a note on type ii convolutional codes the result of a search for the world s second type ii doubly even and self dual convolutional code is reported a rate r 4 8 16 state time invariant convolutional code with free distance d sub free 8 was found to be type ii the initial part of its weight spectrum is better than that of the golay convolutional code gcc generator matrices and path weight enumerators for some other type ii convolutional codes are given by the wrap around technique tail biting versions of 32 18 8 type ii block codes are constructed

a fast algorithm for computing distance spectrum of convolutional codes a fast algorithm for searching a tree fast is presented for computing the distance spectrum of convolutional codes the distance profile of a code is used to limit substantially the error patterns that have to be searched the algorithm can easily be modified to determine the number of nonzero information bits of an incorrect path as well as the length of an error event for testing systematic codes a faster version of the algorithm is given fast is much faster than the standard bidirectional search on a microvax d sub infinity 27 was verified for a rate r 1 2 memory m 25 code in 37 s of cpu time extensive tables of rate r 1 2 encoders are given several of the listed encoders have distance spectra superior to those of any previously known codes of the same rate and memory a conjecture than an r 1 2 systematic convolutional code of memory 2m will perform as well as a nonsystematic convolutional code of memory m is given strong support

on a bifurcation in the quantum system for an approximation to the henon heiles system abstract it is known in classical mechanics that the henon heiles system exhibits bifurcation phenomena in its periodic solutions depending on a parameter these phenomena are usually analyzed by applying the birkhoff gustavson s normal form method the henon heiles hamiltonian is expanded into a series in normal form and then truncated at a finite order it is also known that the bifurcation taking place in the truncated system approximates the one taking place in the original henon heiles system in case of sufficiently small energy in this regard the aim of this article is to find what will happen when the truncated henon heiles system is quantized quantum analogue to the classical bifurcation phenomena are observed as follows degeneracy of energy eigenvalues proves to occur at a certain parameter value at that value the quantum system happens to admit the so 2 d symmetry d being the dihedral group however at a generic parameter value the symmetry reduces to so 2 d2 changes in eigenstates depending on the parameter are observed a symmetry breaking of zero angular momentum state takes place when the parameter passes that particular value

a levinson type algorithm for modeling fast sampled data the standard discrete time autoregressive model is poorly suited for modeling series obtained by sampling continuous time processes at fairly rapid rates large computational errors can occur when the levinson algorithm is used to estimate the parameters of this model because the toeplitz covariance matrix is ill suited for inversion an alternative model is developed based on an incremental difference operator rather than the shift operator it is shown that as the sampling period goes to zero unlike the standard autoregressive parameters the coefficients of this model converge to certain parameters that depend directly on the statistics of the continuous time process a levinson type algorithm for efficiently estimating the parameters of this model is derived numerical examples are given to show that when the sampling interval is small this algorithm is considerably less sensitivity to arithmetic roundoff errors than the levinson algorithm

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

estimation of continuous time stochastic signals from sample covariances the problem of estimating the parameters in stochastic continuous time signals represented as continuous time autoregressive moving average arma processes from discrete time data is considered the proposed solution is to fit the covariance function of the process parameterized by the unknown parameters to sample covariances it is shown that the method is consistent and an expression for the approximate covariance matrix of the estimated parameter vector is derived the derived variances are compared with empirical variances from a monte carlo simulation and with the cramer rao bound it turns out that the variances are close to the cramer rao bound for certain choices of the sampling interval and the number of covariance elements used in the criterion function

a characterization of certain griesmer codes mmd codes in a more general sense let c be an n k d sub q linear code the defect of c is the parameter s s c n k 1 d if k spl ges m 1 spl ges 2 then by the griesmer bound d spl les q sup m q 1 q sub m 1 s m the author s interest is in those linear codes having the maximum minimum distance i e d q sup m q 1 q sup m 1 s m for m 1 we have d q s 1 and the codes are maximum minimum distance mmd codes in the sense of faldum and willems see ibid vol 44 p 1555 58 1998 thus we consider mmd codes in a more general sense we refer to them simply as mmd codes all mmd codes with m 1 are known up to formal equivalence note that two codes are formally equivalent if they have the same weight distribution the author classifies up to formal equivalence the mmd codes with m spl ges 2

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

relative generalized hamming weights of one point algebraic geometric codes security of linear ramp secret sharing schemes can be characterized by the relative generalized hamming weights of the involved codes 26 24 in this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one point algebraic geometric codes as it is demonstrated for hermitian codes our bound is often tight furthermore for these codes the relative generalized hamming weights are often much larger than the corresponding generalized hamming weights

parameters of goppa codes revisited we discuss parameters of goppa 1970 codes such as minimum distance covering radius distance distribution and generalized hamming weights by a variation on the exponential sums method and combinatorial arguments we sharpen known bounds on these parameters

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

the symmetry group of the harmonic oscillator and its reduction the symmetry of the harmonic oscillator is dealt with in the hamiltonian formalism unitary operators representing the symmetry are studied from this point of view of additional interest is reduction of the symmetry group su 4 for the four dimensional harmonic oscillator subspaces are determined from the representation spaces for su 4 so as to give those for so 4

on the weight hierarchy of goethals codes over z sub 4 the rth generalized hamming weight d sub r m j of the goethals code spl gscr sub m j of length 2 sup m over z sub 4 is considered in this correspondence in the case that m spl ges 3 is an odd integer d sub r m j is exactly determined for r 0 5 1 1 5 2 2 5 and 3 0 for a composite m we give an upper bound d sub r m j using the lifting technique

on the stability of a general polynomial in the space of markov parameters abstract markov parameters and the associated stability criterion were first introduced for continuous time real polynomials recently robust stability of such polynomials was considered in markov parameters space where efficient robust stability tests were obtained based on the markov theorem this has motivated the authors to extend the above idea to more general types of polynomials and develop markov parameters and the associated stability criterion for complex continuous time as well as real and complex discrete time polynomials moreover for each polynomial type we present compact relations in order to recover the coefficients of a polynomial corresponding to a given set of markov parameters the stability results presented here may be useful for 1 d and 2 d filtering and control applications

degeneracy of energy levels in a maslov quantized perturbed 1 1 resonant oscillator a quantum counterpart of a hamiltonian pitchfork bifurcation it is widely known in classical theory that nonlinear hamiltonian systems depending on several parameters often exhibit bifurcation in their trajectories against the changes in the parameters a hamiltonian pitchfork bifurcation in the oscillator s periodic trajectories is a typical example in this article to seek a quantum counterpart of that bifurcation the maslov quantization is applied to a perturbed 1 1 resonant oscillator in the birkhoff gustavson normal form with two parameters by using a geometric method the degeneracy of energy levels is found to be taken as a quantum counterpart to that bifurcation the bifurcation set for the hamiltonian pitchfork bifurcation in classical theory is viewed as the classical limit of a bifurcation set for the degeneracy of energy levels in quantum theory

the weight distributions of extended binary primitive bch codes of length 128 in previous work a method was presented to compute the weight distribution of a linear block code by using its trellis diagram in this correspondence the method is improved by using the trellis structure of linear block codes another method with reduced computational complexity is also proposed which uses the invariant property of a code with these methods the weight distributions of all extended binary primitive bch codes of length 128 are computed except for those for which the formulas of the weight distribution are known it turns out that 128 64 22 extended binary primitive bch code is formally self dual the probability of an undetectable error for each code is computed and its monotonicity is examined

reduction of the semisimple 1 1 resonance the method of averaging is often used in hamiltonian systems of two degrees of freedom to find periodic orbits such periodic orbits can be reconstructed from the critical points of an associated reduced hamiltonian on a reduced space this paper details the construction of the reduced space and the reduced hamiltonian for the semisimple 1 1 resonance case the reduced space will be a 2 sphere in r3 and the reduced differential equations will be euler s equations restricted to this sphere the orbit projection from the energy surface in phase space to this sphere will be the hopf map the results of the paper are related to problems in physics on degeneracies due to symmetries of classical two dimensional harmonic oscillators and their quantum analogues for the hydrogen atom

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

improved bounds on the threshold gap in ramp secret sharing in this paper we consider linear secret sharing schemes over a finite field mathbb f q where the secret is a vector in mathbb f q ell and each of the n shares is a single element of mathbb f q we obtain lower bounds on the so called threshold gap g of such schemes defined as the quantity r t where r is the smallest number such that any subset of r shares uniquely determines the secret and t is the largest number such that any subset of t shares provides no information about the secret our main result establishes a family of bounds which are tighter than previously known bounds for ell geq 2 furthermore we also provide bounds in terms of n and q on the partial reconstruction and privacy thresholds a more fine grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size finally we compare our lower bounds with known upper bounds in the asymptotic setting

the quantum normal form and its equivalents a quantum analog called the quantum normal form of the classical birkhoff gustavson normal form is presented the algebraic relationship between the quantum and birkhoff gustavson normal forms has been established by developing the latter using lie transforms it is shown that the birkhoff gustavson normal form can be obtained from the quantum normal form using an anharmonic oscillator and a henon heiles system as test cases the equivalence between the quantum normal form and the rayleigh schrodinger perturbation method is shown this equivalence provides an algebraic connection between the birkhoff gustavson normal form and the rayleigh schrodinger perturbation approach the question of weyl and torus quantizations of the birkhoff gustavson normal form is discussed in the light of the quantum normal form

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

an efficient order recursive algorithm with a lattice structure for estimating continuous time ar process parameters abstract an order recursive algorithm is proposed to solve the normal equation in estimating continuous time ar process parameters the new algorithm requires o m 2 computations where m is the order of the ar process and therefore is computationally efficient this is similar to the levinson type algorithm of pham and le breton however the proposed algorithm has a new associated lattice structure and as a consequence a new stability test for continuous time systems not just ar processes by checking the reflection coefficients of the lattice this can be viewed as a continuous time analogy that is recursive in order of the well known discrete time version its discrete time approximation when sampled data are given and connections with other algorithms are also discussed

on the intractability of permuting a block code to minimize trellis complexity an important problem in the theory and application of block code trellises is to find a coordinate permutation of a given code to minimize the trellis complexity we show that the problem of finding a coordinate permutation that minimizes the number of vertices at a given depth in the minimal trellis for a binary linear block code is np complete

a delta mywe algorithm for parameter estimation of noisy ar processes we develop a delta operator based modified yule walker equation algorithm mywe for parameter estimation of a noisy autoregressive ar process the methodology in developing this new algorithm is similar to the previous works on pure ar processes computer simulation results are given to show the improvement of performance in estimating ar parameters in white noise over the q operator mywe algorithm

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

efficient solution of linear systems of equations with recursive structure abstract inversion algorithms of order n2 for n n matrices with recursive structure are derived in a unified way several known algorithms as well as some new ones for toeplitz type hilbert type and vandermonde type matrices are given

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 weighted coordinates bound and trellis complexity of block codes and periodic packings weighted entropy profiles and a new bound the weighted coordinates bound on the state complexity profile of block codes are presented these profiles and bound generalize the notion of dimension length profile dlp and entropy length profile elf to block codes whose symbols are not drawn from a common alphabet set and in particular group codes likewise the new bound may improve upon the dlp and elf bounds fur linear and nonlinear block codes over fields however it seems that the major contribution of the proposed bound is to the study of trellis complexity of block codes whose different coordinates are drawn from different alphabet sets the label code of lattice and nonlattice periodic packings usually has this property the construction of a trellis diagram for a lattice and some related bounds are generalized to periodic packings by introducing the fundamental module of the packing and using the new bound on the state complexity profile this generalization is limited to a given coordinate system we show that any bounds on the trellis structure of block codes and in particular the bound presented in this work are applicable to periodic packings

generalized bezout s theorem and its applications in coding theory this paper presents a generalized bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more plane curves a new approach to determine a lower bound on the minimum distance for algebraic geometric codes defined from a class of plane curves is introduced based on the generalized bezout theorem examples of more efficient linear codes are constructed using the generalized bezout theorem and the new approach for d 4 the linear codes constructed by the new construction are better than or equal to the known linear codes for d spl ges 5 these new codes are better than the known ag codes defined from whole spaces the klein codes 22 16 5 and 22 15 6 over gf 2 sup 3 and the improved hermitian code 64 56 6 over gf 2 sup 4 are also constructed

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

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

a goppa like bound on the trellis state complexity of algebraic geometric codes for a linear code c of length n and dimension k wolf 1978 noticed that the trellis state complexity s c of c is upper bounded by w c min k n k we point out some new lower bounds for s c in particular if c is an algebraic geometric code then s c spl ges w c g a where g is the genus of the underlying curve and a is the abundance of the code

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

lexicographic codes error correcting codes from game theory lexicographic codes or lexicodes are defined by various versions of the greedy algorithm the theory of these codes is closely related to the theory of certain impartial games which leads to a number of surprising properties for example lexicodes over an alphabet of size b 2 a are closed under addition while if b 2 2 a the lexicodes are closed under multiplication by scalars where addition and multiplication are in the nim sense explained in the text hamming codes and the binary golay codes are lexieodes remarkably simple constructions are given for the steiner systems s 5 6 12 and s 5 8 24 several record breaking constant weight codes are also constructed

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

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

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

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

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

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

chains demi matroids and profiles it is shown that each chain of linear codes has an associated demi matroid a combinatorial structure that extends the notion of a vector matroid of a linear code these demi matroids are proven to determine important properties of the chain and it is shown that linear code chain duality is represented by demi matroid duality in a natural and yet surprising way profiles are defined for arbitrary demi matroids and thus for linear code chains which generalizes previous results in the literature and provides simple and transparent proofs thereof finally applications are given to encryption of messages through wire tap channels of type ii

perturbations of elliptic billiards abstract the dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied in the case of an elliptic billiard table the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points using a discrete analog to the melnikov method we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table from the formulas that we get we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table in addition we also include a characterization of the period two periodic points for any billiard table

trellis complexity and minimal trellis diagrams of lattices this paper presents results on trellis complexity and low complexity trellis diagrams of lattices we establish constructive upper bounds on the trellis complexity of lattices these bounds both improve and generalize the similar results of tarokh and vardy see ibid vol 43 p 1294 1300 1997 we also construct trellis diagrams with minimum number of paths for some important lattices such trellises are called minimal the constructed trellises which are novel in many cases can be employed to efficiently decode the lattices via the viterbi algorithm in particular a general structure for minimal trellis diagrams of d sub n lattices is obtained this structure corresponds to a new code formula for d sub n moreover we develop some important duality results which are used in both deriving the upper bounds and finding the minimal trellises all the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases

on the macwilliams identity for convolutional codes the adjacency matrix associated with a convolutional code collects in a detailed manner information about the weight distribution of the code a macwilliams identity conjecture stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code will be formulated and an explicit formula for the transformation will be stated the formula involves the macwilliams matrix known from complete weight enumerators of block codes the conjecture will be proven for the class of convolutional codes where either the code itself or its dual does not have forney indices bigger than one for the general case the conjecture is backed up by many examples and a weaker version will be established

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

a macwilliams identity for convolutional codes the general case a macwilliams identity for convolutional codes will be established it makes use of the weight adjacency matrices of the code and its dual based on state space realizations the controller canonical form of the codes in question the macwilliams identity applies to various notions of duality appearing in the literature on convolutional coding theory

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

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

on the complexity of decoding lattices using the korkin zolotarev reduced basis upper and lower bounds are derived for the decoding complexity of a general lattice l the bounds are in terms of the dimension n and the coding gain spl gamma of l and are obtained based on a decoding algorithm which is an improved version of kannan s 1983 method the latter is currently the fastest known method for the decoding of a general lattice for the decoding of a point x the proposed algorithm recursively searches inside an n dimensional rectangular parallelepiped cube centered at x with its edges along the gram schmidt vectors of a proper basis of l we call algorithms of this type recursive cube search rcs algorithms it is shown that kannan s algorithm also belongs to this category the complexity of rcs algorithms is measured in terms of the number of lattice points that need to be examined before a decision is made to tighten the upper bound on the complexity we select a lattice basis which is reduced in the sense of korkin zolotarev 1873 it is shown that for any selected basis the decoding complexity using rcs algorithms of any sequence of lattices with possible application in communications spl gamma spl ges 1 grows at least exponentially with n and spl gamma it is observed that the densest lattices and almost all of the lattices used in communications e g barnes wall lattices and the leech lattice have equal successive minima esm for the decoding complexity of esm lattices a tighter upper bound and a stronger lower bound result are derived

a new sphere packing in 20 dimensions we construct a new nonlattice sphere packing in 20 dimensions which is denser than any 20 dimensional packing previously known properties of the new sphere packing are investigated

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 rate 1 2 1 3 and 1 4 binary convolutional encoders with an optimum distance profile tabulations of binary systematic and nonsystematic polynomial convolutional encoders with an optimum distance profile for rate 1 2 1 3 and 1 4 are given the reported encoders are found by computer searches that optimize over the weight spectra the free distances for rate 1 3 and 1 4 are compared with heller s 1968 and griesmer s 1960 upper bounds

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

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

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

the algebraic quantisation of the birkhoff gustavson normal form the author develops an algebraic quantisation method for the birkhoff gustavson normal form for this purpose the weyl quantisation rule is used the method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities the author gives a brief review of the normal form derives some of its general properties and finds a general analytic solution for the fourth degree normal form for hamiltonians of two degrees of freedom in particular this includes the henon heiles system the author compares the results of specific examples with other works the question of canonically invariant quantisation the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed the author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals if the normal form accidentally terminates so that the classical system is integrable then this implies quantum integrability of the normal form hamiltonian

codes satisfying the chain condition the authors considered weight hierarchies of codes satisfying the chain condition they called these chain good first they gave a set of simple necessary conditions for a sequence to be chain good they proved that given one chain good sequence there is an infinite set of chain good sequences that can be constructed from this one sequence finally they used this result to completely describe the sets of chain good sequences of dimensions up to 5

the weight hierarchies of q ary codes of dimension 4 the weight hierarchy of a linear n k q code c over gf q is the sequence d sub 1 d sub 2 d sub k where d sub spl tau is the smallest support of an spl tau dimensional subcode of c the possible weight hierarchies of n 4 q codes are studied in particular the possible weight hierarchies of n 4 3 codes are determined

the twisted squaring construction trellis complexity and generalized weights of bch and qr codes the structure of the twisted squaring construction a generalization of the squaring construction is studied with respect to trellis diagrams and complexity we show that binary affine invariant codes which include the extended primitive bch codes and the extended binary quadratic residue codes are equivalent to twisted squaring construction codes in particular a recursive symmetric reversible design of the bch codes is derived using these constructions the parameters of the minimal trellis diagram of the bch codes are determined including the componentwise state space profile and trellis complexity new designs and permutations that yield low trellis complexity for the quadratic residue codes are presented generalized hamming weights are derived from these constructions as an example the 48 24 12 quadratic residue code is analyzed a strictly componentwise optimal permutation is derived and the corresponding state space profile and complete generalized hamming weight hierarchy are obtained

minimum support weights of binary codes some relations between the minimum support weights are discussed in particular the possible weight hierarchies of codes of dimension 4 are determined