum distance decoding a novel acceptance criterion that is less stringent than previous criteria is developed the criterion accepts the codeword that is closest to the received vector for many cases where previous criteria fail to accept any codeword as a result the performance of generalized minimum distance gmd decoding is better if the new criterion is used for m ary signaling the weights used in gmd decoding are generalized to permit each of the possible m symbol values to have a different weight

writing representations over proper division subrings abstract let e be a division ring and g a finite group of automorphisms of e whose elements are distinct modulo inner automorphisms of e let f e g be the division subring of elements of e fixed by g given a representation a e d d of an f algebra a we give necessary and sufficient conditions for to be writable over f here e d d denotes the algebra of d d matrices over e and a matrix a writes over f if a 1 a a f d d we give an algorithm for constructing an a or proving that no a exists the case of particular interest to us is when e is a field and is absolutely irreducible the algorithm relies on an explicit formula for a and a generalization of hilbert s theorem 90 that arises in galois cohomology the algorithm has applications to the construction of absolutely irreducible group representations especially for solvable groups and to the recognition of class c 5 in aschbacher s matrix group classification scheme m aschbacher on the maximal subgroups of the finite classical groups invent math 76 1984 469 514 mr0746539 shangzhi li on the subgroup structure of classical groups in group theory in china in math appl china ser vol 365 kluwer acad publ dordrecht 1996 pp 70 90 mr1447199 1 13

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

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

universal bound on the performance of lattice codes we present a lower bound on the probability of symbol error for maximum likelihood decoding of lattices and lattice codes on a gaussian channel the bound is tight for error probabilities and signal to noise ratios of practical interest as opposed to most existing bounds that become tight asymptotically for high signal to noise ratios the bound is also universal it provides a limit on the highest possible coding gain that may be achieved at specific symbol error probabilities using any lattice or lattice code in n dimensions in particular it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains the asymptotic as n spl rarr spl infin behavior of the new bound is shown to coincide with the shannon 1948 limit for gaussian channels

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

decoding by embedding correct decoding radius and dmt optimality the closest vector problem cvp and shortest nonzero vector problem svp are the core algorithmic problems on euclidean lattices they are central to the applications of lattices in many problems of communications and cryptography kannan s embedding technique is a powerful technique for solving the approximate cvp yet its remarkable practical performance is not well understood in this paper the embedding technique is analyzed from a bounded distance decoding bdd viewpoint we present two complementary analyses of the embedding technique we establish a reduction from bdd to hermite svp via unique svp which can be used along with any hermite svp solver including among others the lenstra lenstra and lovasz lll algorithm and show that in the special case of lll it performs at least as well as babai s nearest plane algorithm lll aided successive interference cancellation the former analysis helps us to explain the folklore practical observation that unique svp is easier than standard approximate svp it is proven that when the lll algorithm is employed the embedding technique can solve the cvp provided that the noise norm is smaller than a decoding radius 1 2 where 1 is the minimum distance of the lattice and o 2n 4 this substantially improves the previously best known correct decoding bound o 2n focusing on the applications of bdd to decoding of multiple input multiple output systems we also prove that bdd of the regularized lattice is optimal in terms of the diversity multiplexing gain tradeoff and propose practical variants of embedding decoding which require no knowledge of the minimum distance of the lattice and or further improve the error performance

on the complexity of growth of the number of distinct fuzzy switching functions abstract several attempts have been made to enumerate fuzzy switching fsf s and to develop upper and lower bounds for the number of fsf s of n variables in an effort to better understand the properties and the complexity of fsf s previous upper bounds are 2 4n 9 and 2 2 3n 2n 1 7 it has also been shown that the exact numbers of fsf s of n variables for n 0 1 2 3 and 4 are 2 6 8 84 43 918 and 160 297 985 276 respectively this paper will give a brief overview of previous approaches to the problem study some of the properties of fuzzy switching functions and give improved upper and lower bounds for a general n

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

two algorithms for soft decision decoding of reed solomon codes with application to multilevel coded modulations in this paper two symbol level soft decision decoding algorithms for reed solomon codes derived form the ordered statistics os and from the generalized minimum distance gmd decoding methods are presented and analyzed both the os and the gmd algorithms are based on the idea of producing a list of candidate code words among which the one having the larger likelihood is selected as output we propose variants of the mentioned algorithms that allow to finely tune the size of the list in order to obtain the desired decoding complexity the method proposed by agrawal and vardy for computing the error probability of the gmd algorithm is extended to our decoding methods examples are presented where these algorithms are applied to singly extended reed solomon codes over gf 16 used as outer codes in a 128 dimensional coded modulation scheme that attains good performance with manageable decoding complexity

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

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 algebraic soft decision decoding algorithms for bch codes three algebraic soft decision decoding algorithms are presented for binary bose chaudhuri hocquengham bch codes two of these algorithms are based on the bounded distance bd 1 generalized minimum distance gmd decoding presented by berlekamp 1984 and the other is based on chase 1972 decoding a simple algebraic algorithm is first introduced and it forms a common basis for the decoding algorithms presented next efficient bd 1 gmd and bd 2 gmd decoding algorithms are presented it is shown that for binary bch codes with odd designed minimum distance d and length n both the bd 1 gmd and the bd 2 gmd decoding algorithms can be performed with complexity o nd the error performance of these decoding algorithms is shown to be significantly superior to that of conventional gmd decoding by computer simulation finally an efficient algorithm is presented for chase decoding of binary bch codes like a one pass gmd decoding algorithm this algorithm produces all necessary error locator polynomials for chase decoding in one run

on the reliability order based decoding algorithms for binary linear block codes in this correspondence we consider the decoding of binary block codes over the additive white gaussian noise awgn channel with binary phase shift keying bpsk signaling by a reliability order based decoding algorithm robda we mean a soft decision decoding algorithm which decodes to the best most likely codeword of the form that is the sum of the hard decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions examples of robdas include many well known decoding algorithms such as the generalized minimum distance gmd decoding algorithm chase decoding algorithms and the reliability based decoding algorithms proposed by fossorier and lin it is known that the squared error correction radii of robdas can be computed from the minimal squared euclidean distances mseds between the all one sequence and the polyhedra corresponding to the error patterns for the computation of such mseds we give a new method which is more compact than the one proposed by fossorier and lin these results are further used to show that any bounded distance robda is asymptotically optimal the ratio between the probability of decoding error of a bounded distance robda and that of the maximum likelihood ml decoding approaches 1 when the signal to noise ratio snr approaches infinity provided that the minimum hamming distance of the code is greater than 2

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 computation of the class number of an algebraic number field it is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field f the method assumes the truth of the generalized riemann hypothesis in order to estimate the residue of the dedekind zeta function of f at s 1 sufficiently well that h can be determined unambiguously

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

quaternionic manin symbols brandt matrices and hilbert modular forms in this paper we propose a generalization of the algorithm we developed previously along the way we also develop a theory of quaternionic m symbols whose definition bears some resemblance to the classical m symbols except for their combinatorial nature the theory gives a more efficient way to compute hilbert modular forms over totally real number fields especially quadratic fields and we have illustrated it with several examples namely we have computed all the newforms of prime levels of norm less than 100 over the quadratic fields q 29 and q 37 and whose fourier coefficients are rational or are defined over a quadratic field

lattice codes and generalized minimum distance decoding for ofdm systems lattice coding of orthogonal frequency division multiplexing ofdm systems is considered mapping of multilevel construction lattices to ofdm blocks is shown and a methodology for probabilistic analysis of multistage generalized minimum distance gmd decoding of the received ofdm blocks is shown as a case study transmission of points from a 128 dimensional barnes wall lattice is considered tight approximations to the system error rate are obtained and verified by simulation it appears that gmd decoding of lattice encoded ofdm provides high coding gain at low complexity

generalized minimum distance decoding of euclidean space codes and lattices it is shown that multistage generalized minimum distance gmd decoding of euclidean space codes and lattices can provide an excellent tradeoff between performance and complexity we introduce a reliability metric for gaussian channels that is easily computed from an inner product and prove that a multistage gmd decoder using this metric is a bounded distance decoder up to the true packing radius the effective error coefficient of multistage gmd decoding is determined two simple modifications in the gmd decoding algorithm that drastically reduce this error coefficient are proposed it is shown that with these modifications gmd decoding achieves the error coefficient of maximum likelihood decoding for block codes and for generalized construction a lattices multistage gmd decoding of the lattices d sub 4 e sub 8 k sub 12 bw sub 16 and spl lambda sub 24 is investigated in detail for k sub 12 bw sub 16 and spl lambda sub 24 the gmd decoders have considerably lower complexity than the best known maximum likelihood or bounded distance decoding algorithms and appear to be the most practically attractive decoders available for high dimensional codes and lattices spl ges 64 dimensions maximum likelihood decoding becomes infeasible while gmd decoding algorithms remain quite practical as an example we devise a multistage gmd decoder for a 128 dimensional sphere packing with a nominal coding gain of 8 98 db that attains an effective error coefficient of 1365760 this decoder requires only about 400 real operations in addition to algebraic errors and erasures decoding of certain bch and hamming codes it therefore appears to be practically feasible to implement algebraic multistage gmd decoders for high dimensional sphere packings and thus achieve high effective coding gains

woven convolutional codes i encoder properties encoders for convolutional codes with large free distances can be constructed by combining several less powerful convolutional encoders this paper is devoted to constructions in which the constituent convolutional codes are woven together in a manner that resembles the structure of a fabric the general construction is called twill and it is described together with two special cases viz woven convolutional encoders with outer warp and with inner warp the woven convolutional encoders inherit many of their structural properties such as minimality and catastrophicity from their constituent encoders for all three types of woven convolutional codes upper and lower bounds on their free distances as well as lower bounds on the active distances of their encoders are derived

active distances for convolutional codes a family of active distance measures for general convolutional codes is defined these distances are generalizations of the extended distances introduced by thommesen and justesen 1983 for unit memory convolutional codes it is shown that the error correcting capability of a convolutional code is determined by the active distances the ensemble of periodically time varying convolutional codes is defined and lower bounds on the active distances are derived for this ensemble the active distances are very useful in the analysis of concatenated convolutional encoders

maximum likelihood decoding of the leech lattice an algorithm for maximum likelihood decoding of the leech lattice is presented the algorithm involves projecting the points of the leech lattice directly onto the codewords of the 6 3 4 quaternary code the hexacode projection on the hexacode induces a partition of the leech lattice into four cosets of a certain sublattice 24 such a partition into cosets enables maximum likelihood decoding of the leech lattice with 3595 real operations in the worst case and only 2955 operations on the average this is about half the worst case and the average complexity of the best previously known algorithm

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

on unequal error protection of convolutional codes from an algebraic perspective in this paper convolutional codes are studied for unequal error protection uep from an algebraic theoretical viewpoint we first show that for every convolutional code there exists at least one optimal generator matrix with respect to uep the uep optimality of convolutional encoders is then combined with several algebraic properties e g systematic basic canonical and minimal to establish the fundamentals of convolutional codes for uep in addition a generic lower bound on the length of a uep convolutional code is proposed good uep codes with their lengths equal to the derived lower bound are obtained by computer search

a bounded distance decoding algorithm for binary linear block codes achieving the minimum effective error coefficient a new bounded distance bd decoding algorithm is presented for binary linear n k d block codes on additive white gaussian noise channels the algorithm is based on the generalized minimum distance gmd decoding algorithm of forney 1989 using the acceptance criterion of taipale and pursley gmd tp proposed in 1991 it is shown that the gmd tp decoding algorithm is a bd decoding algorithm with effective error coefficient sup n sub d it is also shown that the decision regions of gmd tp are good inner approximations of those of full gmd decoding and therefore full gmd decoding is bd and has an effective error coefficient that is well approximated by sup n sub d moreover by adding a d erasure correction step to gmd decoding the effective error coefficient can be reduced to a sub d the number of minimum weight codewords which is the same as the effective error coefficient of maximum likelihood decoding the decoding algorithm is mainly based on algebraic errors and erasures decoding and therefore has polynomial rather than exponential complexity

bounded distance 1 soft decision reed solomon decoding we present a new reed solomon decoding algorithm which embodies several refinements of an earlier algorithm some portions of this new decoding algorithm operate on symbols of length lgq bits other portions operate on somewhat longer symbols in the worst case the total number of calculations required by the new decoding algorithm is proportional to nr where n is the code s block length and r is its redundancy this worst case workload is very similar to prior algorithms but in many applications average case workload and error correcting performance are both much better the input to the new algorithm consists of n received symbols from gf q and n nonnegative real numbers each of which is the reliability of the corresponding received symbol any conceivable errata pattern has a score equal to the sum of the reliabilities of its locations with nonzero errata values a max likelihood decoder would find the minimum score over all possible errata patterns our new decoding algorithm finds the minimum score only over a subset of these possible errata patterns the errata within any candidate errata pattern may be partitioned into errors and erasures depending on whether the corresponding reliabilities are above or below an erasure threshold different candidate errata patterns may have different thresholds each chosen to minimize its corresponding errata count which is defined as 2 spl middot number of errors number of erasures the new algorithm finds an errata pattern with minimum score among all errata patterns for which errata count spl les r 1 where r is the redundancy of the rs code conventional algorithms also require that the erasure threshold be set a priori the new algorithm obtains the best answer over all possible settings of the erasure threshold conventional cyclic rs codes have length n q 1 and their locations correspond to the nonzero elements of gf q the new algorithm also applies very naturally to rs codes which have been doubly extended by the inclusion of 0 and spl infin as additional locations

minimizing representations over number fields ii computations in the brauer group abstract finding minimal fields of definition for representations of finite groups is one of the most important unsolved problems of computational representation theory while good methods exist for representations over finite fields it is still an open question in the case of number fields continuing and extending previous work we give a practical method for finding defining fields of minimal degree for absolutely irreducible representations the method is based on techniques from galois cohomology and the use of an explicit form of a weak grunwald wang theorem