on the inequivalence of generalized preparata codes if m is odd and sigma in aut gf 2 m is such that x rightarrow x sigma 2 1 is 1 1 there is a 2 m 1 1 2 m l 2m 2 nonlinear binary code p sigma having minimum distance 5 all the codes p sigma have the same distance and weight enumerators as the usual preparata codes which rise as p sigma when x sigma x 2 it is shown that p sigma and p tau are equivalent if and only if tau sigma pm 1 and aut p sigma is determined

performance of reed muller codes and a maximum likelihood decoding algorithm for ofdm the performance of reed muller encoding and a maximum likelihood decoding algorithm for orthogonal frequency division multiplexing is presented the example codes have a tightly bounded peak to mean envelope power ratio while simultaneously enabling powerful error correction we present a maximum likelihood decoder that makes use of a distance preserving map and multiple fast hadamard transforms its operation is described in detail and its performance is assessed under realistic channel conditions

construction of hadamard mathbb z _ 2 mathbb z _ 4 q _ 8 codes for each allowable value of the rank and dimension of the kernel this paper deals with hadamard mathbb z 2 mathbb z 4 q 8 codes which are binary codes after a gray map from a subgroup of direct products of mathbb z 2 mathbb z 4 and q 8 where q 8 is the quaternionic group in a previous work these codes were classified in five shapes in this paper we analyze the allowable range of values for the rank and dimension of the kernel which depends on the particular shape of the code we show that all these codes can be represented in a standard form from a set of generators which can help in understanding the characteristics of each shape the main results we present are the characterization of hadamard mathbb z 2 mathbb z 4 q 8 codes as a quotient of a semidirect product of mathbb z 2 mathbb z 4 linear codes and the construction of hadamard mathbb z 2 mathbb z 4 q 8 codes with each allowable pair of values for the rank and dimension of the kernel

coding for the lee and manhattan metrics with weighing matrices this paper has two goals the first one is to discuss two related packing problems in the lee and manhattan metrics one is to find good codes for error correction i e packings of lee spheres and the other is to transform the space in a way that volumes are preserved and each lee sphere or scaled cross polytope will be transformed into a shape inscribed in a small cube the second goal is to consider weighing matrices for some of these coding problems weighing matrices have been used as building blocks for codes in the hamming metric in various constructions in this paper we will consider mainly two types of weighing matrices namely conference matrices and hadamard matrices to construct codes in the lee and manhattan metric we will show that these matrices have some desirable properties when considered as generator matrices for codes in these metrics

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

zrm codes quaternary zrm r m codes were defined so that their binary images via gray map are reed muller codes for some specific values of in the literature two different definitions of such codes can be found they will be denoted zrm r m and zrm r m codes in this correspondence we show that both definitions are equivalent exactly for those values of r such that their binary images are reed muller codes moreover we prove that for all r these binary images are linear codes in the case of zrm r m but they are not if we use the definition of zrm r m in this last case we compute the rank and the dimension of the kernel of these codes

the frisch scheme in dynamic system identification abstract the use of the frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input output noisy sequences unlike the algebraic case it is shown that in general only a single model is compatible with the data these results are first proposed for single input single output systems and then generalized to the multivariable case

kerdock codes determine unitary 2 designs the non linear binary kerdock codes are known to be gray images of certain extended cyclic codes of length n 2 m over mathbb z 4 we show that exponentiating these mathbb z 4 valued codewords by i triangleq sqrt 1 produces stabilizer states that are quantum states obtained using only clifford unitaries these states are also the common eigenvectors of commuting hermitian matrices forming maximal commutative subgroups mcs of the pauli group we use this quantum description to simplify the derivation of the classical weight distribution of kerdock codes next we organize the stabilizer states to form n 1 mutually unbiased bases and prove that automorphisms of the kerdock code permute their corresponding mcs thereby forming a subgroup of the clifford group when represented as symplectic matrices this subgroup is isomorphic to the projective special linear group psl 2 n we show that this automorphism group acts transitively on the pauli matrices which implies that the ensemble is pauli mixing and hence forms a unitary 2 design the kerdock design described here was originally discovered by cleve et al 2016 but the connection to classical codes is new which simplifies its description and translation to circuits significantly sampling from the design is straightforward the translation to circuits uses only clifford gates and the process does not require ancillary qubits finally we also develop algorithms for optimizing the synthesis of unitary 2 designs on encoded qubits i e to construct logical unitary 2 designs software implementations are available at https github com nrenga symplectic arxiv18a which we use to provide empirical gate complexities for up to 16 qubits

brief paper on the identifiability of errors in variables models with white measurement errors we discuss identifiability of dynamic siso errors in variables eiv models with white measurement errors although this class of models turns out to be generically identifiable it has been pointed out that in certain circumstances there may be two eiv models which are indistinguishable from external input output experiments this lack of global identifiability may be prejudicial to identification and needs better understanding the identifiability conditions found in the literature guarantee uniqueness under certain coprimality assumptions on the rational transfer function of the ideal true system and the spectral density of the noiseless true input unfortunately these conditions are not testable since they concern precisely the unknowns of the problem which are not available to the experimenter we provide new identifiability conditions which are instead expressible in terms of the external description of the observable signals namely their joint power spectral densities

cyclic and negacyclic codes over finite chain rings the structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring r are established when n is not divisible by the characteristic of the residue field r some cases where n is divisible by the characteristic of the residue field r are also considered namely the structure of negacyclic codes of length 2 sup t over spl zopf sub 2 sup m and that of their duals are derived

decoding of linear codes over galois rings we present a method for decoding an arbitrary linear code over a galois ring spl rscr by a process of lifting decoding algorithms for a family of linear codes over a finite field spl kscr forming an spl alpha element chain where spl kscr is the quotient field of spl rscr and spl rscr has characteristic p sup spl alpha as a new result this method also works for linear codes over spl rscr which are nonfree spl rscr modules

duadic codes over z sub 2k duadic codes constitute a well known family of binary cyclic codes they are generalized in this correspondence to the setting of abelian codes over the ring z sub 2k self duality isoduality and type ii properties are studied

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

on minimum lee weights of hensel lifts of some binary bch codes motivated by the paper of calderbank mcguire kumar and helleseth see ibid vol 42 no 1 p 217 26 jan 1996 we prove the following result for any given positive integer l spl ges 3 the minimum lee weights of hensel lifts to z sub 4 of binary primitive bch codes of length 2 sup m 1 and designed distance 2 sup l 1 is just 2 sup l 1 when a m can be divided by l or b m is sufficiently large for hensel lifts of binary primitive bch codes of arbitrary designed distance spl delta spl ges 4 we also prove that their minimum lee weight d sub l spl les 2 log sub 2 spl delta 1 1 when m is sufficiently large moreover a result about minimum lee weights of certain z sub 4 codes defined by galois rings which is similar to the result in calderbank et al is proved

on lee association schemes over z4 and their terwilliger algebra abstract let f 0 1 2 3 and define the set k k 0 k 1 k 2 of relations on f such that x y k i if and only if x y i mod 4 let n be a positive integer we consider the lee association scheme l n over z 4 which is the extension of length n of the initial scheme f k let t denote the terwilliger algebra of l n with respect to the zero codeword of length n we show that t is generated by a homomorphic image of the universal enveloping algebra of the lie algebra sl 3 c and the center z t furthermore we determine the irreducible modules for t using the schur weyl duality

negacyclic codes over z sub 4 of even length this paper generalizes the results from wolfmann see ibid vol 45 p 2527 2532 nov 1999 and vol 47 p 1773 1779 july 2001 classifying all negacyclic codes over z sub 4 of even length using a transform approach it is then shown which linear binary cyclic codes are images of negacyclic codes under the gray map in the process the concatenated structure of both negacyclic codes and binary repeated root cyclic codes is given

several classes of codes and sequences derived from a bbz_ 4 valued quadratic form let m and k be positive integers with m gcd m k being odd for a r and b l the exponential sum x litr ax 2bx2k 1 is studied systematically in this paper where i 1 r gr 4 m is a galois ring l is the teichmuller set of r and tr is the trace function from the galois ring r to z4 through the discussions on the solutions of certain equations and the newly developed theory of z4 valued quadratic forms the distribution of the exponential sum is completely determined as its applications we can determine the lee weight and hamming weight distributions of a class of codes ck over z4 and the correlation distribution of a quaternary sequence family uk respectively furthermore the hamming weight distributions of the binary codes obtained from ck under the most significant bit msb and gray maps are also determined for the msb map sequences of uk the nontrivial maximal correlation value is given and the correlation distribution is determined for the gray map sequences of uk it should be noted that the distribution of the exponential sum for the case gcd m k 1 is obtained for the first time and then the corresponding codes and sequences are novel

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

optimal phases for a family of quadriphase cdma sequences signature sequences with good even even and odd or polyphase correlations are crucial for asynchronous code division multiple access cdma when the data sequence is random the even and odd or polyphase correlations are equally important however for most known signature sequences only their even correlations were analyzed it appears that determining the odd or the polyphase correlations is generally a very hard problem since the odd or the polyphase correlations depend on the phases of the signature sequences sole 1989 boztas hammons and kumar 1992 found a family of quadriphase sequences that are asymptotically optimal these sequences gain a factor spl radic 2 in terms of their maximum periodic even correlations when compared with the best possible binary phase shift keying bpsk sequences we find the optimal phases of these sequences the optimality is in the sense that at these phases the mean square values of the even odd and the polyphase correlations are minimal and achieve the welch 1974 bound equality simultaneously furthermore we show that at these phases the average user interference of these sequences is always smaller than that of the ideal random signature sequences comprehensive analytical and numerical results show that good phase sequences can offer a nonnegligible amount of gain over bad phase sequences at modest and high signal to noise ratios

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

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

linear constructions for dna codes in this paper we translate in terms of coding theory constraints that are used in designing dna codes for use in dna computing or as bar codes in chemical libraries we propose new constructions for dna codes satisfying either a reverse complement constraint a gc content constraint or both that are derived from additive and linear codes over four letter alphabets we focus in particular on codes over gf 4 and we construct new dna codes that are in many cases better sometimes far better than previously known codes we provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search

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

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

p adic valuation of weights in abelian codes over spl zopf p sup d counting polynomial techniques introduced by wilson are used to provide analogs of a theorem of mceliece mceliece s original theorem relates the greatest power of p dividing the hamming weights of words in cyclic codes over gf p to the length of the smallest unity product sequence of nonzeroes of the code calderbank li and poonen presented analogs for cyclic codes over zopf 2d using various weight functions hamming lee and euclidean weight as well as count of occurrences of a particular symbol some of these results were strengthened by wilson who also considered the alphabet zopf pd for p an arbitrary prime these previous results new strengthened versions and generalizations are proved here in a unified and comprehensive fashion for the larger class of abelian codes over zopf pd with p any prime for abelian codes over zopf 4 combinatorial methods for use with counting polynomials are developed these show that the analogs of mceliece s theorem obtained by wilson for hamming weight lee weight and symbol counts and the analog obtained here for euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose fourier transforms have a specified support

decoding of cyclic codes over f sub 2 uf sub 2 we give a simple decoding algorithm to decode linear cyclic codes of odd length over the ring r f sub 2 uf sub 2 0 1 u u u 1 where u sup 2 0 a spectral representation of the cyclic codes over r is given and a bch like bound is given for the lee distance of the codes the ring r shares many properties of z sub 4 and f sub 4 and admits a linear gray map

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

macwilliams type identities on the lee and euclidean weights for linear codes over zl abstract motivated by the works of shiromoto 3 and shi et al 4 we study the existence of the macwilliams type identities with respect to the lee and euclidean weight enumerators for linear codes over the ring z l necessary and sufficient conditions for the existence of the macwilliams type identities with respect to the lee and euclidean weight enumerators for linear codes over z l are given some examples about such macwilliams type identities are also presented

4 phase sequences with near optimum correlation properties two families of four phase sequences are constructed using irreducible polynomials over z sub 4 family a has period l 2 sup r 1 size l 2 and maximum nontrivial correlation magnitude c sub max

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

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

binary images of cyclic codes over z sub 4 we determine all linear cyclic codes over z sub 4 of odd length whose gray images are linear codes or equivalently whose nechaev gray 1989 image are linear cyclic codes or are linear cyclic codes

adaptive algorithms for the simultaneous stabilization and decoupling of time varying multivariable systems the not previously considered problem of decoupling time varying multivariable systems is investigated and two dual purpose algorithms which decouple and regulate the system are derived one algorithm also gives pole assignment and the other minimum variance control the algorithms are shown to stabilize and decouple the closed loop system

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

l 2 optimal identification of mimo errors in variables models from the v gap geometrical interpretation an l 2 optimal identification method is extended to cope with mimo errors in variables eiv model estimation based on a geometrical interpretation for the v gap metric the l 2 optimal approximate models are composed of system and noise models and characterised by a normalised right graph symbol nrgs and its complementary inner factor cif respectively this metric can be evaluated as the supreme of sine values of the maximal principal angles between nrgs frequency responses of two concerned models in order to make full use of the angular cosine formula for complex vectors to reduce computational loads a cif of the nrgs of the perturbed model is introduced and thus the system parameter optimisation can be efficiently solved by sequential quadratic programming methods with the estimated system model the associated noise model can be built by right multiplication of an inner matrix finally a simulation example demonstrates the effectiveness of the proposed identification method

fft based bp decoding of general ldpc codes over abelian groups we introduce a wide class of low density parity check ldpc codes large enough to include ldpc codes over finite fields rings or groups as well as some nonlinear codes a belief propagation decoding procedure with the same complexity as for the decoding of ldpc codes over finite fields is also presented moreover an encoding procedure is developed

a note on a basic exact sequence for the lee and euclidean weights of linear codes over zl abstract this paper is devoted to presenting two counterexamples to shiromoto s 1999 results 2 on macwilliams type identities with respect to lee weight enumerators and euclidean weight enumerators over z l

model reference adaptive and nonadaptive control of linear time varying plants we address the adaptive and nonadaptive model reference control problem for a class of multivariable linear time varying plants namely index invariant ones we show that under appropriate controllability and observability conditions this class of plants admits a fractional description in terms of polynomial differential operators and as such allows for a polynomial equation based controller design we also show that for a model reference control objective the controller can be designed by solving a set of algebraic equations further when the plant parameters are only partially known we employ a gradient based adaptive law with projection and normalization to update the controller parameters and establish the stability and tracking properties of adaptive closed loop plant finally we present a simple example to illustrate the design and realization of both the adaptive and nonadaptive control laws

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

on the correlation distributions of the optimal quaternary sequence family cal u and the optimal binary sequence family cal v recently new optimal families s and u of quaternary sequences have been presented and the optimal binary sequence family v obtained from family s under gray map has been investigated as well the two sequence families u and v are optimal with respect to the well known sidelnikov bound and welch bound but their exact correlation distributions are not known until now in this paper their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of z4 valued quadratic forms

quaternary constant amplitude codes for multicode cdma a constant amplitude code is a code that reduces the peak to average power ratio papr in multicode code division multiple access mc cdma systems to the favorable value 1 in this paper quaternary constant amplitude codes codes over z 4 of length 2 m with error correction capabilities are studied these codes exist for every positive integer m while binary constant amplitude codes cannot exist if m is odd every word of such a code corresponds to a function from the binary m tuples to z4 having the bent property i e its fourier transform has magnitudes 2 m 2 several constructions of such functions are presented which are exploited in connection with algebraic codes over z4 in particular quaternary reed muller kerdock and delsarte goethals codes to construct families of quaternary constant amplitude codes mappings from binary to quaternary constant amplitude codes are presented as well

minimal vectors in linear codes minimal vectors in linear codes arise in numerous applications particularly in constructing decoding algorithms and studying linear secret sharing schemes however properties and structure of minimal vectors have been largely unknown we prove basic properties of minimal vectors in general linear codes then we characterize minimal vectors of a given weight and compute their number in several classes of codes including the hamming codes and second order reed muller codes further we extend the concept of minimal vectors to codes over rings and compute them for several examples turning to applications we introduce a general gradient like decoding algorithm of which minimal vectors decoding is an example the complexity of minimal vectors decoding for long codes is determined by the size of the set of minimal vectors therefore we compute this size for long randomly chosen codes another example of algorithms in this class is given by zero neighbors decoding we discuss relations between the two decoding methods in particular we show that for even codes the set of zero neighbors is strictly optimal in this class of algorithms this also implies that general asymptotic improvements of the zero neighbors algorithm in the frame of gradient like approach are impossible we also discuss a link to secret sharing schemes

families of hadamard bbz_ 2 bbz_ 4 q_ 8 codes a z2z4q8 code is the binary image after a gray map of a subgroup of z2k1 z4k2 q8k3 where q8 is the quaternion group on eight elements such z2z4q8 codes are translation invariant propelinear codes as are the well known z4 linear or z2z4 linear codes in this paper we show that there exist pure z2z4q8 codes that is codes that do not admit any abelian translation invariant propelinear structure we study the dimension of the kernel and rank of the z2z4q8 codes and we give upper and lower bounds for these parameters we give tools to construct a new class of hadamard codes formed by several families of z2z4q8 codes we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are hadamard

the set of observationally equivalent errors in variables models abstract linear dynamic errors in variables or factor models with uncorrelated noise components are considered we give a description of the set of all noise spectra which are oberservationally equivalent with respect to the second moments of observations

a structure theory for linear dynamic errors in variables models we deal with problems connected with the identification of linear dynamic systems in situations when inputs and outputs may be contaminated by noise the case of uncorrelated noise components and the bounded noise case is considered if also the inputs may be contaminated by noise a number of additional complications in identification arise in particular the underlying system is not uniquely determined from the population second moments of the observations a description of classes of observationally equivalent systems is given continuity properties of mappings relating classes of observationally equivalent systems to the spectral densities of the observations are derived and the classes of spectral densities corresponding to a given maximum number of outputs are studied

algebraic construction of cyclic codes over z sub 8 with a good euclidean minimum distance let s 8 denote the set of the eight admissible signals of an 8psk communication system the alphabet s 8 is endowed with the structure of z sub 8 the set of integers taken modulo 8 and codes are defined to be z sub 8 submodules of z sub 8 sup n three cyclic codes over z sub 8 are then constructed their length is equal to 6 8 and 7 and they respectively contain 64 64 and 512 codewords the square of their euclidean minimum distance is equal to 8 16 4 spl radic 2 and 10 2 spl radic 2 respectively the size of the codes of length 6 and 7 can be doubled while the euclidean minimum distance remains the same

linear dynamic errors in variables models some structure theory abstract this paper gives a survey of some recent results on problems of identifiability or more general of the relation between the observations and certain system characteristics for linear dynamic errors in variables models for a large part of the paper the noise components are assumed to be mutually uncorrelated after the general problem statement a rather complete analysis of the single input single output case is given also the case of three variables and the case where the number of inputs is equal to the number of outputs are discussed in detail finally the use of higher order cummulant spectra for identifiability is investigated

cyclic codes over z sub 4 locator polynomials and newton s identities certain nonlinear binary codes contain more codewords than any comparable linear code presently known these include the kerdock 1972 and preparata 1968 codes that can be very simply constructed as binary images under the gray map of linear codes over z sub 4 that are defined by means of parity checks involving galois rings this paper describes how fourier transforms on galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes these methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of z sub 4 linear codes with length 2 sup m m odd and size 2 2 sup m 1 5m 2 the gray image of the code of length 32 is the best 64 2 sup 37 code that is presently known this paper also determines the exact minimum lee distance of the linear codes over z sub 4 that are obtained from the extended binary two and three error correcting bch codes by hensel lifting the gray image of the hensel lift of the three error correcting bch code of length 32 is the best 64 2 sup 32 code that is presently known this code also determines an extremal 32 dimensional even unimodular lattice

a mass formula and rank of spl zopf sub 4 cyclic codes of length 2 sup e in this correspondence we study cyclic codes of length n 2 sup e over the ring r sub 4 spl zopf sub 4 x x sup n 1 in particular we derive a closed form expression for the number of these codes for a given length n we also study the rank of these codes and derive an expression for that furthermore we give an example in which we study all cyclic codes of length 8 we also study all self dual codes of length 8 and 16 and classify them according to their type

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

a linear construction for certain kerdock and preparata codes the nordstrom robinson kerdock and slightly modified preparata codes are shown to be linear over z4 the integers mod 4 the kerdock and preparata codes are duals over z4 and the nordstrom robinson code is self dual all these codes are just extended cyclic codes over z4 this provides a simple definition for these codes and explains why their hamming weight distributions are dual to each other firstand second order reed muller codes are also linear codes over z4 but hamming codes in general are not nor is the golay code

fast ml decoding algorithm for the nordstrom robinson code corresp the extended nordstrom robinson code is an optimum nonlinear double error correcting code of length 16 with considerable practical importance this code is also useful as a rate 1 2 vector quantizer for random waveforms such as speech linear predictive coding lpc residual a fast decoding algorithm is described for maximum likelihood decoding or nearest neighbor search in the squared error sense with 304 additions and 128 comparisons

an exponential number of generalized kerdock codes if n 1 is an odd composite integer then there are at least 2 1 2 n pairwise inequivalent binary error correcting codes of length 2 n size 2 2 n and minimum distance 2 n 1 2 1 2 n 1

a novel numerical method for exact model matching problem with stability in system design the exact model matching problem with stability consists of compensating a given system using a realizable control law of a specified structure in order to ensure the stability of the compensated system and achieve a target closed loop transfer function in this paper we develop a novel numerical method to verify the solvability of the problem for left invertible systems and further construct a desired solution our method has a complexity which is cubic in the state dimension of the system and the desired model and can be implemented in a numerically reliable way

list decoding of generalized reed solomon codes over commutative rings we show that the list decoding procedure of guruswami and sudan may be used to decode generalized reed solomon rs codes defined over commutative rings with identity an algorithm for performing the first of the two phases of this decoding procedure is also given in particular we show that for generalized rs codes over chain rings the probability of picking an incorrect codeword from a given list can be reduced by increasing the size of the code alphabet while keeping the length and minimum hamming distance of the code constant

polynomial equation approach to exact model matching problem in multivariable linear systems a new procedure for solving exact model matching problems in multivariable linear systems is presented the procedure consists of solving a linear polynomial matrix equation conditions for existing dynamical stable and least order solutions of the problem are given it is a generalization for multivariable linear systems of kucera s 1981 approach for single input single output siso systems

a note on one weight and two weight projective mathbb z _ 4 codes in this paper we solve the open problems raised in 8 and present some examples to illustrate the obtained results moreover we work out the diophantine problem by shi and wang and then give the sufficient conditions for the nonexistence of two lee weight projective codes over mathbb z 4 with type 4 k 1 2 k 2