a distinguisher for high rate mceliece cryptosystems the goppa code distinguishing gd problem consists in distinguishing the matrix of a goppa code from a random matrix the hardness of this problem is an assumption to prove the security of code based cryptographic primitives such as mceliece s cryptosystem up to now it is widely believed that the gd problem is a hard decision problem we present the first method allowing to distinguish alternant and goppa codes over any field our technique can solve the gd problem in polynomial time provided that the codes have sufficiently large rates the key ingredient is an algebraic characterization of the key recovery problem the idea is to consider the rank of a linear system which is obtained by linearizing a particular polynomial system describing a key recovery attack it appears that this dimension depends on the type of code considered explicit formulas derived from extensive experimentations for the rank are provided for generic random alternant and goppa codes over any field finally we give theoretical explanations of these formulas in the case of random codes alternant codes over any field of characteristic two and binary goppa codes

sort and match algorithm for soft decision decoding let a q ary linear n k code c be used over a memoryless channel we design a decoding algorithm spl psi sub n that splits the received block into two halves in n different ways first about spl radic n error patterns are found on either half then the left and right hand lists are sorted out and matched to form codewords finally the most probable codeword is chosen among at most n spl radic n codewords obtained in all n trials the algorithm can be applied to any linear code c and has complexity order of n sup 3 spl radic n for any n spl ges q sup n k the decoding error probability p sub n exceeds at most 1 q sup n k n times the probability p sub spl psi c of maximum likelihood decoding for code rates r spl ges 1 2 the complexity order q sup n k 2 grows as square root of general trellis complexity q sup min n k k when used on quantized additive white gaussian noise awgn channels the algorithm spl psi sub n can provide maximum likelihood decoding for most binary linear codes even when n has an exponential order of q sup n k

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

reliability based soft decision decoding with iterative information set reduction the reliability based decoding approach using the reprocessing of the most reliable information set only is extended into the iterative reprocessing of several information sets at the end of each information set reprocessing some information bits are delivered by the decoder consequently information sets with decreasing cardinality values are considered at each iteration a tight upper bound on the error performance achieved by this new method is derived compared to previously proposed competitive approaches this new method reduces the number of candidate codewords needed to achieve practically optimum decoding importantly it also preserves the very simple structured implementation of the order statistic decoding

bicyclic signed digraphs with maximal energy abstract the energy of a digraph sidigraph is defined as the sum of absolute values of real parts of its eigenvalues monsalve and rada 2016 determined the bicyclic digraphs with maximal energy among the set of all bicyclic digraphs with fixed order in this paper we find the bicyclic sidigraphs with maximal energy over the set of all bicyclic sidigraphs with fixed order we also construct few families of non cospectral equienergetic bicyclic sidigraphs

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

on the complexity of minimum distance decoding of long linear codes we suggest a decoding algorithm of q ary linear codes which we call supercode decoding it ensures the error probability that approaches the error probability of minimum distance decoding as the length of the code grows for n spl rarr spl infin the algorithm has the maximum likelihood performance the asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known the algorithm develops the ideas of covering set decoding and split syndrome decoding

locally testable and locally correctable codes approaching the gilbert varshamov bound one of the most important open problems in the theory of error correcting codes is to determine the tradeoff between the rate r and minimum distance delta of a binary code the best known tradeoff is the gilbert varshamov bound and says that for every delta in 0 1 2 there are codes with minimum distance delta and rate r r mathsf gv delta 0 for a certain simple function r mathsf gv cdot in this paper we show that the gilbert varshamov bound can be achieved by codes which support local error detection and error correction algorithms specifically we show the following results 1 local testing for all delta in 0 1 2 and all r there exist codes with length n rate r and minimum distance delta that are locally testable with mathsf quasipolylog n query complexity 2 local correction for all epsilon 0 for all delta sufficiently large and all r there exist codes with length n rate r and minimum distance delta that are locally correctable from delta 2 o 1 fraction errors with o n epsilon query complexity furthermore these codes have an efficient randomized construction and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity our results on locally correctable codes also immediately give locally decodable codes with the same parameters our local testing result is obtained by combining thommesen s random concatenation technique and the best known locally testable codes by kopparty et al our local correction result which is significantly more involved also uses random concatenation along with a number of further ideas the guruswami sudan indyk list decoding strategy for concatenated codes alon edmonds luby distance amplification and the local list decodability local list recoverability and local testability of reed muller codes curiously our final local correction algorithms go via local list decoding and local testing algorithms this seems to be the first time local testability is used in the construction of a locally correctable code

error correcting capabilities of concatenated codes with mds outer codes on memoryless channels with maximum likelihood decoding error correcting capabilities of concatenated codes with maximum distance separable mds outer codes and time varying inner codes used on memoryless discrete channels with maximum likelihood decoding are investigated it is proved that asymptotically the gallager random coding theorem can be obtained for all rates by such codes further the expurgated coding theorem as well is proved to be valid for all rates on regular channels the latter result implies that the gilbert varshamov bound for block codes over any finite field can be obtained asymptotically for all rates by linear concatenated codes

error correction based on partial information we consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword more specifically suppose that we have encoded k data symbols using an n k code with code length n and dimension k during storage some of the codeword coordinates might be corrupted by errors we aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth namely we can only download an alpha proportion of the corrupted codeword for a given alpha our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors a naive scheme is to read alpha n coordinates of the codeword this method used in conjunction with mds codes guarantees recovery from any lfloor alpha n k 2 rfloor errors in this paper we show that we can instead download an alpha proportion from each of the codeword s coordinates for a well designed mds code this method can guarantee recovery from lfloor n k alpha 2 rfloor errors which is 1 alpha times more than the naive method and is also the maximum number of errors that an n k code can correct by downloading only an alpha proportion of the codeword we present two families of such optimal constructions and decoding schemes of which one is based on interleaved reed solomon codes and the other on folded reed solomon codes we further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword we also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an alpha proportion of the corrupted codeword

concatenated codes with convolutional inner codes the minimum distance of concatenated codes with reed solomon outer codes and convolutional inner codes is studied for suitable combinations of parameters the minimum distance can be lower bounded by the product of the minimum distances of the inner and outer codes for a randomized ensemble of concatenated codes a lower bound of the gilbert varshamov type is proved

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

the hardness of solving subset sum with preprocessing the two problems subset sum and linear decoding which have given birth to the knapsack and the mceliece public key cryptosystems are known to be np complete for linear decoding it has been proved that even if one knows the linear code in advance and can preprocess it the existence of a polynomial time decoding algorithm would imply that the polynomial time hierarchy collapses at an early stage it is proved that the same holds for subset sum even if the knapsack is known in advance and can be preprocessed there is no polynomial time algorithm solving it unless the polynomial hierarchy collapses also given is the sketch of a new straightforward proof of this result for linear decoding

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

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

bicyclic digraphs with maximal energy if d is a digraph with n vertices then the energy of d is defined as e d k 1 n re z k where re z 1 re zn are the real parts of the eigenvalues z 1 z n of d in this paper we solve a problem proposed in khan et al 2015 we find the maximal value of the energy over the set of all bicyclic digraphs b n with n vertices

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

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

soft decision decoding using punctured codes let a q ary linear n k code be used over a memoryless channel we design a soft decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s spl isin k n first we take s cyclically consecutive positions starting from any initial point then we cut the subinterval of length s into two parts and examine t most plausible error patterns on either part to obtain codewords of a punctured s k code we try to match the syndromes of both parts finally the designed codewords of an s k code are re encoded to find the most probable codeword on the full length n for any long linear code the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum likelihood ml decoding given sufficiently large t by optimizing s we prove that this near ml decoding can be achieved by using only t spl ap q sup n k k n k error patterns for most long linear codes this optimization also gives about t re encoded codewords as a result we obtain the lowest complexity order of q sup n k k n k known to date for near ml decoding for codes of rate 1 2 the new bound grows as a cubic root of the general trellis complexity q sup min n k k for short blocks of length 63 the algorithm reduces the complexity of the trellis design by a few decimal orders

the complexity of information set decoding information set decoding is an algorithm for decoding any linear code expressions for the complexity of the procedure that are logarithmically exact for virtually all codes are presented the expressions cover the cases of complete minimum distance decoding and bounded hard decision decoding as well as the important case of bounded soft decision decoding it is demonstrated that these results are vastly better than those for the trivial algorithms of searching through all codewords or through all syndromes and are significantly better than those for any other general algorithm currently known for codes over large symbol fields the procedure tends towards a complexity that is subexponential in the symbol size

principal component transforms of triaxial recordings by singular value decomposition polarization analysis can be achieved efficiently by treating a time window of a single station triaxial recording as a matrix and doing a singular value decomposition svd of this seismic data matrix svd of the triaxial data matrix produces an eigenanalysis of the data covariance cross energy matrix and a rotation of the data onto the directions given by the eigenanalysis karhunen loeve transform all in one step svd provides a complete principal components analysis of the data in the analysis time window selection of this time window is crucial to the success of the analysis and is governed by three considerations the window should contain only one arrival the window should be such that the signal to noise ratio is maximized and the window should be long enough to be able to discriminate random noise from signal the svd analysis provides estimates of signal signal polarization directions and noise an f test is proposed which gives the confidence level for the hypothesis of rectilinear pol

the existence of binary linear concatenated codes with reed solomon outer codes which asymptotically meet the gilbert varshamov bound binary linear concatenated codes with reed solomon codes as the outer code and varying nonsystematic inner codes are investigated it is shown that codes exist in this class which asymptotically meet the gilbert varshamov bound

the hardness of decoding linear codes with preprocessing the problem of maximum likelihood decoding of linear block codes is known to be hard the fact that the problem remains hard even if the code is known in advance and can be preprocessed for as long as desired in order to device a decoding algorithm is shown the hardness is based on the fact that existence of a polynomial time algorithm implies that the polynomial hierarchy collapses thus some linear block codes probably do not have an efficient decoder the proof is based on results in complexity theory that relate uniform and nonuniform complexity classes

instantaneous polarization attributes and directional filtering we introduce a systematic definition of instantaneous attributes for an arbitrary multicomponent seismic signal the definition is a natural generalization of known complex trace attributes of a one component signal instantaneous amplitude and all polarization parameters are defined as invariants of instantaneous phase rotation the principal feature of our approach is the unique definition of the instantaneous phase for a signal with any number of components plots of subtle polarization parameters of multicomponent seismic data are easily obtained using conventional seismic plotting routines we illustrate our approach on a synthetic example and apply it to real 3 component wide angle crustal data plots of polarization attributes provide evidence for shear wave splitting in an sg arrival having determined the instantaneous polarization vector we design a new type of time domain spatial directional filter the filter enhances linearly polarized events with specified instantaneous polarization the

an average weight distance enumerator for binary expansions of reed solomon codes an average hamming weight enumerator is derived for the codewords at each hamming distance from a received pattern in the set of all possible binary expansions of a reed solomon code since these codes may be decoded by list decoders such as those studied by sudan 1997 the enumerator can be used to estimate the average number of codewords in the list returned by such a decoder

beast decoding of block codes obtained via convolutional codes beast is a bidirectional efficient algorithm for searching trees in this correspondence beast is extended to maximum likelihood ml decoding of block codes obtained via convolutional codes first it is shown by simulations that the decoding complexity of beast is significantly less than that of the viterbi algorithm then asymptotic upper bounds on the beast decoding complexity for three important ensembles of codes are derived they verify beast s high efficiency compared to other algorithms for high rates the new asymptotic bound for the best ensemble is in fact better than previously known bounds

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

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

optimal sectionalization of a trellis while the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature no systematic method for finding the best sectionalization of a given trellis is presently known we present a polynomial time algorithm which produces the optimal sectionalization of a given trellis t in time o n sup 2 where n is the length of the code generated by t the algorithm is developed in a general setting of certain operations and functions defined on the set of trellises it therefore applies to both linear and nonlinear codes and easily accommodates a broad range of optimality criteria the particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum likelihood trellis decoding is investigated in detail several different methods for decoding a given trellis are discussed and compared in a number of examples finally analysis of the dynamical properties of certain optimal sectionalizations is presented

a new description of trellis codes a trellis code is a sliding window method of encoding a binary data stream as a sequence of real or complex numbers that are input to a noisy transmission channel ungerboeck has constructed simple trellis codes that provide the same noise immunity as is given by increasing the power of uncoded transmission by factors ranging from two to four his method is to specify an underlying convolutional code and a rule mapping by set partitioning that maps the output of this code onto a fixed signal constallation a new description of a trellis code is given that combines these two steps into one the new description is analytic rather than graphical many practical codes can be described very simply and strict bounds on performance can be obtained a method for differential encoding trellis codes is presented that was suggested by the authors representation

concatenated quantum codes can attain the quantum gilbert varshamov bound a family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant the asymptotic quantum gilbert varshamov gv bound states that there exist q ary quantum codes of sufficiently long block length n having fixed rate r with distance at least n h 1 q 2 1 r 2 where h q 2 is the q 2 ary entropy function for q only random quantum codes are known to asymptotically attain the quantum gv bound however random codes have little structure in this paper we generalize the classical result of thommesen to the quantum case thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum gv bound the outer codes are quantum generalized reed solomon codes and the inner codes are random independently chosen stabilizer codes where the rates of the inner and outer codes lie in a specified feasible region

on the bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

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

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

suboptimum decoding of decomposable block codes to decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous however if a code can be decomposed into constituent codes with smaller dimensions and simpler structure it is possible to devise a practical and yet efficient scheme to decode the code this paper investigates a class of decomposable codes their distance and structural properties it is shown that this class includes several classes of well known and efficient codes as subclasses several methods for constructing decomposable codes or decomposing codes are presented a two stage soft decision or hard decision decoding scheme for decomposable codes their translates or unions of translates is devised and its error performance is analyzed for an awgn channel the two stage soft decision decoding is suboptimum error performances of some specific decomposable codes based on the proposed two stage soft decision decoding are evaluated it is shown that the proposed two stage suboptimum decoding scheme provides an excellent trade off between the error performance and decoding complexity for codes of moderate and long block length

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

the existence of concatenated codes list decodable up to the hamming bound it is proven that binary linear concatenated codes with an outer algebraic code specifically a folded reed solomon code and independently and randomly chosen linear inner codes achieve with high probability the optimal tradeoff between rate and list decoding radius in particular for any 0 0 there exist concatenated codes of rate at least 1 h e that are combinatorially list decodable up to a fraction of errors the hamming bound states that the best possible rate for such codes cannot exceed 1 h and standard random coding arguments show that this bound is approached by random codes with high probability a similar result with better list size guarantees holds when the outer code is also randomly chosen the methods and results extend to the case when the alphabet size is any fixed prime power q 2

ellipsoidal lists and maximum likelihood decoding we study an interrelation between the coverings generated by linear n k codes and complexity of their maximum likelihood ml decoding first discrete ellipsoids in the hamming spaces e sub 1 sup n are introduced these ellipsoids represent the sets of most probable error patterns that need to be tested in soft decision ml decoding we show that long linear n k codes surrounded by ellipsoids of exponential size 2 sup n k can cover the whole space e sub 2 sup n then it is proven that ml decoding of most long n k codes needs only about 2 sup n k most probable error patterns to be tested on any quantized memoryless channel finally ml decoding complexity is bounded from above by 2 sup k n k n this substantially reduces the general trellis complexity 2 sup min n k k

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

suboptimal decoding of linear codes partition technique general symmetric channels are introduced and near maximum likelihood decoding in these channels is studied first we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis it is proved that the decoding error probability of suboptimal decoding is bounded above for any q ary code of length n and code rate r by twice the error probability of its maximum likelihood decoding and tends to the latter as n grows second we design a suboptimal trellis like algorithm which reduces the known decoding complexity of the order of q sup n min r 1 r operations to that of q sup nr i r operations for all cyclic codes and virtually all long linear codes we also consider the corresponding bounds for concatenated codes an important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability

instantaneous polarization attributes based on an adaptive approximate covariance method we introduce a method for computing instantaneous polarization attributes from multicomponent signals this is an improvement on the standard covariance method scm because it does not depend on the window size used to compute the standard covariance matrix we overcome the window size problem by deriving an approximate analytical formula for the cross energy matrix in which we automatically and adaptively determine the time window the proposed method uses polarization analysis as applied to multicomponent seismic by waveform separation and filtering

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

concatenated codes with fixed inner code and random outer code we derive lower bounds on the distance and error exponent of the coding scheme described in the title the bounds are compared to the parameters and error performance of a concatenated code family with varying inner codes of equal rates and a fixed minimum distance separable mds code as the outer code letting the inner and outer code lengths approach infinity

distance properties of expander codes the minimum distance of some families of expander codes is studied as well as some related families of codes defined on bipartite graphs the weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the gilbert varshamov gv bound a lower bound on the minimum distances of constructive families of expander codes is derived the relative minimum distance of the expander code is shown to exceed the product bound i e the quantity spl delta sub 0 spl delta sub 1 where spl delta sub 0 and spl delta sub 1 are the minimum relative distances of the constituent codes as a consequence of this a polynomially constructible family of expander codes is obtained whose relative distance exceeds the zyablov bound on the distance of serial concatenations