error correcting codes in projective space the projective space of order n over the finite field bbfq denoted here as pq n is the set of all subspaces of the vector space bbfqn the projective space can be endowed with the distance function d u v dimu dimv 2 dim u v which turns pq n into a metric space with this an n m d code bbc in projective space is a subset of pq n of size m such that the distance between any two codewords subspaces is at least d koetter and kschischang recently showed that codes in projective space are precisely what is needed for error correction in networks an n m d code can correct t packet errors and packet erasures introduced adversarially anywhere in the network as long as 2t 2 d this motivates our interest in such codes in this paper we investigate certain basic aspects of coding theory in projective space first we present several new bounds on the size of codes in pq n which may be thought of as counterparts of the classical bounds in coding theory due to johnson delsarte and gilbert varshamov some of these are stronger than all the previously known bounds at least for certain code parameters we also present several specific constructions of codes and code families in pq n finally we prove that nontrivial perfect codes in pq n do not exist

connections on naturally reductive spaces their dirac operator and homogeneous models in string theory given a reductive homogeneous space m g h endowed with a naturally reductive metric we study the one parameter family of connections t joining the canonical and the levi civita connection t 0 1 2 we show that the dirac operator dt corresponding to t 1 3 is the so called cubic dirac operator recently introduced by b kostant and derive the formula for its square for any t thus generalizing the classical parthasarathy formula on symmetric spaces applications include the existence of a new g invariant first order differential operator on spinors and an eigenvalue estimate for the first eigenvalue of d1 3 this geometric situation can be used for constructing riemannian manifolds which are ricci flat and admit a parallel spinor with respect to some metric connection whose torsion t 0 is a 3 form the geometric model for the common sector of string theories we present some results about solutions to the string equations and a detailed discussion of a 5 dimensional example

absence of absolutely continuous spectrum of floquet operators the spectrum of the floquet operator associated with time periodic perturbations of discrete hamiltonians is considered if the gap between successive eigenvalues j of the unperturbed hamiltonian grows as j j 1 j and the multiplicity of j grows asj with 0 asj tends to infinity then the corresponding floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough

evaluation of the weight distribution of a class of cyclic codes based on index 2 gauss sums the duals of cyclic codes with two zeros have been extensively studied and their weight distributions have recently been evaluated in some cases in this paper we determine the weight distribution of a certain new class of such codes by computations involving index 2 gauss sums

linear authentication codes bounds and constructions in this paper we consider a new class of unconditionally secure authentication codes called linear authentication codes or linear a codes we show that a linear a code can be characterized by a family of subspaces of a vector space over a finite field we then derive an upper bound on the size of the source space when other parameters of the system that is the sizes of the key space and the authenticator space and the deception probability are fixed we give constructions that are asymptotically close to the bound and show applications of these codes in constructing distributed authentication systems

minimal linear codes from characteristic functions minimal linear codes have interesting applications in secret sharing schemes and secure two party computation this paper uses characteristic functions of some subsets of mathbb f q to construct minimal linear codes by properties of characteristic functions we can obtain more minimal binary linear codes from known minimal binary linear codes which generalizes results of ding et al ieee trans inf theory vol 64 no 10 pp 6536 6545 2018 by characteristic functions corresponding to some subspaces of mathbb f q we obtain many minimal linear codes which generalizes results of ieee trans inf theory vol 64 no 10 pp 6536 6545 2018 and ieee trans inf theory vol 65 no 11 pp 7067 7078 2019 finally we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes

codes and designs related to lifted mrd codes lifted maximum rank distance mrd codes which are constant dimension codes are considered it is shown that a lifted mrd code can be represented in such a way that it forms a block design known as a transversal design a slightly different representation of this design makes it similar to a q analog of a transversal design the structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted mrd code codes that attain these bounds are constructed these codes are the largest known constant dimension codes for the given parameters these transversal designs can also be used to derive a new family of linear codes in the hamming space bounds on the minimum distance and the dimension of such codes are given

bounds on list decoding of rank metric codes so far there is no polynomial time list decoding algorithm beyond half the minimum distance for gabidulin codes these codes can be seen as the rank metric equivalent of reed solomon codes in this paper we provide bounds on the list size of rank metric codes in order to understand whether polynomial time list decoding is possible or whether it works only with exponential time complexity three bounds on the list size are proven the first one is a lower exponential bound for gabidulin codes and shows that for these codes no polynomial time list decoding beyond the johnson radius exists second an exponential upper bound is derived which holds for any rank metric code of length n and minimum rank distance d the third bound proves that there exists a rank metric code over bbfqm of length n m such that the list size is exponential in the length for any radius greater than half the minimum rank distance this implies that there cannot exist a polynomial upper bound depending only on n and d similar to the johnson bound in hamming metric all three rank metric bounds reveal significant differences to bounds for codes in hamming metric

the asymptotic behavior of grassmannian codes the iterated johnson bound is the best known upper bound on the size of an error correcting code in the grassmannian gq n k the iterated schonheim bound is the best known lower bound on the size of a covering code in gq n k we prove that both bounds are asymptotically attained for fixed k and fixed radius as n approaches infinity our methods rely on results from the theory of quasi random hypergraphs which are proved using probabilistic techniques we also determine the asymptotics of the size of the best grassmannian codes and covering codes when n k and the radius are fixed as n approaches infinity

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

constant rank codes and their connection to constant dimension codes constant dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding what the maximal cardinality of any constant dimension code with finite dimension and minimum distance is and how to construct the optimal constant dimension code or codes that achieves the maximal cardinality both remain open research problems in this paper we introduce a new approach to solving these two problems we first establish a connection between constant rank codes and constant dimension codes via this connection we show that optimal constant dimension codes correspond to optimal constant rank codes over matrices with sufficiently many rows as such the two aforementioned problems are equivalent to determining the maximum cardinality of constant rank codes and to constructing optimal constant rank codes respectively to this end we then derive bounds on the maximum cardinality of a constant rank code with a given minimum rank distance propose explicit constructions of optimal or asymptotically optimal constant rank codes and establish asymptotic bounds on the maximum rate of a constant rank code

new constructions of subspace codes using subsets of mrd codes in several blocks a basic problem for the constant dimension subspace coding is to determine the maximal possible size mathrm a q n d k of a set of k dimensional subspaces in mathrm f q n such that the subspace distance satisfies d u v 2k 2 dim u cap v geq d for any two different subspaces u and v in this set we present two new constructions of constant dimension subspace codes using subsets of maximum rank distance mrd codes in several blocks this method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting mrd codes in these two constructions subsets of mrd codes with bounded ranks play an essential role the delsarte theorem about the rank distribution of mrd codes is an important ingredient to count codewords in our constructed constant dimension subspace codes we give many new lower bounds for mathrm a q n d k more than 110 new constant dimension subspace codes better than previously best known codes are constructed

almost perfect nonlinear power functions on gf 2 n the niho case almost perfect nonlinear apn mappings are of interest for applications in cryptography we prove for odd n and the exponent d 22r 2r 1 where 4r 1 0modn that the power functions xd on gf 2n is apn the given proof is based on a new class of permutation polynomials which might be of independent interest our result supports a conjecture of niho stating that the power function xd is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequences of degree n and a decimation of that sequence by d takes on precisely the three values 1 1 2 n 1 2

the weight hierarchy of some reducible cyclic codes the generalized hamming weights ghws of linear codes are fundamental parameters the knowledge of which is of great interest in many applications however to determine the ghws of linear codes is difficult in general in this paper we study the ghws for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases this is achieved by extending the idea of yang et al into higher dimension and by employing some interesting combinatorial arguments it shall be noted that these cyclic codes may have arbitrary number of nonzeros

pseudodifferential operators on supermanifolds and the atiyah singer index theorem fermionic quantization or clifford algebra is combined with pseudodifferential operators to simplify the proof of the atiyah singer index theorem for the dirac operator on a spin manifold

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

a lie algebraic approach to the local index theorem on compact homogeneous spaces abstract using a k theory point of view bott related the atiyah singer index theorem for elliptic operators on compact homogeneous spaces to the weyl character formula this article explains how to prove the local index theorem for compact homogenous spaces using lie algebra methods the method follows in outline the proof of the local index theorem due to berline and vergne but the use of kostant s cubic dirac operator in place of the riemannian dirac operator leads to substantial simplifications an important role is also played by the quantum weil algebra of alekseev and meinrenken

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

clifford algebra analogue of the hopf koszul samelson theorem theρ decompositionc g end vρ c p and the g module structure of g

algebraic list decoding of subspace codes subspace codes are collections of subspaces of a certain ambient vector space over a finite field koetter and kschischang introduced subspace codes in order to correct errors and erasures in noncoherent random linear network coding they have also studied a remarkable family of subspace codes obtained by evaluating certain linearized polynomials the koetter kschischang subspace codes are widely regarded as the counterpart of reed solomon codes in the domain of network error correction koetter and kschischang have furthermore devised an algebraic decoding algorithm for these codes analogous to the berlekamp welch decoding algorithm for reed solomon codes in this paper we develop list decoding algorithms for subspace codes thereby providing for certain code parameters a better tradeoff between rate and error correction capability than that of the koetter kschischang codes in a sense we are able to achieve for these codes what sudan was able to achieve for reed solomon codes in order to do so we have to modify and generalize the original koetter kschischang construction in many important respects the end result is this for any integer our list l decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error satisfies l l l 1 2 r where r is the packet rate which is a new parameter introduced in this paper as in the case of sudan s list decoding algorithm this exceeds the previously best known error correction radius 1 r established by koetter and kschischang only for low rates r

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

cartesian authentication codes from functions with optimal nonlinearity in this paper we present several classes of authentication codes using functions with perfect nonlinearity and optimum nonlinearity some of the authentication codes are optimal on the other hand these authentication codes are easy to implement due to their simple algebraic structures

a family of optimal constant composition codes constant composition codes are a special class of constant weight codes with very strong constraints it is hard to construct optimal constant composition codes there are only a few classes of such optimal codes in the literature in this correspondence a family of optimal ternary constant composition codes is constructed from a class of newly discovered perfect nonlinear functions this class of codes is related to a new family of skew hadamard difference sets which are the only examples of such difference sets discovered in the last seventy years

multireceiver authentication code4 models bounds constructions and extensions multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify authenticity of the received message in this paper we give a formal definition of multireceiver authentication codes derive information theoretic and combinatorial lower bounds on their performance and give new efficient and flexible constructions for such codes finally we extend the basic model to the case that multiple messages are sent and the case that the sender can be any member of the group

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

on the continuous spectral component of the floquet operator for a periodically kicked quantum system by a straightforward generalization we extend the work of combescure j stat phys 59 679 1990 from rank 1 to rank n perturbations the requirement for the floquet operator to be pure point is established and compared to that in combescure the result matches that in mccaw and mckeller j math phys 46 032108 2005 the method here is an alternative to that work we show that if the condition for the floquet operator to be pure point is relaxed then in the case of the kicked harmonic oscillator a singularly continuous component of the floquet operator spectrum exists we also provide an in depth discussion of the conjecture presented in the work of combescure of the case where the unperturbed hamiltonian is more general we link the physics conjecture directly to a number theoretic conjecture of vinogradov the method of trigonometrical sums in the theory of numbers interscience london 1954 and show that a solution of vinogradov s conjecture solves the physics conjecture the result i

the dirac operator on homogeneous spaces and representations of reductive lie groups ii soit g un groupe de lie reductif reel connexe et k un sousgroupe compact maximal contenant un sousgroupe connexe compact donne l de g soit d v l operateur de dirac tordu soit kerd v l espace des spineurs harmoniques l une representation unitaire de g on etudie comment kerd v l se decompose en representations irreductibles unitaires de g quand l est un sous groupe compact

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

a generic construction of cartesian authentication codes in this paper a coding theory construction of cartesian authentication codes is presented the construction is a generalization of some known constructions within the framework of this generic construction several classes of authentication codes using certain classes of error correcting codes are described the authentication codes presented in this paper are better than known ones with comparable parameters it is demonstrated that the construction is related to certain combinatorial designs such as difference matrices and generalized hadamard matrices

on the cardinality of systematic authentication codes via error correcting codes in both open and private communication the participants face potential threats from a malicious enemy who has access to the communication channel and can insert messages impersonation attack or alter already transmitted messages substitution attack authentication codes a codes have been developed to provide protection against these threats in this paper we introduce a new distance called the authentication distance a distance and show that an a code can be described as a code for the a distance the a distance is directly related to the probability p sub s of success in a substitution attack we show how to transform an error correcting code into an a code and vice versa we further use these transformations to provide both upper and lower bounds on the size of the information to be authenticated and study their asymptotic behavior as examples of obtained results we prove that the cardinality of the source state space grows exponentially with the number of keys provided p sub s p sub i we generalize the square root bound given by gilbert macwilliams and sloane in 1979 and we provide very efficient constructions using concatenated reed solomon codes

recursive code construction for random networks a modification of ko tter kschischang codes for random networks is presented these codes were also studied by wang in the context of authentication problems the new codes have higher information rate while maintaining the same error correcting capabilities an efficient error correcting algorithm is proposed for these codes

almost perfect nonlinear power functions on gf 2 sup n the welch case we summarize the state of the classification of almost perfect nonlinear apn power functions x sup d on gf 2 sup n and contribute two new cases to prove these cases we derive new permutation polynomials the first case supports a well known conjecture of welch stating that for odd n 2m 1 the power function x sup 2m 3 is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

properties and applications of preimage distributions of perfect nonlinear functions the preimage distributions of perfect nonlinear functions from an abelian group of order n to an abelian group of order 3 or 4 respectively are studied based on the properties of the preimage distributions of perfect nonlinear functions from an abelian group of order 3r to an abelian group of order 3 the weight distributions of the ternary linear codes c pi from the perfect nonlinear functions pi x from f 3 r to itself are determined these results suggest that two open problems proposed by carlet ding and yuan in 2005 and 2006 respectively are answered

optimal ferrers diagram rank metric codes optimal rank metric codes in ferrers diagrams are considered such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space first we consider rank metric anticodes and prove a code anticode bound for ferrers diagram rank metric codes the size of optimal linear anticodes is given four techniques and constructions of ferrers diagram rank metric codes are presented each providing optimal codes for different diagrams and parameters for which no optimal solution was known before the first construction uses maximum distance separable codes on the diagonals of the matrices the second one takes a subcode of a maximum rank distance code and the last two combine codes in small diagrams to a code in a larger diagram the constructions are analyzed and compared and unsolved diagrams are identified

singular continuous floquet operator for periodic quantum systems abstract consider the floquet operator of a time independent quantum system acting on a separable hilbert space periodically perturbed by a rank one kick e i h 0 t e i t where t is the period the coupling constant and h 0 is a pure point self adjoint operator bounded from below under some hypotheses on the vector cyclic w r t h 0 we prove the following if the gaps between the eigenvalues n are such that n 1 n c n for some 0 1 and c 0 then the floquet operator of the perturbed system is purely singular continuous t a e if h 0 is the hamiltonian of the one dimensional rotator on l 2 r t 0 z and the ratio 2 t t 0 2 is irrational then the floquet operator is purely singular continuous as soon as t 0 2 we also establish an integral formula for the family e i h 0 t e i t t 0 r

floquet operators without singular continuous spectrum let u be a unitary operator defined on a infinite dimensional separable complex hilbert space h assume there exists a self adjoint operator a on h such that u au a ci k for some positive constant c and compact operator k then assuming the commutators u au a and a u au admit a bounded extension over h we prove the spectrum of the operator u has no singular continuous component and only a finite number of eigenvalues of finite multiplicity we give a localized version of this result and apply it to study the spectrum of the floquet operator of periodic time dependent kicked quantum systems

floquet hamiltonians with pure point spectrum we consider floquet hamiltonians of the type k f i partial t h 0 beta v omega t whereh0 a selfadjoint operator acting in a hilbert space has simple discrete spectrume1 0 for a given 0 t v t is 2 periodic andr times strongly continuously differentiable as a bounded operator on and are real parameters and the periodic boundary condition is imposed in time we show roughly that providedr is large enough small enough and non resonant then the spectrum ofkf is pure point the method we use relies on a successive application of the adiabatic treatment due to howland and the kam type iteration settled by bellissard and extended by combescure both tools are revisited adjusted and at some points slightly simplified

spectral properties of a periodically kicked quantum hamiltonian we study the spectral properties of the floquet operator for the periodically kicked hamiltonianh t h 0 t nt h 0 being self adjoint and pure point we show that the floquet operator is pure point for almost every if is cyclic forh 0 and has absolutely convergent expansion in the basis of eigenstates ofh 0 when this last condition is not satisfied the floquet operator can have a continuous spectrum as we show by an example

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

localized error correction in projective space in this paper we extend the localized error correction code introduced by l a bassalygo and coworkers from hamming space to projective space for constant dimensional localized error correction codes in projective space we have a lower bound and an upper bound of the capacity which are asymptotically tight when z x n z 2 where x z and n are dimensions of codewords error configurations and the ground space respectively we determine the capacity of nonconstant dimensional localized error correction codes when z n 3

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

parallel multilevel constructions for constant dimension codes constant dimension codes cdcs as special subspace codes have received a lot of attention due to their application in random network coding this paper introduces a family of new codes called rank metric codes with given ranks grmcs to generalize the parallel construction in xu and chen ieee trans inf theory 64 2018 6315 6319 and the classic multilevel construction a singleton like upper bound and a lower bound for grmcs derived from gabidulin codes are given via grmcs two effective constructions for cdcs are presented by combining the parallel construction and the multilevel construction many cdcs with larger size than the previously best known codes are given the ratio between the new lower bound and the known upper bound for 4 delta 2 delta 2 delta q cdcs is calculated it is greater than 0 99926 for any prime power q and any delta geq 3

on the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum we consider quantum hamiltonians of the form h t h v t where the spectrum of h is semibounded and discrete and the eigenvalues behave as en n with 0 0 p 1 and 1 2 we show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and e is small enough more precisely for any initial condition dom h1 2 the diffusion of energy is bounded from above as h t o t where sigma alpha 2 lceil p 1 rceil gamma frac 1 2 as an application we consider the hamiltonian h t p ev t on l2 s1 d which was discussed earlier in the literature by howland

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

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

the non commutative weil algebra for any compact lie group g together with an invariant inner product on its lie algebra g we define the non commutative well algebra w g as a tensor product of the universal enveloping algebra u g and the clifford algebra cl 8 just like the usual we

macdonald s identities and the large n limit of ym 2 on the cylinder the purpose of this paper is to determine the large n asymptotics of the free energy fn a u a of ym2 two dimensional yang mills theory with gauge group gn su n on a cylinder where a is a so called principal element of type mathematically where hgn is the central heat kernel of gn we find that where is an explicit quadratic functional in the limit distribution d of eigenvalues of un depending only on the integral geometry of su 2 the factor of n contradicts some predictions in the physics literature on the large n limit of ym2 on the cylinder due to gross matytsin kazakov wynter and others

equivariant eta invariants on homogeneous spaces let d be a homogeneous dirac operator on the quotient m g h of two compact connected lie groups we construct a deformation tilde d ofd and calculate its equivariant eta invariant eta g tilde d explicitly on the dense subset g 0 of g that acts freely onm on g 0 eta g tilde d and eta g d differ only by a virtual character of g moreover if g supset h is a symmetric pair or if d is the untwisted dirac operator then eta g d eta g tilde d on g 0 we also sketch some applications of eta g tilde d

on the relationships between perfect nonlinear functions and universal hash families in this paper the relationships between perfect nonlinear in brief pn functions and optimal universal hash families are discussed we point out the equivalence of constructions between them i e from pn functions one can obtain optimal universal hash families and vice versa as an application of our construction a message authentication code is proposed which provides better resistance to substitution attack than a known construction given by carlet et al in 2006 more generally the connections between functions with given differential uniformity and some universal hash families are studied

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

hybrid noncoherent network coding we describe a novel extension of subspace codes for noncoherent networks suitable for use when the network is viewed as a communication system that introduces both dimension and symbol errors we show that when symbol erasures occur in a significantly large number of different basis vectors transmitted through the network and when the min cut of the network is much smaller than the length of the transmitted codewords the new family of codes outperforms their subspace code counterparts for the proposed coding scheme termed hybrid network coding we derive two upper bounds on the size of the codes these bounds represent a variation of the singleton and of the sphere packing bound we show that a simple concatenated scheme that consists of subspace codes and reed solomon codes is asymptotically optimal with respect to the singleton bound finally we describe two efficient decoding algorithms for concatenated subspace codes that in certain cases have smaller complexity than their subspace decoder counterparts

singular continuous floquet operator for systems with increasing gaps abstract consider the floquet operator of a time independent quantum system periodically perturbed by a rank one kick acting on a separable hilbert space e ih0te i t where t and are the period and the coupling constant respectively assume the spectrum of the self adjoint operator h0 is pure point simple bounded from below and the gaps between the eigenvalues n grow like n 1 n cnd with d 2 under some hypotheses on the arithmetical nature of the eigenvalues and the vector cyclic for h0 we prove the floquet operator of the perturbed system has purely singular continuous spectrum

logarithm cartesian authentication codes chanson ding and salomaa have recently constructed several classes of authentication codes using certain classes of functions in this paper we further extend that work by constructing two classes of cartesian authentication codes using the logarithm functions the codes constructed here involve the theory of cyclotomy and are better than a subclass of helleseth johansson s codes and bierbrauer s codes in terms of the maximum success probability with respect to the substitution attack

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

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

on the kashiwara vergne conjecture let g be a connected lie group with lie algebra mathfrak g in 1977 duflo constructed a homomorphism of mathfrak g modules text duf colon s mathfrak g to u mathfrak g which restricts to an algebra isomorphism on invariants kashiwara and vergne 1978 proposed a conjecture on the campbell hausdorff series which among other things extends the duflo theorem to germs of bi invariant distributions on the lie group g

subspace codes based on graph matchings ferrers diagrams and pending blocks this paper provides new constructions and lower bounds for subspace codes using ferrers diagram rank metric codes from matchings of the complete graph and pending blocks we present different constructions for constant dimension codes with minimum injection distance 2 or k 1 where k is the constant dimension furthermore we present a construction of new codes from old codes for any minimum distance then we construct nonconstant dimension codes from these codes some examples of codes obtained by these constructions are the largest known codes for the given parameters

constructions of authentication codes from algebraic curves over finite fields we present a new application of algebraic curves over finite fields to the constructions of universal hash families and unconditionally secure codes we show that the constructions derived from the garcia stichtenoth curves yield new classes of authentication codes and universal hash families which are substantially better than those previously known

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients