adaptive control of non linear continuous time systems using neural networks general relative degree and mimo cases multilayer neural networks are used in a non linear adaptive control problem the plant is an unknown feedback linearizable continuous time system with relative degree 1 the single input single output system is studied first and then the methodology is extended to control square multi input multi output systems the control objective is for the plant to track a reference trajectory and the control law is defined in terms of the outputs of the neural networks the parameters of the networks are updated on line according to an augmented tracking error and the network derivatives a local convergence theorem is given on the convergence of the tracking error this control algorithm is applied to control a two input two output relative degree two system

on the construction of equiangular frames from graphs abstract we give details of the 1 1 correspondence between equiangular frames of n vectors for r d and graphs with n vertices this has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory the nontight examples given here which correspond to graphs with more than 2 eigenvalues have the potential for similar applications e g the frame corresponding to the 5 cycle graph is the unique grassmannian frame of 5 vectors in openr 3 further the associated canonical tight frames have a small number of angles in many cases

two constructions of asymptotically optimal codebooks according to the welch bound abstract in a number of applications the codebook that minimizes the maximal cross correlation amplitude imax is often desirable there are two major ingredients in this paper these two contents are to propose two constructions of codebooks the codebooks generated by these constructions asymptotically meet the welch bound what we find is the codebook constructed in section 2 can also asymptotically meet the levenshtein bound the parameters of these codebooks are new

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

equiangular tight frames and signature sets in groups abstract quiangular frames are an important class of finite dimensional frames because of their superior performance and numerous applications the objective of this paper is to present a new tool to construct equiangular tight frames from groups we prove that many equiangular tight frames arise from subsets of groups which we call signature sets subsequently we define quasi signature sets and examine real equiangular tight frames associated with these subsets of groups this approach yields further results and establishes new correspondences we extend these results to complex equiangular tight frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular tight frames that arise from groups in this manner

traveling wavefronts for time delayed reaction diffusion equation ii nonlocal nonlinearity abstract in this paper we study a class of time delayed reaction diffusion equation with local nonlinearity for the birth rate for all wavefronts with the speed c c where c 0 is the critical wave speed we prove that these wavefronts are asymptotically stable when the initial perturbation around the traveling waves decays exponentially as x but the initial perturbation can be arbitrarily large in other locations this essentially improves the stability results obtained by mei so li and shen m mei j w h so m y li s s p shen asymptotic stability of traveling waves for the nicholson s blowflies equation with diffusion proc roy soc edinburgh sect a 134 2004 579 594 for the speed c 2 d m e p d m with small initial perturbation and by lin and mei c k lin m mei on travelling wavefronts of the nicholson s blowflies equations with diffusion submitted for publication for c c with sufficiently small delay time r 0 the approach adopted in this paper is the technical weighted energy method used in m mei j w h so m y li s s p shen asymptotic stability of traveling waves for the nicholson s blowflies equation with diffusion proc roy soc edinburgh sect a 134 2004 579 594 but inspired by gourley s a gourley linear stability of travelling fronts in an age structured reaction diffusion population model quart j mech appl math 58 2005 257 268 and based on the property of the critical wavefronts the weight function is carefully selected and it plays a key role in proving the stability for any c c and for an arbitrary time delay r 0

adaptive neural network tracking control with disturbance attenuation for multiple input nonlinear systems a switching adaptive neural network controller for multiple input nonlinear affine in the control dynamical systems with unknown nonlinearities is designed capable of arbitrarily attenuating l 2 or l infin external disturbances in the absence of disturbances a uniform ultimate boundedness property of the tracking error with respect to an arbitrarily small set around the origin is guaranteed as well as the uniform boundedness of all the signals in the closed loop the proposed switching adaptive controller effectively avoids possible division by zero while guaranteeing the continuity of switching in this way problems connected to existence of solutions and chattering phenomena are alleviated simulations illustrate the approach

stability of traveling wavefronts for a discrete diffusive competition system with three species abstract this paper is concerned with the nonlinear stability of monostable traveling wavefronts for a discrete three species competition diffusion system applying weighted energy method combining with the comparison principle we first show that the traveling wavefronts with large speed are exponentially stable when the initial perturbation around the traveling wave decays exponentially as x but the initial perturbation can be arbitrarily large in other locations then choosing different weight function we improve the stability result to any traveling wavefronts with speed greater than the critical wave speed we have to emphasize that the discrete dispersal operator in the system increases the difficulty of establishing energy estimates

a unified approach to whiteman s and ding helleseth s generalized cyclotomy over residue class rings the theory of cyclotomy dates back to gauss and has a number of applications in combinatorics coding theory and cryptography cyclotomy over a residue class ring bbzv can be divided into classical cyclotomy or generalized cyclotomy depending on v prime or composite in this paper we introduce a generalized cyclotomy of order d over bbzp1e1p2e2 pnen which includes whiteman s and ding helleseth s generalized cyclotomy as special cases here p1 p2 pn are pairwise distinct odd primes satisfying d pi 1 for all 1 i n and e1 e2 en are positive integers we derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers as applications we completely solve an open problem and a conjecture on whiteman s generalized cyclotomy of order four over bbzp1p2 besides we also construct an infinite series of near optimal codebooks over bbzp1p2 as well as some infinite series of asymptotically optimal difference systems of sets over bbzp1e1p2e2 pnen

new constructions of asymptotically optimal codebooks with multiplicative characters in practical applications such as direct spread code division multiple access communications space time codes and compressed sensing and codebooks with small inner product correlation are required it is extremely difficult to construct codebooks achieving the levenshtein bound in this paper two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented these constructions produce complex codebooks asymptotically achieving the levenshtein bound and codebooks asymptotically achieving the welch bound the codebooks presented in this paper have new parameters

frames graphs and erasures two uniform frames and their use for the coding of vectors are the main subject of this paper these frames are known to be optimal for handling up to two erasures in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero here we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost we derive general error bounds for two uniform frames when more than two erasures occur and apply these to concrete examples we show that among the 227 known equivalence classes of two uniform 36 15 frames arising from hadamard matrices there are 5 that give smallest error bounds for up to 8 erasures

traveling wave front for a two component lattice dynamical system arising in competition models abstract we study traveling front solutions for a two component system on a one dimensional lattice this system arises in the study of the competition between two species with diffusion or migration if we divide the habitat into discrete regions or niches we consider the case when the nonlinear source terms are of lotka volterra type and of monostable case we first show that there is a positive constant the minimal wave speed such that a traveling front exists if and only if its speed is above this minimal wave speed then we show that any wave profile is strictly monotone moreover under some conditions we show that the wave profile is unique up to translations for a given wave speed finally we characterize the minimal wave speed by the parameters in the system

achieving the welch bound with difference sets consider a codebook containing n unit norm complex vectors in a k dimensional space in a number of applications the codebook that minimizes the maximal cross correlation amplitude i sub max is often desirable relying on tools from combinatorial number theory we construct analytically optimal codebooks meeting in certain cases the welch lower bound when analytical constructions are not available we develop an efficient numerical search method based on a generalized lloyd algorithm which leads to considerable improvement on the achieved i sub max over existing alternatives we also derive a composite lower bound on the minimum achievable i sub max that is effective for any codebook size n

two classes of codebooks nearly meeting the welch bound in this paper the value of imax c for codebooks c c d constructed by certain almost difference sets d in fqx is determined and expressed in terms of jacobi sums from which it shows that such codebooks nearly meet the welch bound this result is an answer of a question raised by c ding and t feng in 2 we also present another series of codebooks which nearly meet the welch bound where the codebook is constructed by a subset r fq1 fq2 when q1 q2 de is an almost difference set of r

new families of codebooks achieving the levenstein bound in this paper a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced it includes some earlier constructions of codebooks meeting the levenstein bound as special cases with this construction two new families of codebooks achieving the levenstein bound are obtained the codebooks constructed in this paper could have a very small alphabet size

a family of alltop functions that are ea inequivalent to the cubic function sequences with optimal correlation properties are much sought after for applications in communication systems in 1980 alltop ieee trans inf theory 26 3 350 354 1980 described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation subsequently these sequences were used to construct mutually unbiased bases mubs a structure of importance in quantum information theory the key feature of this cubic function is that its difference function is a planar function functions with planar difference functions have been called alltop functions this paper provides a new family of alltop functions and establishes the use of alltop functions for construction of sequence sets and mubs

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

nonlinear h almost disturbance decoupling the l2 gain almost disturbance decoupling problem for siso nonlinear systems is formulated sufficient conditions are identified for the existence of a parametrized state feedback controller such that the l2 gain from disturbances to output can be made arbitrarily small by increasing its gain the controller is explicity constructed using a lyapunov based recursive scheme sufficient conditions for the solvability of the h almost disturbance decoupling problem and the explicit construction of teh controller are given for a more restrictive class of nonlinear systems

stability of non monotone critical traveling waves for reaction diffusion equations with time delay abstract this paper is concerned with the stability of critical traveling waves for a kind of non monotone time delayed reaction diffusion equations including nicholson s blowflies equation which models the population dynamics of a single species with maturation delay such delayed reaction diffusion equations possess monotone or oscillatory traveling waves the latter occurs when the birth rate function is non monotone and the time delay is big it has been shown that such traveling waves x c t exist for all c c and are exponentially stable for all wave speed c c 13 where c is called the critical wave speed in this paper we prove that the critical traveling waves x c t monotone or oscillatory are also time asymptotically stable when the initial perturbations are small in a certain weighted sobolev norm the adopted method is the technical weighted energy method with some new flavors to handle the critical oscillatory waves finally numerical simulations for various cases are carried out to support our theoretical results

constructions of quasi complementary sequence sets associated with characters an boldsymbol m boldsymbol k boldsymbol n delta max quasi complementary sequence set qcss is referred to as a set of boldsymbol m 2 d matrices of order boldsymbol k times boldsymbol n with periodic tolerance delta max in a multicarrier code division multiple access mc cdma communication system the set size boldsymbol m of a qcss is equal to the maximum number of users it can support and the periodic tolerance delta max determines the interference performance for the application of a qcss it is desirable that the set size should be as large as possible and the periodic tolerance should be as small as possible in this paper a framework of a periodic qcss is proposed from additive and multiplicative characters associated with a specific integer set it is then discovered that the parameters of the obtained qcss are determined by the employed integer set from this discovery new classes of periodic qcsss with large set sizes and low periodic tolerances are constructed and associated with some known integer sets

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

nonbinary kasami sequences over gf p the correlation values and the distribution of these correlation values for the small set of nonbinary kasami sequences over gf p p prime are presented the correlation results are an extension of the binary results and have p 2 correlation levels this nonbinary kasami set is asymptotically optimum with respect to its correlation properties these sequences are obtained as in the binary case from a large primitive polynomial of degree n 2 m and a small primitive polynomial of degree m that yields a sequence length of p sup n 1 and maximum nontrivial correlation value of 1 p sup m nonbinary kasami sequences are directly implemented using shift registers and are applicable for code division multiple access systems

a construction of codebooks associated with binary sequences an n k codebook is a set of n unit norm code vectors in a k dimensional vector space for its applications it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible meeting the welch bound equality strictly or asymptotically in this paper an n k codebook is constructed from a k n partial matrix with k n where each code vector is equivalent to a column of the matrix to obtain the k n matrix k rows are selected from a j n matrix associated with a binary sequence of length j and hamming weight k where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence it is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of transform of the binary sequence thus constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its transform is as small as possible from the discovery new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from fourier and hadamard matrices associated with binary sequences

complex codebooks from combinatorial designs codebooks also called signal sets meeting the welch bounds are desirable in code division multiple access cdma systems in 2003 binary codebooks meeting welch s bounds were constructed using difference sets by ding golin and kloslashve recently a generic construction of codebooks with cyclic difference sets meeting welch s bound on the maximum cross correlation amplitude was developed by xia in this correspondence the idea of xia is extended and related constructions of optimal codebooks with both cyclic and noncyclic difference sets are presented these codebooks are optimal in the sense that they also meet this welch bound in addition complex codebooks that almost meet welch s bound on the maximum cross correlation amplitude are also constructed with almost difference sets

the stone weierstrass theorem and its application to neural networks the stone weierstrass theorem and its terminology are reviewed and neural network architectures based on this theorem are presented specifically exponential functions polynomials partial fractions and boolean functions are used to create networks capable of approximating arbitrary bounded measurable functions a modified logistic network satisfying the theorem is proposed as an alternative to commonly used networks based on logistic squashing functions

equiangular tight frames from paley tournaments abstract we prove the existence of equiangular tight frames having n 2 d 1 elements drawn from either c d or c d 1 whenever n is either 2 k 1 for k n or a power of a prime such that n 3 mod 4 we also find a simple explicit expression for the prime power case by establishing a connection to a 2 d element equiangular tight frame based on quadratic residues

generators and irreducible polynomials over finite fields weil s character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field an application to the distribution of irreducible polynomials is given which confirms an asymptotic version of a conjecture of hansen mullen

optimal codebooks from binary codes meeting the levenshtein bound in this paper a generic construction of codebooks based on binary codes is introduced with this generic construction a few previous constructions of optimal codebooks are extended and a new class of codebooks almost meeting the levenshtein bound is presented exponentially many codebooks meeting or almost meeting the levenshtein bound from binary codes are obtained in this paper the codebooks constructed in this paper have alphabet size 4 as a byproduct three bounds on the parameters of binary codes are derived

coherence optimization and best complex antipodal spherical codes vector sets with optimal coherence according to the welch bound cannot exist for all pairs of dimension and cardinality if such an optimal vector set exists it is an equiangular tight frame and represents the solution to a grassmannian line packing problem best complex antipodal spherical codes bcascs are the best vector sets with respect to the coherence by extending methods used to find best spherical codes in the real valued euclidean space the proposed approach aims to find bcascs and thereby a complex valued vector set with minimal coherence there are many applications demanding vector sets with low coherence examples are not limited to several techniques in wireless communication or to the field of compressed sensing within this contribution existing analytical and numerical approaches for coherence optimization of complex valued vector spaces are summarized and compared to the proposed approach the numerically obtained coherence values improve previously reported results the drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases

stability of traveling wavefronts for a discrete diffusive lotka volterra competition system abstract in this paper we study a discrete diffusive lotka volterra competition system it is known that this system has traveling wavefronts we prove that the traveling wavefronts are exponentially stable when the initial perturbation around the traveling wavefronts decays exponentially as x but can be arbitrarily large in other locations the approach we use here is the comparison principle and the weighted energy method

feedback linearization using neural networks abstract for a class of single input single output continuous time nonlinear systems a multilayer neural network based controller that feedback linearizes the system is presented control action is used to achieve tracking performance for a state feedback linearizable but unknown nonlinear system the control structure consists of a feedback linearization portion provided by two neural networks plus a robustifying portion that keeps the control magniture bounded a stability proof is given in the sense of lyapunov it is shown that all the signals in the closed loop system are uniformly ultimately bounded no off line learning phase is needed initialization of the network weights is straightforward

a new approach to constructing quadratic pseudo planar functions over mathbb f _ 2 n planar functions over finite fields give rise to finite projective planes they were also used in the constructions of des like iterated ciphers error correcting codes and codebooks they were originally defined only in finite fields with odd characteristic but recently zhou introduced pesudo planar functions in even characteristic which yields similar applications all known pesudo planar functions are quadratic and hence they give presemifields in this paper a new approach to constructing quadratic pseudo planar functions is given then five explicit families of pseudo planar functions are constructed one of which is a binomial two of which are trinomials and the other two are quadrinomials all known pesudo planar functions are revisited some of which are generalized these functions not only lead to projective planes relative difference sets and presemifields but also give optimal codebooks meeting the levenstein bound complete sets of mutually unbiased bases and compressed sensing matrices with low coherence

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

near optimal partial hadamard codebook construction using binary sequences obtained from quadratic residue mapping in this paper a new class of n k near optimal partial hadamard codebooks is proposed the construction of the proposed codebooks from hadamard matrices is based on binary row selection sequences which are generated by quadratic residue mapping of p ary m sequences the proposed codebooks have parameters n p n and k p 1 2p n sqrt n 1 for an odd prime p and an even positive integer n we prove that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the welch bound equality for sufficiently large p and derive their inner product distribution

on meeting the peak correlation bounds in this paper we study the problem of meeting peak periodic or aperiodic correlation bounds for complex valued sets of sequences to this end the welch levenstein and exponential bounds on the peak inner product of sequence sets are considered and used to provide compound peak correlation bounds in both periodic and aperiodic cases the peak aperiodic correlation bound is further improved by using the intrinsic dimension deficiencies associated with its formulation in comparison to the compound bound the new aperiodic bound contributes an improvement of more than 35 for some specific values of the sequence length n and set cardinality m we study the tightness of the provided bounds by using both analytical and computational tools in particular novel algorithms based on alternating projections are devised to approach a given peak periodic or aperiodic correlation bound several numerical examples are presented to assess the tightness of the provided correlation bounds as well as to illustrate the effectiveness of the proposed methods for meeting these bounds

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

stable adaptive neuro control design via lyapunov function derivative estimation a new approach to the tracking problem for affine in the control nonlinear dynamical systems whose nonlinearities are assumed to be unknown is presented in this paper the philosophy of the developed technique is based on estimating the derivative of an unknown lyapunov function exploiting the approximation capabilities of the linear in the weights neural network structures a novel resetting strategy guarantees the boundedness away from zero of certain signals the uniform ultimate boundedness of the tracking error to an arbitrarily small set plus the boundedness of all other signals in the closed loop is guaranteed

a generic construction of complex codebooks meeting the welch bound codebooks also called signal sets meeting the welch bound on the maximum correlation amplitude are called mwbe codebooks and are desirable in code division multiple access systems two different but related constructions of mwbe codebooks from difference sets were developed by xia and ding recently the objectives of this correspondence are to present a generic construction of mwbe codebooks that contains the previous two constructions as special cases and describe new mwbe codebooks that cannot be produced by the earlier two constructions

existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay this paper is concerned with the existence uniqueness and globally asymptotic stability of traveling wave fronts in the quasi monotone reaction advection diffusion equations with nonlocal delay under bistable assumption we construct various pairs of super and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front up to translation which is monotonically increasing and globally asymptotically stable with phase shift the influence of advection on the propagation speed is also considered comparing with the previous results our results recovers and or improves a number of existing ones in particular these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay some new results are obtained

new constructions of codebooks nearly meeting the welch bound with equality an n k codebook c is a set of n unit norm complex vectors in bbck optimal codebooks meeting the welch bound with equality are desirable in a number of areas however it is very difficult to construct such optimal codebooks there have been a number of attempts to construct codebooks nearly meeting the welch bound with equality i e the maximal cross correlation amplitude imax c is slightly higher than the welch bound equality but asymptotically achieves it for large enough n in this paper using difference sets and the product of abelian groups we propose new constructions of codebooks nearly meeting the welch bound with equality our methods yield many codebooks with new parameters in some cases our constructions are comparable to known constructions

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

cryptographic properties of the welch gong transformation sequence generators welch gong wg transformation sequences are binary sequences of period 2 sup n 1 with two level autocorrelation these sequences were discovered by golomb gong and gaal 1998 and they verified the validity of their construction for 5 spl les n spl les 20 later no chung and yun 1998 found another way to construct the wg sequences and verified their result for 5 spl les n spl les 20 dillon 1998 first proved this result for odd n and finally dobbertin and dillon 1999 proved it for even n in this paper we investigate a two faced property of the wg transformation sequences for application in stream ciphers and pseudorandom number generators one is to present the randomness or unpredictability of the wg transformation sequences the other is to exhibit the security properties of the wg transformations regarded as boolean functions in particular we prove that the wg transformation sequences in addition to the known two level autocorrelation and three level cross correlation with m sequences have the ideal 2 tuple distribution and large linear span increasing exponentially with n moreover it can be implemented efficiently this is the first type of pseudorandom sequences with good correlation statistic properties large linear span and efficient implementation when wg transformations are regarded as boolean functions they have high nonlinearity we derive a criterion for the boolean representation of wg transformations to be r resilient and show that they are at least 1 resilient under some basis of the finite field gf 2 sup n an algorithm to find such bases is given the degree and linear span of wg transformations are presented as well