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

the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

upper bounds on generalized distances we derive new asymptotic bounds for generalized distances our approach extends the classical hamming plotkin and elias bounds the latter bound involves extending the definition of generalized distances to nonlinear codes

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

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 trellis based recursive maximum likelihood decoding algorithm for binary linear block codes this paper presents an efficient trellis based maximum likelihood decoding algorithm for binary linear block codes this algorithm is recursive in nature and is devised based on the structural properties and optimum sectionalization of a code trellis the complexity of the proposed decoding algorithm is analyzed numerical results show that the proposed decoding algorithm significantly reduces the decoding complexity a recursive method for finding the optimum sectionalization of a trellis in terms of computational complexity is given

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

generalized hamming weights of nonlinear codes and the relation to the z sub 4 linear representation we give a new definition of generalized hamming weights of nonlinear codes and a new interpretation connected with it these generalized weights are determined by the entropy length profile of the code we show that this definition characterizes the performance of nonlinear codes on the wire tap channel of type ii the new definition is invariant under translates of the code it satisfies the property of strict monotonicity and the generalized singleton bound we check the relations between the generalized weight hierarchies of z sub 4 linear codes and their binary image under the gray map we also show that the binary image of a z sub 4 linear code is a symmetric not necessarily rectangular code moreover if this binary image is a linear code then it admits a twisted squaring construction

dual mode predictive control algorithm for constrained hammerstein systems in the process industry there exist many systems which can be approximated by a hammerstein model moreover these systems are usually subjected to input magnitude constraints in this paper a multi channel identification algorithm mcia is proposed in which the coefficient parameters are identified by least squares estimation lse together with a singular value decomposition svd technique compared with traditional single channel identification algorithms the present method can enhance the approximation accuracy remarkably and provide consistent estimates even in the presence of coloured output noises under relatively weak assumptions on the persistent excitation pe condition of the inputs then to facilitate the following controller design this mcia is converted into a two stage single channel identification algorithm ts scia which preserves most of the advantages of mcia with this ts scia as the inner model a dual mode non linear model predictive control nmpc algorithm is developed in detail over a finite horizon an optimal input profile found by solving a open loop optimal control problem drives the non linear system state into the terminal invariant set afterwards a linear output feedback controller steers the state to the origin asymptotically in contrast to the traditional algorithms the present method has a maximal stable region a better steady state performance and a lower computational complexity finally simulation results on a heat exchanger are presented to show the efficiency of both the identification and the control algorithms

a table of state complexity bounds for binary linear codes this article contains a table of bounds on the state complexity of binary linear codes with length smaller than 25 general results on the state complexity of binary linear codes with low dimension or low minimum distance are included

functional series modeling approach to identification of nonstationary stochastic systems the problem of identification of a nonstationary stochastic system is considered and an estimation method based on the functional series modeling fsm of the system parameter trajectory is proposed for its solution it is shown that the parameter matching properties of fsm estimators can be described in terms of the appropriately defined time varying impulse and frequency responses it is suggested and verified by means of computer simulation that the averaged frequency characteristics associated with fsm estimators can yield useful information about their parameter matching abilities

the perfect binary one error correcting codes of length 15 part ii properties a complete classification of the perfect binary one error correcting codes of length 15 as well as their extensions of length 16 was recently carried out in p r j o stergard and o pottonen the perfect binary one error correcting codes of length 15 part i classification ieee trans inf theory vol 55 pp 4657 4660 2009 in the current accompanying work the classified codes are studied in great detail and their main properties are tabulated the results include the fact that 33 of the 80 steiner triple systems of order 15 occur in such codes further understanding is gained on full rank codes via switching as it turns out that all but two full rank codes can be obtained through a series of such transformations from the hamming code other topics studied include non systematic codes embedded one error correcting codes and defining sets of codes a classification of certain mixed perfect codes is also obtained

optimal choice of time scaling factor for linear system approximations using laguerre models clowes 1965 illustrated how to select the optimal time scaling factor for approximations of rational transfer functions using a series expansion of laguerre functions the present paper extends this result to approximations for general l sub 2 stable linear systems as a direct application of this extended result we are able to obtain approximations for irrational transfer functions which minimize the frequency domain error in the l sub 2 sense also we give empirical solutions for the optimal time scaling factor for approximations of first order plus delay systems which can then be applied to a larger class of overdamped systems

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

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

entropy length profiles bounds on the minimal covering of bipartite graphs and trellis complexity of nonlinear codes the trellis representation of nonlinear codes is studied from a new perspective we introduce the new concept of entropy length profile elp this profile can be considered as an extension of the dimension length profile dlp to nonlinear codes this elaboration of the dlp the entropy length profiles appears to be suitable to the analysis of nonlinear codes additionally and independently we use well known information theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs we use these bounds in conjunction with the elp notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of nonlinear block codes represented by any trellis diagram we lay down no restrictions on the trellis structure and we do not confine the scope of our results to proper or one to one trellises only the basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords we also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords this approach is pursued further to derive similar bounds on the branch complexity profile to the best of our knowledge the proposed upper bounds are the only upper bounds that address nonlinear codes the novel lower bounds are tighter than the existing bounds the new quantities and bounds reduce to well known results when applied to linear codes

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

optimal and near optimal encoders for short and moderate length tail biting trellises the results of an extensive search for short and moderate length polynomial convolutional encoders for time invariant tail biting representations of block codes at rates r 1 4 1 3 1 2 and 2 3 are reported the tail biting representations found are typically as good as the best known block codes

on the trellis complexity of certain binary linear block codes the trellis complexity s c of an n k d code c is investigated in the case where the weights of nonzero codewords in c are confined to d spl middotspl middotspl middot 2d 1 spl cup n it is shown that s c spl ges k 1 furthermore s c k 1 if the code is self complementary if the nonzero weights are confined to d spl middotspl middotspl middot 2d 3 then s c k

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

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

computationally efficient soft decision decoding of linear block codes based on ordered statistics soft decision decoding of a linear block code using the most reliable basis corresponding to each received word is investigated based either on probabilistic properties or on the structure of the code considered three improvements to the algorithm devised by fossorier and lin see ibid vol 41 no 9 p 1379 1396 1995 are presented these modifications allow large computation savings or significant decoding speedup with little error performance degradation first a reduced probabilistic list of codeword candidates is associated with order i reprocessing of a given code it results in a large reduction of the maximum number of computations with a very small degradation in performance then a probabilistic stopping criterion is introduced for order 0 reprocessing this new test significantly decreases the average number of computations when appropriately implemented finally the application of the algorithm to coset decoding is considered for u u v constructed codes in addition to the conventional coset decoding a new adaptive practically optimum coset decoding method is presented where at each reprocessing stage the number of surviving cosets decreases suboptimum closest coset decoding is also investigated it is shown that two stage decoding with the algorithm of fossorier and lin offers a large variety of choices since the reprocessing order of each stage can be determined independently

nonparametric identification of wiener systems by orthogonal series a wiener system i e a system comprising a linear dynamic and a nonlinear memoryless subsystems connected in a cascade is identified both the input signal and disturbance are random white and gaussian the unknown nonlinear characteristic is strictly monotonous and differentiable and therefore the problem of its recovering from input output observations of the whole system is nonparametric it is shown that the inverse of the characteristic is a regression function and a class of orthogonal series nonparametric estimates recovering the regression is proposed and analyzed the estimates apply the trigonometric legendre and hermite orthogonal functions pointwise consistency of all the algorithms is shown under some additional smoothness restrictions the rates of their convergence are examined and compared an algorithm to identify the impulse response of the linear subsystem is proposed

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

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

ordered binary decision diagrams and minimal trellises ordered binary decision diagrams obdds are graph based data structures for representing boolean functions they have found widespread use in computer aided design and in formal verification of digital circuits minimal trellises are graphical representations of error correcting codes that play a prominent role in coding theory this paper establishes a close connection between these two graphical models as follows let spl cscr be a binary code of length n and let f sub c x sub 1 x sub n be the boolean function that takes the value 0 at x sub 1 x sub n if and only if x sub 1 x sub n spl isin spl cscr given this natural one to one correspondence between boolean functions and binary codes we prove that the minimal proper trellis for a code spl cscr with minimum distance d 1 is isomorphic to the single terminal obdd for its boolean indicator function f sub c x sub 1 x sub n prior to this result the extensive research during the past decade on binary decision diagrams in computer engineering and on minimal trellises in coding theory has been carried out independently as outlined in this work the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines

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

the preparata and goethals codes trellis complexity and twisted squaring constructions the trellis complexity of the preparata and goethals codes is examined it is shown that at least for a given set of permutations these codes are rectangular upper bounds on the state complexity profiles of the preparata and goethals codes are given the upper bounds on the state complexity of the preparata and goethals codes are determined by the dimension length profiles dlp of the extended primitive double and triple error correcting bch codes respectively a twisted squaring construction for the preparata and goethals codes is given based on the double and triple error correcting extended primitive bch codes respectively

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

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

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

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

on the trellis representation of the delsarte goethals codes in this correspondence the trellis representation of the kerdock and delsarte goethals codes is addressed it is shown that the states of a trellis representation of dg m spl delta under any bit order are either strict sense nonmerging or strict sense nonexpanding except maybe at indices within the code s distance set for spl delta spl ges 3 and for m spl ges 6 the state complexity s sub max dg m spl delta is found for all values of m and spl delta a formula for the number of states and branches of the biproper trellis diagram of dg m spl delta is given for some of the indices and upper and lower bounds are given for the remaining indices the formula and the bounds refer to the delsarte goethals codes when arranged in the standard bit order

on a method of atkinson for evaluating domain integrals in the boundary element method we have shown how to couple a numerical method due to atkinson for computing particular solutions to a class of elliptic differential equations with a variety of boundary element methods which alleviates the problem of domain discretization for solving inhomogeneous equations when the inhomogeneous term is of the form typically used in the dual reciprocity method atkinson s formula is such that only boundary integrals need to be approximated numerically if this method is coupled with the method of potentials it results in a computational technique which requires neither boundary nor domain discretization some numerical results are given validating our approach

distance verification for classical and quantum ldpc codes the techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes then these techniques are considered for classical and quantum stabilizer low density parity check ldpc codes new complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of ldpc codes these bounds are expressed in terms of the erasure correcting capacity of the corresponding ensemble we also present a new irreducible cluster technique that can be applied to any ldpc code and takes advantage of parity checks sparsity for both the classical and quantum ldpc codes this technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances which also include all known families of the quantum stabilizer ldpc codes

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

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

a quasi interpolation method for solving stiff ordinary differential equations based on the idea of quasi interpolation and radial basis functions approximation a numerical method is developed to quasi interpolate the forcing term of differential equations by using radial basis functions a highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions this overcomes the ill conditioning problem resulting from using the radial basis functions as a global interpolant error estimation is given for a particular second order stiff differential equation with boundary layer the result of computations indicates that the method can be applied to solve very stiff problems with the use of multiquadric a special class of radial basis functions it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation copyright 2000 john wiley sons ltd

laguerre series approximation of infinite dimensional systems laguerre fourier approximations of stable systems are shown to exhibit many desirable properties for various classes of infinite dimensional systems specifically time domain supremum and l1 norm convergence results and frequency domain h norm convergence results are given for laguerre fourier approximations it is also shown that the theory of laguerre polynomials solves explicitly the problem of determining laguerre fourier approximations for a large class of delay systems furthermore it is believed that these results are important for the study of orthonormal series identification as a general technique for identification of infinite dimensional systems

density length profiles and trellis complexity of lattices the density length profile dcp of a lattice spl lambda is analogous to the dimension length profile of a linear code the dlp is a geometrical invariant of spl lambda that includes the coding gain of spl lambda duality results analogous to those of linear block codes are derived for lattices bounds on the dlp may be derived from bounds on hermite s constants these hold with equality for many dense lattices in turn the dlp lowerbounds the state complexity profile of a minimal trellis diagram for spl lambda in any coordinate system it is shown that this bound can be met for the e sub 8 lattice by a laminated lattice construction with a novel trellis diagram bounds and constructions for other important low dimensional lattices are given two laminated lattice constructions of the leech lattice yield trellis diagrams with maximum state space sizes 1024 and 972

equivalence of laguerre based model order reduction and moment matching the equivalence between the classical moment matching and the laguerre based reduction approaches is shown in addition this equivalence is generalized to include a family of coefficients known as generalized markov parameters

technical communique decentralized adaptive control of linear interconnected systems based on laguerre series representation this paper investigates the problem of completely decentralized adaptive control of large scale systems when the subsystems are stable a new approach which eliminates the assumptions about the interconnection terms the time delay the order etc is proposed each subsystem is modeled by an orthonormal laguerre network put in state space form a simple decentralized predictive control law is proposed the potential of such a scheme is illustrated by simulation results

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

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

reliability based soft decision decoding with multiple biases in this paper a new reliability based soft decision decoding algorithm is presented this algorithm repeatedly uses biased reliability values to construct the most reliable basis mrb as a result this new method makes use of multiple information sets in a stochastic way compared to previously proposed competitive approaches this new method produces a more efficient mrb reprocessing type algorithm to achieve near maximum likelihood decoding mld performance with a proper choice of the bias value it can be combined with any mrb reprocessing type algorithm and in each case a tight performance analysis can be derived

deterministic adaptive control based on laguerre series representation the behaviour of adaptive controllers in the presence of unmodelled dynamics and the need for reduced a priori information have led us to abandon the usual arma transfer function representation for a representation by an orthonormal series the appeal of our new approach is that it eliminates the need for assumptions about the plant order and the time delay the plant is modelled by an orthonormal laguerre network put in state space form a simple predictive control law is proposed an explicit deterministic adaptive controller is then designed simulations show that it is easy to use able to handle non minimum phase plants and more robust than the conventional model based approach although we chose laguerre functions other orthonormal functions may be used we have already tested some with success

on binary linear codes which satisfy the two way chain condition general constructions of binary linear codes satisfying the two way chain condition are found sufficient conditions for their existence are determined some of these codes are proven to be z sub 4 linear an expression for their state complexity profile is also found cases when such codes have covering radius r t n k are shown

decentralized adaptive control of interconnected systems the adaptive control of a class of large scale systems formed of an arbitrary interconnection of subsystems with unknown parameters nonlinearities and bounded disturbances is investigated it is first shown that no matter how weak the interconnections are a decentralized adaptive control scheme can become unstable approaches are then developed for stabilization and tracking using new decentralized adaptive controllers in the case where the relative degree n of the transfer function of each decoupled subsystem is less than or equal to two sufficient conditions are established which guarantee boundedness and exponential convergence of the state and parameter errors to bounded residual sets in the absence of disturbances and interconnections the decentralized adaptive control schemes guarantee exact convergence of the tracking errors to zero the effectiveness of the proposed adaptive schemes is demonstrated using a simple example

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

siso app searches in lattices with tanner graphs an efficient low complexity soft output detector for general lattices is presented based on their tanner graph tg representations closest point searches in lattices can be performed as nonbinary belief propagation on associated tgs soft information output is naturally generated in the process the algorithm requires no backtrack cf classic sphere decoding and extracts extrinsic information a lattice s coding gain enables equivalence relations between lattice points which can be thereby partitioned in cosets total and extrinsic a posteriori probabilities at the detector s output further enable the use of soft detection information in iterative schemes the algorithm is illustrated via two scenarios that transmit a 32 point uncoded super orthogonal so constellation for multiple input multiple output mimo channels carved from an 8 dimensional nonorthogonal lattice d4 d4 it achieves maximum likelihood performance in quasistatic fading and performs close to interference free transmission and identically to list sphere decoding in independent fading with coordinate interleaving and iterative equalization and detection latter scenario outperforms former despite absence of forward error correction coding because the inherent lattice coding gain allows for the refining of extrinsic information the lattice constellation is the same as the one employed in the so space time trellis codes first introduced for 2 2 mimo by ionescu et al then independently by jafarkhani and seshadri algorithmic complexity is log linear in lattice dimensionality versus cubic in classic sphere decoders

recent developments in the numerical evaluation of particular solutions in the boundary element method abstract we survey recent work on the numerical evaluation of particular solutions in the boundary elements method bem particularly rbf approximations and related mathematical convergence theory current research suggests at least in r 2 that the best choice for bases is the generalized multiquadrics numerical results are presented to support this observation

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

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

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

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

trellis decoding complexity of linear block codes in this partially tutorial paper we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity maximum state and edge dimensions total span length and total vertices edges and mergers we obtain bounds on these complexities as extensions of well known dimension length profile dlp bounds codes meeting these bounds minimize all the complexity measures simultaneously conversely a code attaining the bound for total span length vertices or edges must likewise attain it for all the others we define a notion of uniform optimality that embraces different domains of optimization such as different permutations of a code or different codes with the same parameters and we give examples of uniformly optimal codes and permutations we also give some conditions that identify certain cases when no code or permutation can meet the bounds in addition to dlp based bounds we derive new inequalities relating one complexity measure to another which can be used in conjunction with known bounds on one measure to imply bounds on the others as an application we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths

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

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

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

the extraction and complexity limits of graphical models for linear codes two broad classes of graphical modeling problems for codes can be identified in the literature constructive and extractive problems the former class of problems concern the construction of a graphical model in order to define a new code the latter class of problems concern the extraction of a graphical model for a fixed given code the design of a new low density parity check code for some given criteria e g target block length and code rate is an example of a constructive problem the determination of a graphical model for a classical linear block code that implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem this work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes the primary focus of this work is a study of the space of all graphical models for a fixed given code the tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound the forest inducing cut set bound fi csb the proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools e g the csb and can be viewed as a generalization of the square root bound for tail biting trellises to graphical models with arbitrary cyclic topologies searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations that are shown to span this space finally heuristics for extracting novel graphical models for linear block codes using these transformations are investigated

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

high resolution distributed sampling of bandlimited fields with low precision sensors the problem of sampling a discrete time sequence of spatially bandlimited fields with a bounded dynamic range in a distributed communication constrained processing environment is studied a central unit having access to the data gathered by a dense network of low precision sensors is required to reconstruct the field snapshots to maximum accuracy both deterministic and stochastic field models are considered for stochastic fields results are established in the almost sure sense the feasibility of having a flexible tradeoff between the oversampling rate sensor density and the analog to digital converter adc precision while achieving an exponential accuracy in the number of bits per nyquist interval per snapshot is demonstrated this exposes an underlying conservation of bits principle the bit budget per nyquist interval per snapshot the rate can be distributed along the amplitude axis sensor precision and space sensor density in an almost arbitrary discrete valued manner while retaining the same exponential distortion rate characteristics

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

numerical computation of decentralized fixed modes abstract in this paper we use an algebraic characterization of fixed modes of a decentralized linear multivariable system to show that the fixed modes are related to the blocking zeros of certain subsystems derived from the given decentralized system a numerical algorithm is then presented which enables us to compute the fixed modes in a reliable manner examples are provided to illustrate the main results of the paper

asymptotically good codes have infinite trellis complexity the trellis complexity s c of a block code c is defined as the logarithm of the maximum number of states in the minimal trellis realization of the code the parameter s c governs the complexity of maximum likelihood decoding and is considered a fundamental descriptive characteristic of the code in a number of recent works we derive a new lower bound on s c which implies that asymptotically good codes have infinite trellis complexity more precisely for i spl ges 1 let c sub i be a code over an alphabet of size q of length n sub i rate r sub i and minimum distance d sub i the infinite sequence of codes c sub c sub 2spl middotspl middotspl middot such that n sub ispl rarrspl infin when i spl rarrspl infin is said to be asymptotically good if both r sub i and d sub in sub i are bounded away from zero as i spl rarrspl infin we prove that the complexity s c sub i increases linearly with n sub i in any asymptotically good sequence of codes

proof of a conjecture of mceliece regarding the expansion index of the minimal trellis we prove a conjecture of mceliece establishing that for each fixed order of positions of a linear code c the minimal trellis minimizes the quantity e v 1 where e and v stand for the number of edges and vertices in the trellis respectively as a consequence it follows that the minimal trellis uniquely minimizes the total number of operations required for viterbi decoding of a given code moreover we show that these results extend to the general class of rectangular codes namely the class of codes that admit a biproper trellis presentation which includes all group codes and many other useful nonlinear codes

structurally fixed modes a notion of structurally fixed modes is introduced in the framework of decentralized control systems to identify the modes that cannot be shifted by decentralized feedback regardless of the numerical values of system parameters in this way a qualitative characterization of decentralized fixed modes is provided which is suitable for delineating decentrally stabilizable systems the characterization leads naturally to the notion of fixed modes with respect to arbitrary feedback structure constraints

trellis complexity of short linear codes an extended table of shuurman s bounds on the state complexity of short binary linear codes is presented some new lower and upper bounds are obtained most of the newly found codes are based on the so called double zero tail termination dzt construction

an efficient algorithm for constructing minimal trellises for codes over finite abelian groups we present an efficient algorithm for computing the minimal trellis for a group code over a finite abelian group given a generator matrix for the code we also show how to compute a succinct representation of the minimal trellis for such a code and present algorithms that use this information to compute efficiently local descriptions of the minimal trellis this extends the work of kschischang and sorokine see ibid vol 41 no 6 p 1926 37 1995 who treated the case of linear codes over fields an important application of our algorithms is to the construction of minimal trellises for lattices a key step in our work is handling codes over cyclic groups c sub p spl alpha where p is a prime such a code can be viewed as a module over the ring z sub p spl alpha because of the presence of zero divisors in the ring modules do not share the useful properties of vector spaces we get around this difficulty by restricting the notion of linear combination to a p linear combination and by introducing the notion of a p generator sequence which enjoys properties similar to those of a generator matrix for a vector space

some decomposable codes the a x b x a b x construction codes with decomposable structure allow the use of multistage decoding procedures to achieve suboptimum bounded distance error performance with reduced decoding complexity this correspondence presents some new decomposable codes including a class of distance 8 codes that are constructed based on the a x b x a b x construction method some existing best codes are shown to be decomposable and hence can be decoded with multistage decoding