on the structure of classical and quantum flows abstract in the framework of quantum probability stochastic flows on manifolds and the interaction representation of quantum physics become unified under the notion ofmarkov cocycle we prove a structure theorem for weakly continuous markov cocycles which shows that they are solutions of quantum stochastic differential equations on the largest subalgebra contained in the domain of the generator of the markov semigroup canonically associated to the cocycle the result is applied to prove that any markov cocycle on the clifford bundle of a compact riemannian manifold whose structure maps preserve the smooth sections and satisfy some natural compatibility conditions uniquely determines a family of smooth vector fields and a connection with the property that the cocycle itself is induced by the stochastic flow along the paths of the classical diffusion on the manifold defined by these vector fields and by the ito stochastic parallel transport associated to the connection we use the language and techniques of quantum probability but even when restricted to the classical case our results seem to be new

maximal families of bent sequences in a recent paper olsen scholtz and welch osw describe the use of bent functions to construct families of sequences with asymptotically optimal correlation properties we show that the sequences produced by the osw construction possess the claimed correlation properties if and only if the underlying bent functions are pairwise orthogonal consequently the cardinality of a family of osw sequences of length 2 n 1 cannot exceed 2 n 2 we also describe methods of selecting a maximal set of orthogonal bent functions

several new infinite families of bent functions and their duals bent functions are optimal combinatorial objects since their introduction substantial efforts have been directed toward their study in the last three decades a complete classification of bent functions is elusive and looks hopeless today therefore not only their characterization but also their generation are challenging problems this paper is devoted to the construction of bent functions first we provide several new effective constructions of bent functions self dual bent functions and antiself dual bent functions second we provide seven new infinite families of bent functions by explicitly calculating their dual

generalized kasami sequences the large set in this correspondence new binary sequence families fk of period 2n 1 are constructed for even n and any k with gcd k n 2 if n 2 is odd or gcd k n 1 if n 2 is even the distribution of their correlation values is completely determined these families have maximum correlation 2n 2 1 and family size 23n 2 2n 2 for odd n 2 or 23n 2 2n 2 1 for even n 2 the proposed families include the large set of kasami sequences where the k is taken as k n 2 1

new families of m ary sequences with low correlation constructed from sidel nikov sequences in this correspondence for a positive integer m and a prime p such that m pn 1 three families of m ary sequences using the m ary sidel nikov sequences with period pn 1 are constructed two small families contain pn 1 2 m 2 or pn m 3 m ary sequences and both of their maximum magnitudes of correlation values are upper bounded by 2 radic pn 6 the largest family has its maximum magnitude of correlation values upper bounded by 3 radic pn 5 and the family size is m 1 2 2n 1 1 m 1 for p 2 or m 1 2 pn 3 2 m m 1 2 for an odd prime p

new families of almost perfect nonlinear power mappings a power mapping f x x sup d over gf p sup n is said to be differentially k uniform if k is the maximum number of solutions x spl isin gf p sup n of f x a f x b where a b spl isin gf p sup n and a spl ne 0 a 2 uniform mapping is called almost perfect nonlinear apn we construct several new infinite families of nonbinary apn power mappings

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

on quadratic bent functions in polynomial forms in this correspondence we construct some new quadratic bent functions in polynomial forms by using the theory of quadratic forms over finite fields the results improve some previous work moreover we solve a problem left by yu and gong in 2006

a construction of a new family of m ary sequences with low correlation from sidel nikov sequences in this paper a new family of m ary sequences of period pn 1 is proposed the proposed family is constructed by the addition of cyclic shifts of an m ary sidel nikov sequence and its reverse sequence the number of sequences contained in this family is about m 1 2 times of their period and the maximum magnitude of their correlation values is upper bounded by 4 pn 5

systematic methods of constructing bent functions and 2 rotation symmetric bent functions in this paper we first present two systematic constructions of bent functions by modifying the truth tables of rothaus s bent function and maiorana mcfarland s bent function respectively the number of the newly constructed bent functions by modifying the truth table of rothaus s bent function is also determined the methods of constructing self dual bent functions are then given after the dual functions of these bent functions being determined finally as an application two constructions of 2 rotation symmetric bent functions are presented in this paper

on correlations of a family of generalized geometric sequences in this correspondence we study families of generalized geometric sequences formed bp applying a feedforward function to certain sums of decimated m sequences with elements in a finite field we compute their correlation functions which for certain families turn out to be close to the square root of the period the size of these families equals their period we also show that in the binary case the linear complexities of these sequences are much larger than those of cascaded geometric sequences although in these cases the maximum correlations are larger

on the cross correlation of a p ary m sequence of period p 2m 1 and its decimated sequences by p m 1 2 2 p 1 in this paper for an odd prime we investigate into the cross correlation of a p ary m sequence m t of period p n 1 and its d decimated sequences m dt l 0 l m 1 2 where d pm 1 2 2 p l n 2m and m is an odd integer there are pm 1 2 distinct decimated sequences m dt l since gcd d pn 1 pm 1 2 it is shown that the magnitude of the cross correlation values is upper bounded by p 1 2 pn 2 1 we also construct the sequence family f from these sequences where the family size is pm and the correlation magnitude is upper bounded by pm 1 2 pn 2 1

generalized sure for exponential families applications to regularization stein s unbiased risk estimate sure was proposed by stein for the independent identically distributed i i d gaussian model in order to derive estimates that dominate least squares ls recently the sure criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean squared error mse however its use has been limited to the i i d case which precludes many important applications in this paper we begin by deriving a sure counterpart for general not necessarily i i d distributions from the exponential family this enables extending the sure design technique to a much broader class of problems based on this generalization we suggest a new method for choosing regularization parameters in penalized ls estimators we then demonstrate its superior performance over the conventional generalized cross validation and discrepancy approaches in the context of image deblurring and deconvolution the sure technique can also be used to design estimates without predefining their structure however allowing for too many free parameters impairs the estimate s performance to address this inherent tradeoff we propose a regularized sure objective and demonstrate its use in the context of wavelet denoising

quasifree stochastic cocycles and quantum random walks the theory of quasifree quantum stochastic calculus for infinite dimensional noise is developed within the framework of hudson parthasarathy quantum stochastic calculus the question of uniqueness for the covariance amplitude with respect to which a given unitary quantum stochastic cocycle is quasifree is addressed and related to the minimality of the corresponding stochastic dilation the theory is applied to the identification of a wide class of quantum random walks whose limit processes are driven by quasifree noises

multiplicative characters the weil bound and polyphase sequence families with low correlation power residue and sidelnikov sequences are polyphase sequences with low correlation and variable alphabet sizes represented by multiplicative characters in this paper sequence families constructed from the shift and addition of the polyphase sequences are revisited initially 0 1 is assumed for multiplicative characters to represent power residue and sidelnikov sequences in a simple form the weil bound on multiplicative character sums is refined for the assumption where the character sums are equivalent to the correlations of sequences represented by multiplicative characters general constructions of polyphase sequence families that produce some of known families as the special cases are then presented the refined weil bound enables the efficient proofs on the maximum correlation magnitudes of the sequence families from the constructions it is shown that m ary known sequence families with large size can be partitioned into m 1 disjoint subsequence families with smaller maximum correlation magnitudes more generalized constructions are also considered by the addition of multiple cyclic shifts of power residue and sidelnikov sequences

some new results on the cross correlation of m sequences the determination of the cross correlation between an m sequence and its decimated sequence has been a longstanding research problem considering a ternary m sequence of period 33r 1 we determine the cross correlation distribution for decimations d 3r 2 and d 32r 2 where gcd r 3 1 meanwhile for a binary m sequence of period 22lm 1 we make an initial investigation for the decimation d 22lm 1 2m 1 2s where l 2 is even and 0 s 2m 1 it is shown that the cross correlation takes at least four values furthermore we confirm the validity of two famous conjectures due to sarwate et al and helleseth in this case

constructions of quadratic bent functions in polynomial forms in this correspondence the constructions and enumerations of all bent functions represented by a polynomial form of f x spl sigma sub i 1 sup n 2 1 c sub i tr x sup 1 2 i c sub n 2 tr sub 1 sup n sup 2 x sup 1 2 n 2 c sub i spl isin sub 2 f sub 2 are presented for special cases of n using an iterative approach the construction of bent functions of n variables with degree n 2 is also provided using the constructed quadratic bent functions

three families of monomial functions with three valued walsh spectrum let bbb f p be a finite field with p elements where p is a prime let n ge 2 be an integer and d be the least positive integer satisfying p d equiv 1 pmod n let q p 2sd for some integers s in some special cases we obtain the explicit evaluation of the following exponential sums s a b sum x in bbb f q zeta p mathrm tr q p ax q 1 n bx as applications walsh spectrums of the monomial functions mathrm tr q p x q 1 n in three cases are investigated our results show that walsh spectrums of the monomial functions have at most four five or seven distinct values furthermore three families of the monomial functions with three valued walsh spectrums are presented see corollaries 12 21 31 and 32 consequently certain previously known results by li and yue and moisio are extended

generalized plateaued functions and admissible plateaued functions plateaued functions are very important crypto graphic functions due to their various desirable cryptographic characteristics we point out that plateaued functions are more general than bent functions that is functions with maximum nonlinearity some boolean plateaued functions have large nonlinearity which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers and contributes when they are the component functions of the substitution boxes in block ciphers to protection against linear cryptanalysis p ary plateaued functions have attracted recently some attention in the literature and many activities on generalized p ary functions have been carried out this paper increases our knowledge on plateaued functions in the general context of generalized p ary functions we first introduce two new versions of plateaued functions which we shall call generalized plateaued functions and admissible plateaued functions the generalized plateaued functions extend the standard notion of plateaued p ary functions to those whose outputs are in the ring mathbb z p k next we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components finally we provide for the first time two constructions of generalized plateaued functions in particular we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes

prime phase sequences with periodic correlation properties better than binary sequences for the case where p is an odd prime n or 2 is an integer and omega is a complex primitive pth root of unity a construction is presented for a family of p sup n p phase sequences symbols of the form omega sup i where each sequence has length p sup n 1 and where the maximum nontrivial correlation value c sub max does not exceed 1 square root p sup n a complete distribution of correlation values is provided as a special case of this construction a previous construction due to sidelnikov 1971 is obtained the family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs the present design has the additional advantage of not requiring an alphabet of size larger than three the new sequences are suitable for achieving code division multiple access and are easily implemented using shift registers they wee discovered through an application of deligne s bound 1974 on exponential sums of the weil type in several variables the sequences are also shown to have strong identification with certain bent functions

markovian cocycles on operator algebras adapted to a fock filtration abstract the introduction of a feller type condition allows the study of markovian cocycles adapted to a fock filtration to be extended from von neumann algebras to c algebras it is shown that every such cocycle which along with its adjoint cocycle is pointwise strongly continuous and whose associated semigroups are norm continuous weakly satisfies a quantum stochastic differential equation qsde the matrix of coefficients of this equation may thereby be considered as the generator of the cocycle the qsde is satisfied strongly in any of the following cases when the cocycle is completely positive and contractive or the driving quantum noise is finite dimensional or the c algebra is finite dimensional and the cocycle generator is bounded applying the algebra results to fock adapted markovian cocycles on a hilbert space we obtain similar characterisations in particular a contraction cocycle whose markov semigroup is norm continuous strongly satisfies a qsde a representation of cocycles in terms of a family of associated semigroups is central to the present analysis providing the connection with qsdes through a parallel work 2000 j m lindsay and s j wills probab theory related fields 116 505 543

a generalization of the family of p ary decimated sequences with low correlation let p be a prime and n a positive integer let e pn 1 and n pn 1 e in this paper we construct a family s of e2n p ary sequences each member of s has period n and the magnitudes of correlations of members of s are upper bounded by 2 pn 2 en 1

on the equivalence between stein and de bruijn identities this paper focuses on illustrating 1 the equivalence between stein s identity and de bruijn s identity and 2 two extensions of de bruijn s identity first it is shown that stein s identity is equivalent to de bruijn s identity under additive noise channels with specific conditions second for arbitrary but fixed input and noise distributions under additive noise channels the first derivative of the differential entropy is expressed by a function of the posterior mean and the second derivative of the differential entropy is expressed in terms of a function of fisher information several applications over a number of fields such as signal processing and information theory are presented to support the usefulness of the developed results in this paper

spectral families of quantum stochastic integrals we develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation conservation and annihilation processes in fock space these give a non commutative generalisation of classical stochastic integrals driven by poisson random measures a stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations as an application we construct stochastic flows on operator algebras driven by levy processes with finite levy measure

quantum stochastic positive evolutions characterization construction dilation a characterization of the unbounded stochastic generators of quantum completely positive flows is given this suggests the general form of quantum stochastic adapted evolutions with respect to the wiener diffusion poisson jumps or general quantum noise the corresponding irreversible heisenberg evolution in terms of stochastic completely positive cp maps is constructed the general form and the dilation of the stochastic completely dissipative cd equation over the algebra is discovered as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators a unitary quantum stochastic cocycle dilating the subfiltering cp flows over is reconstructed

generic construction of bent functions and bent idempotents with any possible algebraic degrees as a class of optimal combinatorial objects bent functions have important applications in cryptography sequence design and coding theory bent idempotents are a subclass of bent functions and of great interest since they can be stored in less space and allow faster computation of the walsh hadamard transform the objective of this paper is to present a generic construction of bent functions from known ones it includes the previous constructions of bent functions by mesnager and xu et al as special cases and produces new bent functions which cannot be produced by earlier ones in particular it also generates infinite families of bent idempotents over f 2 2m of any algebraic degree between 2 and m this together with a recent construction by su and tang gives a positive answer to an open problem on bent idempotents proposed by carlet in addition an infinite family of anti self dual bent functions is obtained in which the sum of any three distinct functions is again an anti self dual bent function in this family this solves an open problem recently proposed by mesnager

rank reduction and james stein estimation this correspondence addresses the problem of estimating the signal in a signal plus gaussian noise model when it is known that the signal lies in a given subspace an alternative to rank reduction is presented the new estimator has the remarkable property of having a smaller mean square error than that of the maximum likelihood also least squares estimator for all parameter values

some results on cross correlation distribution between a p ary m sequence and its decimated sequences for an odd prime p and two positive integers m k such that m gcd k m 3 is odd let d be a positive integer satisfying d p k 1 2 mod p m 1 i n this paper the cross correlation between a p ary m sequence and its d decimated sequences is investigated and the cross correlation distribution is completely determined the relationship between the decimations d considered in this paper and some known ones is also studied this paper generalizes some previous results and also gives new decimations which lead to low cross correlation

cyclic codes and sequences from generalized coulter matthews function in this paper we will study the exponential sum sigmaxisinfq x alphax pk 1 2 betax that is related to the generalized coulter matthews function x pk 1 2 with k gcd as applications we obtain the following the correlation distribution of a p ary m sequence and a decimated m sequence of degree pk 1 2 the weight distribution of the cyclic code whose dual has two zeros pi 1 and pi pk 1 2

new families of binary sequences with low correlation for a positive integer n new families s and u of binary sequences of period 2 sup n 1 with low correlations are proposed where for some positive integer e s is defined for odd n e and u for even n e the family s has four valued correlations and is a generalization of the family of gold like sequences introduced by s boztas and p v kumar see ibid vol 40 p 532 7 1994 the family u which is also a generalization of the sequence family defined by p udaya polyphase and frequency hopping sequences obtained from finite rings ph d dissertation dept elec eng indian inst technol kanpur 1992 has six valued correlations the relationship between gold like sequences and gold sequences is the same as the relationship between the family s and the family constructed from the binary sequences partially contributed by r gold see ibid vol it 14 p 154 6 1968 t kasami see coordinated sci lab univ of illinois urbana champaign tech rep r 285 ad 632574 1966 and welch using a lifting idea no j s and kumar p v ibid vol 35 p 371 9 1989 for the families s and u families of binary sequences with the same correlation distributions and large linear span are also constructed

a non commutative martingale representation theorem for non fock quantum brownian motion abstract a non commutative theory of stochastic integration is constructed in which the integrators are the components of the quantum brownian motion with non unit variance unlike the unit variance fock case there is a kunita watanabe type representation theorem for processes which are martingales with respect to the generated filtration

further results on generalized bent functions and their complete characterization this paper contributes to increase our knowledge on generalized bent functions including generalized bent boolean functions and generalized p ary bent functions with odd prime p by bringing new results on their characterization and construction in arbitrary characteristic more specifically we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions this enables us to completely characterize generalized bent functions and mathbb z p k bent functions by some affine space associated with the generalized bent functions we also present the relationship between generalized bent boolean functions with an odd number of variables and generalized bent boolean functions with an even number of variables based on the well known maiorana mcfarland class of boolean functions we present some infinite classes of generalized bent boolean functions in addition we introduce a class of generalized hyperbent functions that can be seen as generalized dillon s ps functions finally we solve an open problem related to the description of the dual function of a weakly regular generalized bent boolean function with an odd number of variables via the walsh hadamard transform of their component functions and we generalize these results to the case of odd prime

a new binary sequence family with low correlation and large size for odd n 2l 1 and an integer spl rho with 1 spl les spl rho spl les l a new family s sub o spl rho of binary sequences of period 2 sup n 1 is constructed for a given spl rho s sub o spl rho has maximum correlation 1 2 sup n 2 spl rho 1 2 family size 2 sup n spl rho and maximum linear span n n 1 2 similarly a new family of s sub e spl rho of binary sequences of period 2 sup n 1 is also presented for even n 2l and an integer spl rho with 1 spl les spl rho l where maximum correlation family size and maximum linear span are 1 2 sup n 2 spl rho 2 sup n spl rho and n n 1 2 respectively the new family s sub o spl rho or s sub e spl rho contains boztas and kumar s construction or udaya s as a subset if m sequences are excluded from both constructions as a good candidate with low correlation and large family size the family s sub o 2 is discussed in detail by analyzing its distribution of correlation values

stochastic integral representation theorem for quantum semimartingales abstract the quantum stochastic integral of ito type formulated by hudson and parthasarathy is extended to a wider class of adapted quantum stochastic processes on boson fock space an ito formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and wick exponentials of the basic martingales in quantum stochastic calculus

a new family of p ary sequences with low correlation constructed from decimated sequences in this paper for an odd prime p and an integer n geq 3 a new family of p ary sequences of period p n 1 over 2 with low correlation is constructed the new family is constructed by shifts and additions of two decimated sequences of a p ary m sequence and its family size is 2 p n 1 the complete correlation distribution of this new family is derived it is also shown that the family is optimal with respect to the parameter theta rm rms which denotes the root mean square of all nontrivial correlations compared with the known sequence families our sequence family is new and has a larger family size which is four times of its period

gradient of mutual information in linear vector gaussian channels this paper considers a general linear vector gaussian channel with arbitrary signaling and pursues two closely related goals i closed form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system and ii fundamental connections between information theory and estimation theory generalizing the fundamental relationship recently unveiled by guo shamai and verdu spl acute we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output gradients and derivatives with respect to other parameters are then found via the differentiation chain rule

ternary m sequences with three valued cross correlation function new decimations of welch and niho type we show that the cross correlation between two ternary m sequences of period 3 sup n 1 that differ by the decimation d 2 spl middot 3 sup m 1 where n 2m 1 takes on three different values we conjecture the same result for the decimation d 2 spl middot 3 sup r 1 where n is odd and r is defined by the condition 4r 1 spl equiv 0 mod n these two new cases form in a sense ternary counterparts of two previously confirmed binary cases the conjectures of welch and niho 1972

d form sequences families of sequences with low correlation values and large linear spans large families of binary sequences with low correlation values and large linear span are critical for spread spectrum communication systems the author describes a method for constructing such families from families of homogeneous functions over finite fields satisfying certain properties he then uses this general method to construct specific families of sequences with optimal correlations and exponentially better linear span than no sequences no 1988

on the correlation distribution of the coulter matthews decimation the distribution of the cross correlation between the ternary m sequence s sub t of period n 3 sup m 1 and the decimated sequences s sub dt and s sub dt 1 of period 3 sup m 1 2 where d 3 sup k 1 2 with k odd and gcd k m 1 is determined the method to find this distribution is related to the result by coulter and matthews that f x x sup d is a planar function over gf 3 sup m

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

cross correlation of p ary gmw sequences the cross correlation function of p ary p prime gmw sequences is investigated the well known m sequences are incorporated in these sequences gmw sequences also have the two level autocorrelation function but their linear span may be much larger than those of m sequences the calculation of the cross correlation function of gmw sequences is reduced to the cross correlation of the m sequences because many results on this kind of sequence are known in the literature only the cross correlation function of an m sequence with a gmw sequence has been known up to now

james stein state filtering algorithms in 1961 james and stein discovered a remarkable estimator that dominates the maximum likelihood estimate of the mean of a p variate normal distribution provided the dimension p is greater than two this paper extends the james stein estimator and highlights the benefits of applying these extensions to adaptive signal processing problems the main contribution of this paper is the derivation of the james stein state filter jssf which is a robust version of the kalman filter the jssf is designed for situations where the parameters of the state space evolution model are not known with any certainty in deriving the jssf we derive several other results we first derive a james stein estimator for estimating the regression parameter in a linear regression a recursive implementation which we call the james stein recursive least squares js rls algorithm is derived the resulting estimate although biased has a smaller mean square error than the traditional rls algorithm finally several heuristic algorithms are presented including a james stein version of the yule walker equations for ar parameter estimation

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

a wide family of nonlinear filter functions with a large linear span a broad collection of pseudorandom sequence generators is introduced these generators are based on nonlinear functions applied to maximal length lfsrs and are not constrained in the length l of the lfsrs or the nonlinear order k of the functions the linear span of the resulting sequences is proved to be as large as the linear span of the sequences obtained from a product of equidistant phases that is to say at least lk a count of the number of functions is evaluated and compared with the corresponding count of products of equidistant phases

entropy and the fourth moment phenomenon we develop a new method for bounding the relative entropy of a random vector in terms of its stein factors our approach is based on a novel representation for the score function of smoothly perturbed random variables as well as on the de bruijn s identity of information theory when applied to sequences of functionals of a general gaussian field our results can be combined with the carbery wright inequality in order to yield multidimensional entropic rates of convergence that coincide up to a logarithmic factor with those achievable in smooth distances such as the 1 wasserstein distance in particular our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components with explicit rates of convergence in total variation that are independent of the order of the associated wiener chaoses the results proved in the present paper are outside the scope of other existing techniques such as for instance the multidimensional stein s method for normal approximations

stochastic integral representations of second quantization operators we give a necessary and sufficient condition for the second quantization operator h of a bounded operator h on l2 r or for its differential second quantization operator h to have a representation as a quantum stochastic integral this condition is exactly that h writes as the sum of a hilbert schmidt operator and a multiplication operator we then explore several extensions of this result we also examine the famous counterexample due to journe and meyer and explain its representability defect

on the existence of square dot matrix patterns having a specific three valued periodic correlation function the author examines square matrices of size n containing dot patterns satisfying the following two restrictions 1 each column contain precisely one dot and 2 if the pattern is moved around over a plane tied by the same pattern when in all positions except the home position there is at most one overlap in dots from differing viewpoints there matrices are the characteristic functions of either a certain class of relative difference sets or else a select subset of bent functions also the existence of such an n n matrix implies the existence of a finite projective plane of order n a family of constructions for such matrices is available when n is prime a polynomial equation characterizing such matrices and resembling the hall polynomial equation of cyclic difference sets is presented analogs of known existence tests for cyclic difference sets are then applied to rule out existence for most nonprime values of n it is shown how such patterns can be used to provide hopping patterns for a frequency hopped multiple access system

cross correlation distribution of p ary m sequence of period p 4k 1 and its decimated sequences by left p 2k 1 over 2 right 2 for an odd prime p n 4k and d p2k 1 2 2 there are p2k 1 2 distinct decimated sequences s dt l 0lesl p2k 2 of a p ary m sequence s t of period pn 1 because gcd d pn 1 p2k 1 2 in this paper it is shown that the cross correlation function between s t and s dt l 0lesl p2k 1 2 takes the values in 1 1 radic pn 1 radic pn 1 2radic pn and their cross correlation distribution is also derived

on the crosscorrelation of sequences over gf p with short periods we investigate the crosscorrelation function cd t spl sigma sub i 1 p sup n 1 spl zeta a sup i t a sup di where spl zeta is a complex primitive pth root of unity a sub i i spl isin n sub 0 is a maximal linear shift register sequence of length p sup n 1 and p is an odd prime for p 3 n odd and d p sup n 1 4 p sup n 1 2 we show that 2 spl middot spl radic p sup n is an upper bound for the absolute value of 1 c sub d t for any odd prime p and p sup k 1 where n gcd n k is not divisible by 4 we determine the maximum absolute value of c sub d t and the number of values of c sub d t

on lower bounds to the maximum correlation of complex roots of unity sequences it is shown how the welch bound 1974 on the maximum correlation of families of complex sequences of fixed norm can be modified to provide an improved bound for the case when the sequence symbols are roots of unity as in the welch bound the improved bound is based on a useful expression for the even correlation moments an analysis of the ratio of successive even moments using this expression is shown to yield a small improvement over a similarly derived bound due to v m sidelnikov 1971 interestingly the expression for the moments reduces in the binary q 2 case to a version of the pless power moment identities the derivation also provides insight into the problem of optimal sequence design

minimal linear equivalent analysis of a variable memory binary sequence generator starting from a memory based nonlinear generator due to p r geffe 1973 a binary sequence generator mem bsg consisting of three linear feedback shift registers and a variable memory is defined and analyzed the mem bsg is shown to be convenient for generating fast binary sequences of large period and linear complexity which may be useful in spread spectrum and cryptographic applications

frequency hopping code sequence designs having large linear span in frequency hopping spread spectrum multiple access communication systems it is desirable to use sets of hopping patterns that in addition to having good hamming correlation properties and large period are also derived from sequences having large linear span here two such frequency hopping code sequence designs that are based on generalized bent functions and generalized bent sequences are presented the hamming correlation properties of the designs are optimal in the first case and close to optimal in the second in terms of the alphabet size p required to be prime in both cases the period and family size of the two designs are given by p sup 2 p and p sup n p sup n 2 1 n an even integer respectively the finite field sequences underlying the patterns in the first design have linear span exceeding p whereas still larger linear spans when compared to the sequence period can be obtained using the second design method

bat chirps with good properties zak space construction of perfect polyphase sequences previously a discretization of the linear fm chirp of length n kl 2 l and kl isin z was given and the conditions for its minimal zak space support were derived chirps satisfying these conditions are known as finite chirps in this work subsets of finite chirps of length n l 2 l a prime are examined the investigation leads to a new zak space construction of general polyphase sequence sets of size l 1 with optimal auto and cross correlation properties known as perfect sequence sets it is shown that perfect sequence sets are closely related to sets of finite chirps and in particular include the sets of zadoff chu sequences which are identical with subsets of finite chirps and the sets of generalized frank sequences which are identical with sets of modulations of finite chirps as special cases the entire collection of perfect sequence sets is then given by a partition of the set of perfect auto correlation sequences obtained by right coset decomposition of the group of all permutations with respect to a certain cyclic group the construction suggests several further generalizations that can be obtained by operating exclusively on subgroups of the permutation group

a w transform based criterion for the existence of bounded extensions of e operators abstract let h be the symmetric fock space over a hilbert space h and e h h the exponential mapping by an e operator we mean an operator defined on e h for an e operator a the composition mapping a e is called its w transform in this paper we obtain a criterion based on the w transform for checking whether or not an e operator becomes a bounded linear operator on the fock space

large families of sequences with near optimal correlations and large linear span in order to build spread spectrum communication systems based on the cdma paradigm it is necessary to have large families of binary sequences with low pairwise correlation values for these systems to have resistance to certain cryptanalytic attacks and resistance to jamming the sequences must have large linear span we describe certain families of sequences that have these desirable properties the sequences are based on families of quadratic forms over finite fields

systematic constructions of rotation symmetric bent functions 2 rotation symmetric bent functions and bent idempotent functions rotation symmetric bent functions and their generation two rotation symmetric bent functions are two classes of cryptographically significant boolean functions however few constructions have been presented in the literature which either have restriction on integer n or have algebraic degree no more than 4 in this paper for any even integer n ge 4 three classes of bent functions are presented respectively most notably the proposed n variable rotation symmetric bent functions and two rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to n 2 besides we obtain bent idempotent functions with the maximal algebraic degree n 2

niho type cross correlation functions via dickson polynomials and kloosterman sums suppose that n 2k is even we study the cross correlation function between two m sequences for niho type decimations d 2 sup k 1 s 1 we develop a new technique to study the value distribution of these cross correlation functions which makes use of dickson polynomials as a first application we derive here the distribution of the six valued cross correlation function for s 3 and odd k up to a term which depends on kloosterman sums in addition applying simpler methods we prove a theorem providing niho type decimations with four valued cross correlation functions and their distribution we conjecture that the latter result actually covers all such decimations

value distributions of exponential sums from perfect nonlinear functions and their applications in this paper we present a unified way to determine the values and their multiplicities of the exponential sums sigmaxisinf q zetap tr af x bx a bisinfq q pm pges3 for all perfect nonlinear functions f which is a dembowski ostrom polynomial or p 3 f x 3 k 1 2 where k is odd and k m 1 as applications we determine 1 the correlation distribution of the m sequence alambda tr gammalambda lambda 0 1 and the sequence blambda tr f gammalambda lambda 0 1 over fp where gamma is a primitive element of fq and 2 the weight distributions of the linear codes over fp defined by f

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

annihilation derivative creation derivative and representation of quantum martingales on the basis of the quantum white noise theory we introduce the notion of creation and annihilation derivatives of fock space operators and study the differentiability of white noise operators we define the hitsuda skorohod quantum stochastic integrals by the adjoint actions of quantum stochastic gradients and show explicit formulas for their creation and annihilation derivatives as an application we derive direct formulas for the integrands in the quantum stochastic integral representation of a regular quantum martingale

an extremal inequality motivated by multiterminal information theoretic problems we prove a new extremal inequality motivated by the vector gaussian broadcast channel and the distributed source coding with a single quadratic distortion constraint problems as a corollary this inequality yields a generalization of the classical entropy power inequality epi as another corollary this inequality sheds insight into maximizing the differential entropy of the sum of two dependent random variables

a new family of quadriphase sequences for cdma the construction of a family of quadriphase sequences that are suitable for code division multiple access spread spectrum cma ss communication is described for a given parameter n where n 2 mod 4 the period of the sequence is 2 sup n 1 chips the set of sequences asymptotically achieves welch s lower bound on simultaneous cross correlation and autocorrelation magnitudes all of them being bounded by a value 2 sup n 2 1 the cardinality of this family of sequencies is 2 sup n 2

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power

optimal codes for minimax criterion on error detection nonlinear quadratic codes that are optimal for the minimax error detection are presented characteristic functions for these codes are asymptotically bent for a given block size n and the number of codewords mod c mod these codes minimize max q e e not 0 where q e is the conditional error masking probability given the error pattern e the codewords are blocks of n symbols from gf q encoding and decoding procedures for the codes are described

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

it formulae for gamma target mutual information and relative entropy in this paper we introduce new stein identities for gamma target distribution as well as a new non linear channel specifically designed for gamma inputs from these two ingredients we derive an explicit and simple formula for the derivative of the input output mutual information of this non linear channel with respect to the channel quality parameter this relation is reminiscent of the well known link between the derivative of the input output mutual information of additive gaussian noise channel with respect to the signal to noise ratio and the minimum mean square error the proof relies on a rescaled version of de bruijn identity for gamma target distribution together with a stochastic representation for the gamma specific fisher information finally we are able to derive precise bounds and asymptotics for the input output mutual information of the non linear channel with gamma inputs

natural exponential families with quadratic variance functions statistical theory the normal poisson gamma binomial negative binomial and nefghs distributions are the six univariate natural exponential families nef with quadratic variance functions qvf this sequel to morris 1982 treats certain statistical topics that can be handled within this unified nef qvf formulation including unbiased estimation bhattacharyya and cramer rao lower bounds conditional distributions and moments quadratic regression conjugate prior distributions moments of conjugate priors and posterior distributions empirical bayes and g 2 minimax marginal distributions and their moments parametric empirical bayes and characterizations

on the covering structures of two classes of linear codes from perfect nonlinear functions in this paper the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions from fqm to itself are determined using a unified approach all the minimal codewords of these codes are characterized according to their weights which suggests that their covering structures are determined finally all the minimal access sets of the secret sharing schemes based on their dual codes are obtained

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

representation of mutual information via input estimates a relationship between information theory and estimation theory was recently shown for the gaussian channel relating the derivative of mutual information with the minimum mean square error this paper generalizes the link between information theory and estimation theory to arbitrary channels giving representations of the derivative of mutual information as a function of the conditional marginal input distributions given the outputs we illustrate the use of this representation in the efficient numerical computation of the mutual information achieved by inputs such as specific codes or natural language

signal set design with constrained amplitude spectrum and specified time bandwidth product a new method for designing signals with a given time bandwidth product such that all the signals in the set have a flat amplitude spectrum and have low cross correlation function values is introduced it is shown that these signals lie on a signal parameter space ellipse for a given duration and bandwidth it is possible to trade off the cardinality of the set for better cross correlation properties between remaining members of the set proof that the cross correlation between these signals is bounded from above by a factor proportional to the square root of inverse distance between these signals along one of the axes of the ellipse is given it is also shown that the cross correlation between two signals that are furthest apart is bounded by the square root of inverse time bandwidth product of the signals in the set an example design is then given and a comparison of this signal set to some well known signal sets illustrates that it performs better when the metric is maximum cross correlation value the proposed signal set also has the advantage that it can be designed for any time bandwidth product instead of discrete values

bent function sequences in this paper we construct a new family of nonlinear binary signal sets which achieve welch s lower bound on simultaneous cross correlation and autocorrelation magnitudes given a parameter n with n 0 pmod 4 the period of the sequences is 2 n 1 the number of sequences in the set is 2 n 2 and the cross auto correlation function has three values with magnitudes leq 2 n 2 1 the equivalent linear span of the codes is bound above by sum i 1 n 4 left stackrel n i right these new signal sets have the same size and correlation properties as the small set of kasami codes but they have important advantages for use in spread spectrum multiple access communications systems first the sequences are balances which represents only a slight advantage second the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly hopped from sequence to sequence for code division multiple access operation most importantly the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it this high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing

hessian and concavity of mutual information differential entropy and entropy power in linear vector gaussian channels within the framework of linear vector gaussian channels with arbitrary signaling the jacobian of the minimum mean square error and fisher information matrices with respect to arbitrary parameters of the system are calculated in this paper capitalizing on prior research where the minimum mean square error and fisher information matrices were linked to information theoretic quantities through differentiation the hessian of the mutual information and the entropy are derived these expressions are then used to assess the concavity properties of mutual information and entropy under different channel conditions and also to derive a multivariate version of an entropy power inequality due to costa

non commutative chaotic expansion of hilbert schmidt operators on fock space it is known from a simple algebraic computation that every hilbert schmidt operator on the fock space admits a maassen meyer kernel maassen meyer kernels are a non commutative extension of the usual notion of chaotic expansion of random variables using an extension of the non commutative stochastic integrals which allows to define these integrals on the whole fock space we prove that a hilbert schmidt operator on fock space is the sum of a series of iterated non commutative stochastic integrals with respect to the basic theree quantum noises in this way we recover its maassen meyer kernel which can be completely described from the operator itself

constructions of bent negabent functions and their relation to the completed maiorana mcfarland class the problem of constructing bent negabent functions that do not belong to the completed maiorana mcfarland class emerges implicitly through a series of construction methods proposed recently these approaches manage to optimize the algebraic degree of bent negabent functions but all of the constructed bent negabent functions belong to the completed maiorana mcfarland class in this paper we use the indirect sum construction proposed by carlet in 2004 for constructing the bent negabent functions that are not provably contained in this class which is the first significant attempt in this direction to achieve this we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members then embedding these functions in the framework of the indirect sum construction we are able to specify sufficient conditions for bent negabent functions not being contained in the completed maiorana mcfarland class

stochastic integral representation of bounded quantum martingales in fock space it is a classical theorem of kunita and watanabe 6 that every square integrable martingale adapted to the standard brownian motion can be uniquely expressed as the stochastic integral of a nonanticipating square integrable brownian functional with respect to the same brownian motion the central aim of this paper is to establish a similar integral represen tation of a quantum martingale with respect to the annihilation creation and gauge processes in the context of quantum stochastic calculus in fock space as developed in 11 in the case of non fock quantum brownian motion such an integral representation was achieved by hudson and lindsay 2 33 with much less difficulty owing to the non existence of the so called gauge processes as special cases of our main result we obtain the differentials of hilbert schmidt and unitary martingales 4 as an application the uniqueness of fermion martingales in boson fock space is established the first author wishes to thank hudson and lindsay for several useful conversations on this subject 2

decimations of the frank heimiller sequences the frank heimiller sequence of period n2with n phases is constructed by writing the n by n discrete fourier transform matrix by rows the periodic autocorrelation sidelobes for such a sequence are all zero small families of n phase sequences with optimal periodic cross correlations can be formed by judicious selection of decimations of the frank heimiller sequence

on the weight distributions of two classes of cyclic codes let q pm where p is an odd prime mges2 and 1lesklesm 1 let tr be the trace mapping from fq to fp and zetap e2pii p be a primitive pth root of unity in this paper we determine the value distribution of the following exponential sums sigmaxisinf qchi alphaxp k 1 betax2 alpha betaisinfq where chi x zetap tr x is the canonical additive character of fq as applications we have the following 1 we determine the weight distribution of the cyclic codes c1 and c2 over fpt with parity check polynomial h2 x h3 x and h1 x h2 x h3 x respectively where t is a divisor of d gcd m k and h1 x h2 x and h3 x are the minimal polynomials of pi 1 pi 2 and pi p k 1 over fpt respectively for a primitive element pi of fq 2 we determine the correlation distribution between two m sequences of period q 1 moreover we find a new class of p ary bent functions this paper extends the results in feng and luo 2008