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

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

shift register synthesis modulo m the berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence or linear feedback shift register that can generate the sequence we present an algorithm which generalizes berlekamp s to the case when the elements of the sequence are integers modulo m where m is an arbitrary but known integer details will be published elsewhere

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

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

on the decoding of reed solomon and bch codes over integer residue rings we present a decoding procedure for reed solomon rs and bch codes defined over an integer residue ring pgz sub q where q is a power of a prime p the proposed decoding procedure as for rs and bch codes over fields consists of four major steps 1 calculation of the syndromes 2 calculation of the elementary symmetric functions by a modified berlekamp massey 1968 1969 algorithm for commutative rings 3 calculation of the error location numbers and 4 calculation of the error magnitudes the proposed decoding procedure also applies to the synthesis of a shortest linear feedback shift register lfsr capable of generating a prescribed finite sequence of elements lying in a commutative ring with identity

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

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

cross term forwarding for systems with time delay this paper presents a lyapunov based control design method for nonlinear systems with time delay the key step is construction of a composite lyapunov functionalv for a stable cascade this functional consists of lyapunov functionals for the subsystems and a cross term defined by an infinite integral the ldquol g v typerdquo control law is then employed to achieve global asymptotic stability the design procedure can be applied recursively to stabilize complex feedforward structures when applied to a linear cascade it gives a formula for the control law even if some parts of the system can only be stabilized through channels with multiple non commensurate or distributed delays

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

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

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 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

consta abelian codes over galois rings we study n length consta abelian codes a generalization of the well known abelian codes and constacyclic codes over galois rings of characteristic p sup a where n and p are coprime a twisted discrete fourier transform dft is used to generalize transform domain results of abelian and constacyclic codes to consta abelian codes further we characterize consta abelian codes invariant under two kinds of monomials whose underlying permutations are effected by i multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions all the codes studied here belong to the class of quasi twisted codes which are known to contain some good codes we show that the dual of a consta abelian code invariant under the two monomials is also a consta abelian code closed under both monomials

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

paperfolding morphisms planefilling curves and fractal tiles an interesting class of automatic sequences emerges from iterated paperfolding the sequences generate curves in the plane with an almost periodic structure we generalize the results obtained by davis and knuth on the self avoiding and planefilling properties of these curves giving simple geometric criteria for a complete classification finally we show how the automatic structure of the sequences leads to self similarity of the curves which turns the planefilling curves in a scaling limit into fractal tiles for some of these tiles we give a particularly simple formula for the hausdorff dimension of their boundary

morphic words beatty sequences and integer images of the fibonacci language abstract morphic words are letter to letter images of fixed points x of morphisms on finite alphabets there are situations where these letter to letter maps do not occur naturally but have to be replaced by a morphism we call this a decoration of x theoretically decorations of morphic words are again morphic words but in several problems the idea of decorating the fixed point of a morphism is useful we present two of such problems the first considers the so called aa sequences where is a quadratic irrational a is the beatty sequence defined by a n n and aa is the sequence a a n the second example considers homomorphic embeddings of the fibonacci language into the integers which turns out to lead to generalized beatty sequences with terms of the form v n p n q n r where p q and r are integers

truncated predictor feedback control for exponentially unstable linear systems with time varying input delay abstract the stabilization of exponentially unstable linear systems with time varying input delay is considered in this paper we extend the truncated predictor feedback tpf design method which was recently developed for systems with all poles on the closed left half plane to be applicable to exponentially unstable linear systems assuming that the time varying delay is known and bounded the design approach of a time varying state feedback controller is developed based on the solution of a parametric lyapunov equation an explicit condition is derived for which the stability of the closed loop system with the proposed controller is guaranteed it is shown that for the stability of the closed loop system the maximum allowable time delay in the input is inversely proportional to the sum of the unstable poles in the plant the effectiveness of the proposed method is demonstrated through numerical examples

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

abelian codes over galois rings closed under certain permutations we study n length abelian codes over galois rings with characteristic p sup a where n and p are relatively prime having the additional structure of being closed under the following two permutations i permutation effected by multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions a code is t quasi cyclic t qc if t is an integer such that cyclic shift of a codeword by t positions gives another codeword we call the abelian codes closed under the first permutation as unit invariant abelian codes and those closed under the second as quasi cyclic abelian qca codes using a generalized discrete fourier transform gdft defined over an appropriate extension of the galois ring we show that unit invariant abelian and qca codes can be easily characterized in the transform domain for t 1 qca codes coincide with those that are cyclic as well as abelian the number of such codes for a specified size and length is obtained and we also show that the dual of an unit invariant t qca code is also an unit invariant t qca code unit invariant abelian hence unit invariant cyclic and t qca codes over galois field f sub p sup l and over the integer residue rings are obtainable as special cases

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

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

hamming metric decoding of alternant codes over galois rings the standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence we extend this technique to decode alternant codes over galois rings we consider the module m a b as spl equiv b mod x sup r of all solutions to the key equation where s is the syndrome polynomial and r is the number of rows in a parity check matrix for the code in decoding we seek a particular solution spl sigma spl omega spl isin m which we prove can be found in a grobner basis for m we present an iterative algorithm which generates a grobner basis modulo x sup k 1 from a given basis modulo x sup k at the rth step a grobner basis for m is found and the required solution recovered

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

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

global asymptotic stabilization of feedforward systems with delay in the input the problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved no limitation on the size of the delay is imposed explicit expressions of bounded control laws are determined

global and semi global stabilization of linear systems with multiple delays and saturations in the input this paper studies stabilization problems for linear systems with multiple delays in the input two types of delays are considered the first type of delays is constant delays which can be arbitrarily large while the second type is time varying with an arbitrarily large bound with the first type of delays under the condition that the open loop system is absolutely controllable with all its eigenvalues on the imaginary axis globally stabilizing state and output feedback laws are constructed based on the solution to a family of parametric riccati equations which can be obtained explicitly through the solution of a parametric linear matrix equation with the second type of delays under the condition that the open loop system is absolutely controllable with all its eigenvalues on the imaginary axis being zero global state and output feedback laws are explicitly constructed based on the solution to a similar family of parametric riccati equations when the input is also subject to magnitude saturation it is shown that semiglobal stabilization instead of global stabilization can still be achieved numerical examples illustrate the effectiveness of the proposed approach

on the key equation we consider the set m a b a bh mod x2t of all solutions of the key equation for alternant codes where h is the syndrome polynomial in decoding these codes a particular solution m is sought subject to and being relatively prime and satisfying certain degree conditions we prove that these requirements specify uniquely as the minimal element of m analogous to the monic polynomial of minimal degree generating an ideal of f x with respect to a certain term order and that as such may be determined from an appropriate grobner basis of m motivated by this and other variations of the key equation such as that appropriate to errors and erasures decoding we derive a general algorithm for solving the congruence a bg mod xn for a range of term orders defined by the conditions on the particular solution required our techniques provide a unified approach to the solution of these key equations

global asymptotic stabilization for chains of integrators with a delay in the input the problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved no limitation on the size of the delay is imposed to validate the approach a third order example is presented

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

generalized rational interpolation over commutative rings and remainder decoding we propose a new decoding procedure for bose chaudhuri hocquenghem bch and reed solomon rs codes over z sub m where m is a product of prime powers our method generalizes the remainder decoding technique for rs codes originally introduced by welch and berlekamp and retains its key feature of not requiring the prior evaluation of syndromes it thus represents a significant departure from other algorithms that have been proposed for decoding linear block codes over integer residue rings our decoding procedure involves a welch berlekamp wb type algorithm for solving a generalized rational interpolation problem over a commutative ring r with identity the solution to this problem includes as a special case the solution to the wb key equation over r which is central to our decoding procedure a remainder decoding approach for decoding cyclic codes over z sub m up to the hartmann tzeng bound is also presented

on distributed delay in linear control laws part i discrete delay implementations this note proposes two approaches to approximate distributed delay in linear control laws and furthermore to implement it in the z domain and in the s domain the h sup spl infin norm of the approximation error converges to 0 when the number n of approximation steps approaches spl infin hence the instability problem due to the approximation error which has been widely studied in recent years does not exist provided that n is large enough moreover the static gain is guaranteed so that no extra efforts are needed to retain the steady state performance as by products two new formulas for the forward and backward rectangular rules are obtained these formulas are more accurate than the conventional ones when the integrand has an exponential term

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

stabilization of exponentially unstable discrete time linear systems by truncated predictor feedback abstract predictor state feedback solves the problem of stabilizing a discrete time linear system with input delay by predicting the future state with the solution of the state equation and thus rendering the closed loop system free of delay the solution of the state equation contains a term that is the convolution of the past control input with the state transition matrix thus the implementation of the resulting predictor state feedback law involves iterative calculation of the control signal a truncated predictor feedback law results when the convolution term in the state prediction is discarded when the feedback gain is constructed from the solution of a certain parameterized lyapunov equation the truncated predictor feedback law has been shown to achieve asymptotic stabilization of a system that is not exponentially unstable in the presence of an arbitrarily large delay by tuning the value of the parameter small enough in this paper we extend this result to exponentially unstable systems stability analysis leads to a bound on the delay and a range of the values of the parameter for which the closed loop system is asymptotically stable as long as the delay is within the bound the corresponding output feedback result is also derived

a parametric lyapunov equation approach to the design of low gain feedback low gain feedback has found several applications in constrained control systems robust control and nonlinear control low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic riccati equation are the eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter while the are approach requires the solution of an are for each value of the parameter this note proposes an alternative approach to low gain feedback design based on the solution of a parametric lyapunov equation such an approach possesses the advantages of both the eigenstructure assignment approach and the are based approach it also avoids the possible numerical stiffness in solving a parametric are and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach

negacyclic codes of length 2 sup s over galois rings codes over the ring of integers modulo 4 have been studied by many researchers negacyclic codes such that the length n of the code is odd have been characterized over the alphabet zopf4 and furthermore have been generalized to the case of the alphabet being a finite commutative chain ring in this paper we investigate negacyclic codes of length 2s over galois rings the structure of negacyclic codes of length 2s over the galois rings gr 2a m as well as that of their duals are completely obtained the hamming distances of negacyclic codes over gr 2a m in general and over zopf2 a in particular are studied among other more general results the hamming distances of all negacyclic codes over zopf2 a of length 4 8 and 16 are given the weight distributions of such negacyclic codes are also discussed

decoding a class of lee metric codes over a galois ring we investigate a class of lee 1958 metric alternant codes with symbols in z sub pn establishing a lower bound on the minimum lee distance where certain restrictions are placed on the code parameters corresponding to this bound we have devised a decoding algorithm which is implemented over a finite field the algorithm proceeds by finding a grobner basis of the module m of solutions to a key equation we obtain a necessary characterization of the solution module by solving iteratively a linear sequence over a galois ring and show that the particular solution sought by the decoder is minimal in m the required solution can then be found in an appropriate grobner basis of m

stability criteria of linear systems with multiple input delays under truncated predictor feedback abstract the model reduction technique as a feedback design approach for linear systems with input delays achieves finite spectrum assignment of the closed loop system in a delay free form the resulting predictor type feedback law contains a distributed term that involves the convolution between the past input and the state transition matrix and hence causes difficulty in its implementation for the purpose of easy implementation the truncated predictor feedback tpf simplifies the predictor feedback by discarding the distributed term and delay independent truncated predictor feedback further eliminates the delay dependent exponential factor in this paper we consider the stabilization of a general linear system with multiple time varying input delays by tpf or delay independent tpf stability criteria expressed in terms of input delays and the feedback parameter are derived through lyapunov krasovskii stability analysis and then numerically studied under both feedback laws

brief paper robust consensus of multi agent systems with diverse input delays and asymmetric interconnection perturbations the consensus problem of second order multi agent systems with diverse input delays is investigated based on the frequency domain analysis decentralized consensus conditions are obtained for the multi agent system with symmetric coupling weights then the robustness of the symmetric system with asymmetric perturbation is studied a bound of the largest singular value of the perturbation matrix is obtained as the robust consensus condition simulation examples illustrate the design procedure of consensus protocols and validate the correctness of the results

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

linear systems with delayed controls a reduction linear systems with delayed control action are transformed into systems without delays under an absolute continuity condition the new system is an ordinary differential control equation in the general case the new system is a measure differential control system it is shown how the controllability stabilization and various optimization problems can be analyzed via the reduced systems

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

the frobenius problem for homomorphic embeddings of languages into the integers abstract let s be a map from a language l to the integers satisfying s v w s v s w for all v w l the classical frobenius problem asks whether the complement of s l in the natural numbers will be infinite or finite and in the latter case the value of the largest element in this complement this is also known as the coin problem and l is the full language consisting of all words over a finite alphabet we solve the frobenius problem for the golden mean language any sturmian language and the thue morse language we also consider two dimensional embeddings

forwarding backstepping and finite spectrum assignment for time delay systems the nonlinear control ideas and results of forwarding and backstepping have been recently extended to systems with time delay in this paper we discuss how to apply these tools to design robust controls for linear delay systems we first show that the integral cross term forwarding produces a feedback law that achieves finite spectrum of the closed loop system combining it with backstepping provides a structured design method for a large class of systems with time delay robustness of the controller to uncertainties including uncertain time delay is illustrated by an example

global stabilization of oscillators with bounded delayed input the problem of globally asymptotically stabilizing by bounded feedback of an oscillator with an arbitrary large delay in the input is solved a first solution follows from a general result on the global stabilization of null controllable linear systems with delay in the input by bounded control laws with a distributed term next it is shown through a lyapunov analysis that the stabilization can be achieved as well when neglecting the distributed terms it turns out that this main result is intimately related to the output feedback stabilization problem

computing abelian complexity of binary uniform morphic words although a lot of research has been done on the factor complexity also called subword complexity of morphic words obtained as fixed points of iterated morphisms there has been little development in exploring algorithms that can efficiently compute their abelian complexity the factor complexity counts the number of factors of a given length n while the abelian complexity counts that number up to letter permutation we propose and analyze a simple o n algorithm for quickly computing the exact abelian complexities for all indices from 1 up to n when considering binary uniform morphisms using our algorithm we also analyze the structure in the abelian complexity for that class of morphisms our main result implies in particular that the infinite word over the alphabet 1 0 1 constructed from the consecutive forward differences of the abelian complexity of a fixed point of a binary uniform morphism is in fact an automatic sequence with the same morphic length since the proof produces morphisms that typically contain many redundant letters we present an efficient algorithm to eliminate them in order to simplify the morphisms and to see the patterns produced more clearly

on asymptotic stabilizability of linear systems with delayed input this paper examines the asymptotic stabilizability of linear systems with delayed input by explicit construction of stabilizing feedback laws it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable i e all the open loop poles are on the closed left half plane a simple example shows that such results would not be true if the open loop system is exponentially unstable it is further shown that such systems when subject to actuator saturation are semiglobally asymptotically stabilizable by linear state or output feedback

properties of the parametric lyapunov equation based low gain design with applications in stabilization of time delay systems this technical note studies some properties of the recently developed parametric lyapunov equation based low gain feedback design method as applications of these new properties alternative and simpler solutions are proposed to the global stabilization problem for a class of linear systems with input delay and the semi global stabilization problem when the systems are in addition subject to actuator saturation besides the simplicity in their construction the new solutions can also be easily scheduled online to achieve global result in the presence of input saturation