new block quadrature rules for the approximation of matrix functions abstract golub and meurant have shown how to use the symmetric block lanczos algorithm to compute block gauss quadrature rules for the approximation of certain matrix functions we describe new block quadrature rules that can be computed by the symmetric or nonsymmetric block lanczos algorithms and yield higher accuracy than standard block gauss rules after the same number of steps of the symmetric or nonsymmetric block lanczos algorithms the new rules are block generalizations of the generalized averaged gauss rules introduced by spalevic applications to network analysis are presented

linear codes from some 2 designs a classical method of constructing a linear code over mathrm gf q with a t design is to use the incidence matrix of the t design as a generator matrix over mathrm gf q of the code this approach has been extensively investigated in the literature in this paper a different method of constructing linear codes using specific classes of 2 designs is studied and linear codes with a few weights are obtained from almost difference sets difference sets and a type of 2 designs associated to semibent functions two families of the codes obtained in this paper are optimal the linear codes presented in this paper have applications in secret sharing and authentication schemes in addition to their applications in consumer electronics communication and data storage systems a coding theory approach to the characterization of highly nonlinear boolean functions is presented

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

a note on the maximization of matrix valued hankel determinants with applications in this note we consider the problem of maximizing the determinant of moment matrices of matrix measures the maximizing matrix measure can be characterized explicitly by having equal matrix valued weights at the zeros of classical one dimensional orthogonal polynomials the results generalize classical work of schoenberg indag math 62 1959 282 to the case of matrix measures as a statistical application we consider several optimal design problems in linear models which generalize the classical weighing design problems

matrix orthogonal polynomials on the unit circle the properties of matrix orthogonal polynomials on the unit circle are investigated beginning with their recurrence formulas the techniques of scattering theory and banach algebras are used in the investigation a matrix generalization of a theorem of baxter is proved

some applications of the hermite matrix polynomials series expansions this paper deals with hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems properties of hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well known methods results are applied to compute accurate approximations of certain differential systems in terms of hermite matrix polynomials

the weight distribution of a class of linear codes from perfect nonlinear functions in this correspondence the weight distribution of a class of linear codes based on perfect nonlinear functions also called planar functions is determined the class of linear codes under study are either optimal or among the best codes known and have nice applications in cryptography

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

cmv block matrices for symmetric matrix measures on the unit circle abstract in this contribution we study the relationship between the cmv block matrices associated with two symmetric positive definite matrix measures supported on the unit circle we also consider certain transformations of an orthogonality matrix measure supported on the unit circle and deduce connection formulas for the corresponding orthogonal families and their verblunsky matrix coefficients

on the hypergeometric matrix function abstract this paper deals with the study of the hypergeometric function with matrix arguments f a b c z conditions for matrices a b c so that the series representation of the hypergeometric function be convergent for z 1 and satisfies a matrix differential equation are given after the study of beta and gamma matrix functions an integral representation of f a b c z is obtained for the case where b c and c b are positive stable matrices with bc cb

gaussian quadrature formulae for matrix weights we study gaussian quadrature formulae for a matrix weight we firstly show how to generate gaussian quadrature formulae such that the sum of the rank of the quadrature matrix coefficients is a given nonnegative integer l the nodes are the eigenvalues of certain perturbation of the truncated n jacobi matrix of size l or equivalently the zeros of certain matrix combination of two consecutive orthonormal polynomials and the quadrature coefficients are the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind we secondly prove that any gaussian quadrature formula for a matrix weight must be of this form

five families of three weight ternary cyclic codes and their duals as a subclass of linear codes cyclic codes have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper five families of three weight ternary cyclic codes whose duals have two zeros are presented the weight distributions of the five families of cyclic codes are settled the duals of two families of the cyclic codes are optimal

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

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

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

biorthogonal matrix polynomials related to jacobi matrix polynomials in this paper we define the pair of biorthogonal matrix polynomials suggested by the jacobi matrix polynomials biorthogonality property matrix generating functions and matrix recurrence relations are given

efficient orthogonal matrix polynomial based method for computing matrix exponential the matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields this paper presents an efficient method for computing matrix exponentials based on hermite matrix polynomial expansions hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods a backward error analysis of the approximation in exact arithmetic is given this analysis is used to provide a theoretical estimate for the optimal scaling of matrices two algorithms based on this method have been implemented as matlab functions they have been compared with matlab functions funm and expm obtaining greater accuracy in the majority of tests a careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs reaching in numerical tests relative higher average costs than expm of only 4 43 for the final hermite selected order and obtaining better accuracy results in the 77 36 of the test matrices the matlab implementation of the best hermite matrix polynomial based algorithm has been made available online

jacobi matrix differential equation polynomial solutions and their properties in this paper jacobi matrix polynomials are introduced starting from the hypergeometric matrix function the differential equation satisfied by them is presented a rodrigues formula orthogonality and a three terms matrix recurrence relationship are then developed for jacobi matrix polynomials

vibration confinement in a general beam structure during harmonic excitations abstract in the present paper the green s function is utilized to provide a simple exact and direct analytical method for analysis of a beam structure of generic boundary conditions with attached springs and or finite masses the aim is to confine the vibration in a certain part of the structure the beam equation is based on the timoshenko beam theory with corrections for shear deformation and rotary inertia effects in the analysis the beam is driven by a harmonic external excitation the attached springs are modeled as simple reactions that provide transverse forces to the beam while each added mass provides a transverse force in addition to a moment at its location these forces moments act as secondary forces moments that reduce the response caused by the external force numerical simulations are conducted to find the optimal masses and or springs that confine the vibration in a certain chosen region the results were compared to that obtained from euler bernoulli beam theory beams that are excited by a bi harmonic force as well as dual excitation forces are analyzed in addition the case when the beams are excited near resonances is discussed also a method is proposed to impose a node at any desired location along the structure

the second spectrum of timoshenko beam theory further assessment a review of contributions and views on the second spectrum of timoshenko beam theory tbt over the past two decades together with some new results are presented it is shown that the timoshenko frequency equation factorises not solely for hinged hinged end conditions as is often claimed but also for guided guided and guided hinged these new cases may be regarded as portions of a multi span hinged hinged beam a higher derivative lagrangian that leads directly to the well known fourth order timoshenko beam equation is reviewed a simple relationship between the so called ostrogradski energy and the mechanical energy is derived for hinged hinged end conditions it is shown that the ostrogradski energy is positive for the first spectrum but negative for the second within some branches of physics this would be sufficient to conclude that the second spectrum is unphysical a numerical example presented by levinson and cooke is re examined using both tbt and exact plane stress elastodynamic theory agreement is excellent for the first spectrum however the second spectrum predictions are not in consistent agreement with any single mode of vibration for long wavelength it is very close to the second asymmetric mode but as wavelength shortens it becomes closer to the second symmetric then the third asymmetric modes the conclusion remains unchanged the second spectrum predictions of tbt should be disregarded

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

orthogonal matrix polynomials and applications orthogonal matrix polynomials on the real line or on the unit circle have properties which are natural generalizations of properties of scalar orthogonal polynomials appropriately modified for the matrix calculus we show that orthogonal matrix polynomials both on the real line and on the unit circle appear at various places and we describe some of them the spectral theory of doubly infinite jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line scalar orthogonal polynomials with a sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line orthogonal matrix polynomials on the unit circle are related to unitary block hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrix valued functions parallel algorithms for this purpose have been implemented using pvm and some examples are worked out

the second frequency spectrum of timoshenko beams abstract a second spectrum of frequencies was reported in early analytical work on the vibrations of timoshenko beams however in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded in fact from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies in this paper a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately guidelines are provided which help to identify and classify the frequencies into two separate spectra

gaussian quadrature for matrix valued functions on the real line abstract parallel versions of the block chebyshev algorithm to generate the recursion coefficients of orthonormal matrix polynomials and the gaussian quadrature algorithm to approximate matrix integrals on the real line are implemented on an sp1

perturbations in the nevai matrix class of orthogonal matrix polynomials in this paper we study a jacobi block matrix and the behavior of the limit of its entries when a perturbation of its spectral matrix measure by the addition of a dirac delta matrix measure is introduced 2001 elsevier science inc all rights reserved

proofs of two conjectures on ternary weakly regular bent functions in this paper we study ternary monomial functions of the form f x trn axd where x isin bbf 3 n and trn bbf 3 nrarr bbf 3 is the absolute trace function using a lemma of hou stickelberger s theorem on gauss sums and certain ternary weight inequalities we show that certain ternary monomial functions arising in the 2006 ieee transactions on information theory paper vol 52 pp 2018 2032 2006 are weakly regular bent thus settling a conjecture of helleseth and kholosha we also prove that the coulter matthews bent functions are weakly regular

exponential sums from half quadratic forms and their applications in this paper we consider a class of exponential sums from some half quadratic binomials the exponential sums are proven to be eleven valued with maximal magnitude 1 2 q sqrt q except for the trivial value q as applications first we investigate the autocorrelation and cross correlation distribution among the sequences in a sequence family second we determine the weight distributions of several classes of linear codes some of the dual codes have minimum distance four which are optimal with respect to the hamming bound our results extend the result by choi et al and show that some correlation values in it do not occur

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

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

a method to solve the nonlinear eigenvalue problem of timoshenko plane frames with rigid offsets and end releases abstract the study reported here makes use of the dynamic stiffness matrix dsm for timoshenko beams hence considering rotatory inertia and shear deflection to obtain the exact solution of the associated free vibration problem the natural frequencies and vibration modes in the presence of end releases and rigid offsets are all exactly obtained for different combinations of boundary conditions and in plane arrangements of members this is possible by solving the resulted nonlinear and transcendental eigenvalue problem using two general methods one based on the wittrick williams algorithm adapted to handle end release and rigid offset and the other on the power deflation secant method the latter offered here as a novelty is proven to be an efficient technique judging from the examples given using both algorithms and the finite element method it is concluded that the power deflation method is a powerful tool to solve the nonlinear eigenvalue problem leading to exact solutions of the free vibration of complex skeleton like structures

a series expansion approach to interpreting the spectra of the timoshenko beam abstract the timoshenko beam model results in two fourth order partial differential equations in time and space consequently solving the boundary value problem yields two independent sequences of natural frequencies and two corresponding sequences of mode shapes a particular natural frequency and its corresponding mode shape describe one particular solution to the boundary value problem of the timoshenko beam from an eigenfunction expansion sense all these possible solutions have to be considered in the complete series expansion of the solution however the question of whether these two independent sequences of natural frequencies implies the existence of two distinct spectra of frequencies has been a long standing topic of debate and hitherto has not been resolved completely the object of this paper is to provide answers to some of the issues raised by this debate in this context the complete solution in a series form to the timoshenko beam is investigated and it is shown for the first time that a particular mode shape of the solution is naturally expressed by an ordered pair of characteristic values rather than a single characteristic value this representation facilitates the progressive ordering of all the natural frequencies of the system and their respective mode shapes in a single set and eliminates the remaining argument for the two spectra interpretation

seven classes of three weight cyclic codes cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms compared with linear block codes in this paper seven classes of three weight cyclic codes over gf p whose duals have two zeros are presented where p is an odd prime the weight distributions of the seven classes of cyclic codes are settled some of the cyclic codes are optimal in the sense that they meet certain bounds on linear codes the application of these cyclic codes in secret sharing is also considered

on matrix integration of matrix polynomials this paper is the result of having read a series of recent papers on the quadrature formula for matrix integrals which caused a strong want of clarifying the circumstances for this purpose we have had to revise orthogonality for matrix polynomials being supported by a desire of using means adequate to needs and at the same time of trying to simplify the set up

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

a note on the matrix valued q d algorithm and matrix orthogonal polynomials on 0 1 and 0 in this note a matrix version of the q d algorithm is introduced it is shown that the algorithm may be used to obtain the coefficients of the recurrence relations for matrix orthogonal polynomials on the interval 0 and 0 1 from its moment generating functional the algorithm is illustrated by several examples which generalize classical orthogonal polynomials on the real line

linear codes with two or three weights from weakly regular bent functions linear codes with a few weights have applications in consumer electronics communication data storage system secret sharing authentication codes association schemes and strongly regular graphs this paper first generalizes the method of constructing two weight and three weight linear codes of ding et al and zhou et al to general weakly regular bent functions and determines the weight distributions of these linear codes it solves an open problem proposed by ding et al furthermore this paper constructs new linear codes with two or three weights and presents their weight distributions they contain some optimal codes meeting certain bound on linear codes

orthogonal matrix polynomials and quadrature formulas we prove that the nodes of a quadrature formula for a matrix weight with the highest degree of precision must necessarily be the zeros of a certain orthonormal matrix polynomial with respect to the matrix weight and the quadrature coefficients are then the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind we also extend this result for quadrature formulas with degree of precision one unit smaller than the highest possible

cyclic codes and sequences the generalized kasami case in this paper the large family of generalized kasami sequences has been studied in particular the cross correlation distribution among these sequences has been explicitly calculated meanwhile the weight distributions of two classes of cyclic codes could also be determined this paper generalizes the results from several previous papers

explicit solutions for second order operator differential equations with two boundary value conditions ii abstract boundary value problems for second order operator differential equations with two boundary value conditions are studied explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques analogous expressions to the ones known for the scalar case are obtained

on a multivariable extension of jacobi matrix polynomials the classical jacobi matrix polynomials only for commutative matrices were first studied by defez et al e defez l jodar a law jacobi matrix differential equation polynomial solutions and their properties comput math appl 48 2004 789 803 the main aim of this paper is to construct a multivariable extension with the help of the classical jacobi matrix polynomials jmps generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived furthermore general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented

algebraic constructions of constant composition codes constant composition codes are a special class of constant weight codes they include permutation codes as a subclass in this correspondence two classes of optimal constant composition codes are constructed both constructions employ perfect nonlinear functions but they are different

linear codes from perfect nonlinear mappings and their secret sharing schemes in this paper error correcting codes from perfect nonlinear mappings are constructed and then employed to construct secret sharing schemes the error correcting codes obtained in this paper are very good in general and many of them are optimal or almost optimal the secret sharing schemes obtained in this paper have two types of access structures the first type is democratic in the sense that every participant is involved in the same number of minimal access sets in the second type of access structures there are a few dictators who are in every minimal access set while each of the remaining participants is in the same number of minimal access sets

monomial and quadratic bent functions over the finite fields of odd characteristic considered are p ary bent functions having the form f x tr sub n spl sigma sub i 0 sup s a sub i x sup di a new class of ternary monomial regular bent function with the dillon exponent is discovered the existence of dillon bent functions in the general case is an open problem of deciding whether a certain kloosterman sum can take on the value 1 also described is the general gold like form of a bent function that covers all the previously known monomial quadratic cases the weak regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given finally some criteria for an arbitrary quadratic function to be bent are proven

systematic authentication codes from highly nonlinear functions recently highly nonlinear functions have been successfully employed to construct authentication codes with and without secrecy in this paper we construct four classes of systematic authentication codes from perfect nonlinear functions and almost perfect nonlinear functions the systematic authentication codes presented in this paper are either better than existing codes or as good as the best codes known

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams

quadrature formulas for matrix measures a geometric approach a geometric approach to quadrature formulas for matrix measures is presented using the relations between the representations of the boundary points of the moment space generated by all matrix measures and quadrature formulas simple proofs of existence and uniqueness of quadrature formulas of maximal degree of precision are given several new quadrature formulas for matrix measures supported on a compact interval are presented and several examples are discussed additionally a special construction of degenerated quadrature formulas is discussed and some results regarding the location of the zeros of polynomials orthogonal with respect to a matrix measure on a compact interval are obtained as a by product

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