stable cohomology of spaces of non singular hypersurfaces abstract we prove that the rational cohomology of the space x d n of non singular complex homogeneous polynomials of degree d in n variables stabilizes to the cohomology of gl n c for d sufficiently large

binary kloosterman sums modulo 256 and coefficients of the characteristic polynomial kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory we use stickelberger s theorem and the gross koblitz formula to determine the value of the binary kloosterman sum at a modulo 64 modulo 128 and modulo 256 in terms of coefficients of the characteristic polynomial of a

homological stability for topological chiral homology of completions abstract by proving that several new complexes of embedded disks are highly connected we obtain several new homological stability results our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed e n algebras using this we prove a special case of a conjecture of vakil and wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally

hyperbent functions kloosterman sums and dickson polynomials this paper is devoted to the study of hyperbent functions in n variables i e bent functions which are bent up to a change of primitive roots in the finite field gf 2n our main purpose is to obtain an explicit trace representation for some classes of hyperbent functions we first exhibit an infinite class of monomial functions which is not hyperbent this result indicates that kloosterman sums on f2 m cannot be zero at some points for functions with multiple trace terms we express their spectra by means of dickson polynomials we then introduce a new tool to describe these hyperbent functions the effectiveness of this new method can be seen from the characterization of a new class of binomial hyperbent functions

on the correlation distribution for a niho decimation let p be a prime n 2m and d 3p m 2 with m geq 2 and gcd d p n 1 1 in this paper the correlation distribution between a p ary m sequence of period p n 1 and its d decimation sequence is investigated in a unified approach some results for the binary case are extended to the general case it is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field mathbb f p n for an arbitrary odd prime p it seems difficult to solve these two problems however for p 3 by studying the weight distribution of the ternary zetterberg code and counting the numbers of solutions of some equations over mathbb f 3 n the two problems are solved and thus the corresponding correlation distribution for d is completely determined it is noteworthy that this is the first time that the correlation distribution for a non binary niho decimation has been determined since 1976

how many weights can a cyclic code have upper and lower bounds on the largest number of weights in a cyclic code of given length dimension and alphabet are given an application to irreducible cyclic codes is considered sharper upper bounds are given for the special cyclic codes called here strongly cyclic whose nonzero codewords have period equal to the length of the code asymptotics are derived on the function gamma k q that is defined as the largest number of nonzero weights a cyclic code of dimension k over mathbb f q can have and an algorithm to compute it is sketched the nonzero weights in some infinite families of reed muller codes either binary or q ary as well as in the q ary hamming code are determined two difficult results of independent interest

parameters of goppa codes revisited we discuss parameters of goppa 1970 codes such as minimum distance covering radius distance distribution and generalized hamming weights by a variation on the exponential sums method and combinatorial arguments we sharpen known bounds on these parameters

the moments of a kloosterman sum and the weight distribution of a zetterberg type binary cyclic code in this correspondence we give the moments of a kloosterman sum over fq in terms of the frequencies of weights in the binary zetterberg code of length q 1 which are known by the work of schoof and van der vlugt the method is illustrated by giving explicit formulae for the moments up to the tenth moment as a corollary the weight distribution of a zetterberg type binary cyclic code is obtained

on spectra of bch codes derives an estimate for the error term in the binomial approximation of spectra of bch codes this estimate asymptotically improves on the bounds by sidelnikov 1971 kasami et al 1985 and sole 1990

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

on almost perfect nonlinear functions over mmb f_2 n we investigate some open problems on almost perfect nonlinear apn functions over a finite field of characteristic 2 we provide new characterizations of apn functions and of apn permutations by means of their component functions we generalize some results of nyberg 1994 and strengthen a conjecture on the upper bound of nonlinearity of apn functions we also focus on the case of quadratic functions we contribute to the current works on apn quadratic functions by proving that a large class of quadratic functions cannot be apn

hyperbent functions via dillon like exponents this paper is devoted to hyperbent functions with multiple trace terms including binomial functions via dillon like exponents we show how the approach developed by mesnager to extend the charpin gong family which was also used by wang and coworkers to obtain another similar extension fits in a much more general setting to this end we first explain how the original restriction for charpin gong criterion can be weakened before generalizing the mesnager approach to arbitrary dillon like exponents afterward we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of mesnager and wang and coworkers but also to characterize the hyperbentness of several new infinite classes of boolean functions we go into full details only for a few of them but provide an algorithm and the corresponding software to apply this approach to an infinity of other new families finally we compare the asymptotic and practical performances of different characterizations including these in terms of hyperelliptic curves and actually build hyperbent functions in cases which could not be attained through naive computations of exponential sums

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

linear and quadratic ranges in representation stability abstract we prove two general results concerning spectral sequences of fi modules these results can be used to significantly improve stable ranges in a large portion of the stability theorems for fi modules currently in the literature we work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range previously the best stable ranges known in these examples were exponential up to an additive constant our work on congruence subgroups verifies a conjecture of djament

consistent structures of invariant quadrature rules for the n simplex in this paper we develop a technique to obtain in a systematic way the consistency conditions for the n dimensional simplex t n for any dimension n and degree of precision d the introduction of a convenient basis of invariant polynomials provides a powerful tool to analyze and obtain consistent structures we also present tables listing the optimal consistent structures for dimensions n 2 8 and degree of precision up to d 23 this paper is devoted only to structures no quadrature rules are presented here

optimal addition of knots to cubature formulae for planar regions a method is described to add knots to a cubature formula of degree 2k 1 for an integral over a symmetric region to obtain a cubature formula of degree 2k 1 this method is used to construct cubature formulae for the square the circle the hexagon and the entire plane

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

weight formulas for ternary melas codes in this paper we derive a formula for the frequencies of the weights in ternary melas codes and we illustrate this formula by computing a table of examples

construction of fully symmetric cubature formulae of degree 4 k 3 for fully symmetric planar regions abstract a new method is described for the construction of cubature formulae of degree 4k 3 for two dimensional symmetric regions this method is a generalisation of the t method some formulae of degree 5 9 13 and 17 for the square the circle and the entire plane are constructed

cyclic codes and reducible additive equations we prove a weil serre type bound on the number of solutions of a class of reducible additive equations over finite fields using the trace representation of cyclic codes this enables us to write a general estimate for the weights of cyclic codes we extend wolfmann s weight bound to a larger classes of cyclic codes in particular our result is applicable to any cyclic code over fp and fp 2 where p is an arbitrary prime examples indicate that our bound performs very well against the bose chaudhuri hocquenghem bch bound and that it yields the exact minimum distance in some cases

weil serre type bounds for cyclic codes we give a new method in order to obtain weil serre type bounds on the minimum distance of arbitrary cyclic codes over fpe of length coprime to p where e ges 1 is an arbitrary integer in an earlier paper we obtained weil serre type bounds for such codes only when e 1 or e 2 using lengthy explicit factorizations which seems hopeless to generalize the new method avoids such explicit factorizations and it produces an effective alternative using our method we obtain weil serre type bounds in various cases by examples we show that our bounds perform very well against bose chaudhuri hocquenghem bch bound and they yield the exact minimum distance in some cases

homological stability for configuration spaces of manifolds let cn m be the configuration space of n distinct ordered points in m we prove that if m is any connected orientable manifold closed or open the homology groups hi cn m are representation stable in the sense of church and farb arxiv 1008 1368 applying this to the trivial representation we obtain as a corollary that the unordered configuration space bn m satisfies classical homological stability for each i hi bn m hi bn 1 m for n i this improves on results of mcduff segal and others for open manifolds applied to closed manifolds this provides natural examples where rational homological stability holds even though integral homological stability fails

a duality theorem for the weight distribution of some cyclic codes a close relationship between the weight enumerators of two cyclic codes is presented the weights of the codewords are expressed in terms of kloosterman sums the weil carlitz uchiyama estimate for kloosterman sums is used to obtain bounds for the minimum distance of the codes a relation with negacyclic codes is discussed

improved cubature formulae of high degrees of exactness for the square the method of constructing minimal cubature rules with high algebraic degrees of exactness is developed by adapting a powerful algorithm for solving the system of nonlinear equations as a result new cubature formulae of degrees 15 17 19 21 and 23 are derived for the square they lead to lower numbers of knots and or to better quality with respect to those known previously the formulae obtained should be considered as the most efficient for the calculation of two dimensional integrals with a high precision