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

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 complex bingham quartic distribution and shape analysis summary the complex bingham distribution was introduced by kent as a tractable model for landmark based shape analysis it forms an exponential family with a sufficient statistic which is quadratic in the data however the distribution has too much symmetry to be widely useful in particular under high concentration it behaves asymptotically as a normal distribution but where the covariance matrix is constrained to have complex symmetry to overcome this limitation and to provide a full range of asymptotic normal behaviour we introduce a new complex bingham quartic distribution by adding a selection of quartic terms to the log density in the simplest case this new distribution corresponds to kent s fb5 distribution asymptotic and saddlepoint methods are developed for the normalizing constant to facilitate maximum likelihood estimation examples are given to show the usefulness of this new distribution

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

spherical homoscedastic distributions the equivalency of spherical and normal distributions in classification many feature representations as in genomics describe directional data where all feature vectors share a common norm in other cases as in computer vision a norm or variance normalization step where all feature vectors are normalized to a common length is generally used these representations and pre processing step map the original data from p to the surface of a hypersphere sp 1 such representations should then be modeled using spherical distributions however the difficulty associated with such spherical representations has prompted researchers to model their spherical data using gaussian distributions instead as if the data were represented in p rather than sp 1 this opens the question to whether the classification results calculated with the gaussian approximation are the same as those obtained when using the original spherical distributions in this paper we show that in some particular cases which we named spherical homoscedastic the answer to this question is positive in the more general case however the answer is negative for this reason we further investigate the additional error added by the gaussian modeling we conclude that the more the data deviates from spherical homoscedastic the less advisable it is to employ the gaussian approximation we then show how our derivations can be used to define optimal classifiers for spherical homoscedastic distributions by using a kernel which maps the original space into one where the data adapts to the spherical homoscedastic model we can derive non linear classifiers with potential applications in a large number of problems we conclude this paper by demonstrating the uses of spherical homoscedasticity in the classification of images of objects gene expression sequences and text data

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

optshrink an algorithm for improved low rank signal matrix denoising by optimal data driven singular value shrinkage the truncated singular value decomposition of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low rank matrix here we consider the unobservable denoising problem of how to best approximate a low rank signal matrix buried in noise by optimal re weighting of the singular vectors of the measurement matrix we exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the independent identically distributed gaussian noise case our analysis brings into sharp focus the shrinkage and thresholding form of the optimal weights the nonconvex nature of the associated shrinkage function on the singular values and explains why matrix regularization via singular value thresholding with convex penalty functions such as the nuclear norm will always be suboptimal we validate our theoretical predictions with numerical simulations develop an implementable algorithm optshrink that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

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

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

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

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

complete diagrammatics of the single ring theorem using diagrammatic techniques we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices the formalism allows us to map two distinct areas of free random variables hermitian positive definite operators and non normal r diagonal operators we also rederive the haagerup larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach

saddlepoint approximations for the bingham and fisher bingham normalising constants the fisher bingham distribution is obtained when a multivariate normal random vector is conditioned to have unit length its normalising constant can be expressed as an elementary function multiplied by the density evaluated at 1 of a linear combination of independent noncentral chi 1 2 random variables hence we may approximate the normalising constant by applying a saddlepoint approximation to this density three such approximations implementation of each of which is straightforward are investigated the first order saddlepoint density approximation the second order saddlepoint density approximation and a variant of the second order approximation which has proved slightly more accurate than the other two the numerical and theoretical results we present show that this approach provides highly accurate approximations in a broad spectrum of cases

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

on optimal tests for rotational symmetry against new classes of hyperspherical distributions motivated by the central role played by rotationally symmetric distributions in directional statistics we consider the problem of testing rotational symmetry on the hypersphere we adopt a semipar

infinitely divisible distributions for rectangular free convolution classification and matricial interpretation in a previous paper benaych georges in related convolution 2006 we defined the rectangular free convolution here we investigate the related notion of infinite divisibility which happens to be closely related the classical infinite divisibility there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of infinitely divisible distributions which preserves limit theorems we give an interpretation of this correspondence in terms of random matrices we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their infinitely divisible correspondents when the dimensions go from one to infinity in a ratio

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

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

sobolev tests of goodness of fit of distributions on compact riemannian manifolds classes of coordinate invariant omnibus goodness of fit tests on compact riemannian manifolds are proposed the tests are based on gine s sobolev tests of uniformity a condition for consistency is given the tests are illustrated by an example on the rotation group s0 3

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

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

saddlepoint approximations for the normalizing constant of fisher bingham distributions on products of spheres and stiefel manifolds in an earlier paper kume wood 2005 showed how the normalizing constant of the fisher bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation in this sequel we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent fisher bingham distributions on cartesian products of spheres and fisher bingham distributions on stiefel manifolds in each case the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables both first order and second order saddlepoint approximations are considered computational algorithms numerical results and theoretical properties of the approximations are presented in the challenging high dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered some key words directional data fisher matrix distribution kent distribution orientation statistics

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

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

isotropic probabilistic models for directions planes and referential systems although simple probabilistic models for directions in two dimensional 2d and three dimensional 3d spaces are well known respectively the von mises and fisher distributions probabilistic models for directions planes and referential systems in an n dimensional n d space are either not popular or not available the results for directions in 2d and 3d can be generalized for i arbitrary directions ii planes iii referential systems and iv symmetric random tensors and in turn all these results can be generalized to n d spaces in this paper we will focus on and build up isotropic probabilistic models for directions planes and referential systems in n d space

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

limit distributions of random matrices abstract we study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation in particular we are concerned with the ensemble of symmetric blocks of independent hermitian random matrices which are asymptotically free asymptotically free from diagonal deterministic matrices and whose norms are uniformly bounded almost surely this class contains symmetric blocks of unitarily invariant hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability we show that the associated matricially free gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices including non hermitian gaussian matrices and matrices of wishart type

some minimization problems for the free analogue of the fisher information abstract we consider the free non commutative analogue introduced by d voiculescu of the concept of fisher information for random variables we determine the minimal possible value of a a if a is a non commutative random variable subject to the constraint that the distribution of a a is prescribed more generally we obtain the minimal possible value of a ij a ij 1 i j d if a ij 1 i j d is a family of non commutative random variables such that the distribution of a a is prescribed where a is the matrix a ij d i j 1 the d d generalization is obtained from the case d 1 via a result of independent interest concerning the minimal value of a ij a ij 1 i j d when the matrix a a ij d i j 1 and its adjoint have a given joint distribution a version of this result describes the minimal value of b ij 1 i j d when the matrix b b ij d i j 1 is selfadjoint and has a given distribution we then show how the minimization results obtained for lead to maximization results concerning the free entropy also defined by voiculescu

on the asymptotic distribution of block modified random matrices we study random matrices acting on tensor product spaces which have been transformed by a linear block operation using operator valued free probability theory under some mild assumptions on the linear map acting on the blocks we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution moreover using recent results on operator valued subordination we present an algorithm that computes numerically but in full generality the limiting eigenvalue distribution of the modified matrices our analytical results cover many cases of interest in quantum information theory we unify some known results and we obtain new distributions and various generalizations

rao score tests for goodness of fit and independence summary rao s score statistic is a standard tool for carrying out hypothesis tests a unified approach is given which encompasses many common tests when a suitable group structure is present it is shown how the derivation of the test statistic can be simplified an omnibus test of multivariate normality is derived as an example of group invariance as another application of rao score tests a general method is given for constructing tests of independence

analogues on the sphere of the affine equivariant spatial median robust estimation of location for data on the unit sphere sp 1 is an important problem in directional statistics even though the influence functions of the sample mean direction and other location

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

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

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

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

finite dimensional statistical inference in this paper we derive the explicit series expansion of the eigenvalue distribution of various models namely the case of noncentral wishart distributions as well as correlated zero mean wishart distributions the tools used extend those of the free probability framework which have been quite successful for high dimensional statistical inference when the size of the matrices tends to infinity also known as free deconvolution this contribution focuses on the finite gaussian case and proposes algorithmic methods to compute the moments cases where asymptotic results fail to apply are also discussed

continuous family of invariant subspaces for r diagonal operators we show that every r diagonal operator x has a continuous family of invariant subspaces relative to the von neumann algebra generated by x this allows us to find the brown measure of x and to find a new conceptual proof that voiculescu s s transform is multiplicative our considerations base on a new concept of r diagonality with amalgamation for which we give several equivalent characterizations

phase coupling estimation from multivariate phase statistics coupled oscillators are prevalent throughout the physical world dynamical system formulations of weakly coupled oscillator systems have proven effective at capturing the properties of real world systems and are compelling models of neural systems however these formulations usually deal with the forward problem simulating a system from known coupling parameters here we provide a solution to the inverse problem determining the coupling parameters from measurements starting from the dynamic equations of a system of symmetrically coupled phase oscillators given by a nonlinear langevin equation we derive the corresponding equilibrium distribution this formulation leads us to the maximum entropy distribution that captures pairwise phase relationships to solve the inverse problem for this distribution we derive a closed form solution for estimating the phase coupling parameters from observed phase statistics through simulations we show that the algorithm performs well in high dimensions d 100 and in cases with limited data as few as 100 samples per dimension in addition we derive a regularized solution to the estimation and show that the resulting procedure improves performance when only a limited amount of data is available because the distribution serves as the unique maximum entropy solution for pairwise phase statistics phase coupling estimation can be broadly applied in any situation where phase measurements are made under the physical interpretation the model may be used for inferring coupling relationships within cortical networks

robust principal component analysis for power transformed compositional data geochemical surveys collect sediment or rock samples measure the concentration of chemical elements and report these typically either in weight percent or in parts per million ppm there are usually a large number of elements measured and the distributions are often skewed containing many potential outliers we present a new robust principal component analysis pca method for geochemical survey data that involves first transforming the compositional data onto a manifold using a relative power transformation a flexible set of moment assumptions are made which take the special geometry of the manifold into account the kent distribution moment structure arises as a special case when the chosen manifold is the hypersphere we derive simple moment and robust estimators ro of the parameters which are also applicable in high dimensional settings the resulting pca based on these estimators is done in the tangent space and is related to the power transformation method used in correspondence analysis t

randomized isometric linear dispersion space time block coding for the df relay channel this paper presents a randomized linear dispersion space time block code for decode and forward synchronous relays the coding matrices are obtained as a set of columns or rows of randomly generated haar distributed unitary matrices with respect to independent and identically distributed i i d generated codes this particular isometric structure reduces the intersymbol interference generated within each relay the gain over i i d codes in terms of spectral efficiency is analyzed for both the lmmse and the ml receivers under the assumption of frequency flat quasi static fading in this setting the spectral efficiency is a random quantity since it depends on the random coding matrices however it is proven that the spectral efficiency converges in probability to a deterministic quantity when the dimensions of the matrices tend to infinity while keeping constant their ratio i e the coding rate consequently when the random coding matrices are large enough the presented system behaves as a deterministic one this result is achieved by means of the rectangular r transform a powerful tool of free probability theory which allows determining the distribution of the singular values of a sum of rectangular matrices

bayesian regression using a prior on the model fit the r2 d2 shrinkage prior prior distributions for high dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients which is inherently difficult we instead propose a ne

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

estimating the relative rotation of two tectonic plates from boundary crossings abstract let 1 s be unknown vectors on the sphere and ao be an unknown rotation suppose that uij are estimates of points lying on the great circle normal to i and ik are estimates of points lying on the great circle normal to ao i this article discusses a method to construct a confidence region for ao this problem arises in the reconstruction of the relative motion of two tectonic plates on opposing sides of a rift the boundary on each side is represented by a collection of great circle segments and ao is the rotation that takes one boundary into the other the data consist of measured crossing points uij and ik of the various segments on the opposing boundaries the analysis completes an analysis of hellinger 1981 the errors in tectonic data are quite concentrated and the problem reduces to linear regression once this is realized many interesting problems such as triple junctions or multiple time periods can be examined the analysis is aided substantially by a parameterization of t

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

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

a continuous semigroup of notions of independence between the classical and the free one in this paper we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non commutative random variables these notions are related to the liberation process introduced by d voiculescu to each notion of independence correspond new convolutions of probability measures for which we establish formulae and of which we compute simple examples we prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence

rectangular random matrices related convolution we characterize asymptotic collective behavior of rectangular random matrices the sizes of which tend to infinity at different rates it appears that one can compute the limits of all noncommutative moments thus all spectral properties of the random matrices we consider because when embedded in a space of larger square matrices independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra therefore we can define a rectangular free convolution which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values this convolution is linearized by cumulants and by an analytic integral transform that we called the rectangular r transform

probabilistic model for two dependent circular variables summary motivated by problems in molecular biology and molecular physics we propose a five parameter torus analogue of the bivariate normal distribution for modelling the distribution of two circular random variables the conditional distributions of the proposed distribution are von mises the marginal distributions are symmetric around their means and are either unimodal or bimodal the type of shape depends on the configuration of parameters and we derive the conditions that ensure a specific shape the utility of the proposed distribution is illustrated by the modelling of angular variables in a short linear peptide

the fisher bingham distribution on the sphere abstract the fisher distribution is the analogue on the sphere of the isotropic bivariate normal distribution in the plane the purpose of this paper is to propose and analyze a spherical analogue of the general bivariate normal distribution estimation hypothesis testing and confidence regions are also discussed author

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

the lebesgue decomposition of the free additive convolution of two probability distributions we prove that the free additive convolution of two borel probability measures supported on the real line can have a component that is singular continuous with respect to the lebesgue measure on mathbb r only if one of the two measures is a point mass the density of the absolutely continuous part with respect to the lebesgue measure is shown to be analytic wherever positive and finite the atoms of the free additive convolution of borel probability measures on the real line have been described by bercovici and voiculescu in a previous paper

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

the complex bingham distribution and shape analysis summary a complex version of the bingham distribution is defined on the unit complex sphere in ck various statistical properties including asymptotic normality under high concentration are derived symmetries in the distribution make it a natural tool for the analysis of the shape of landmark data in two dimensions strengths and weaknesses of this approach are investigated links and contrasts with bookstein co ordinates are discussed a hybrid approach to principal component analysis based on complex and real co ordinates is suggested

scaled von mises fisher distributions and regression models for paleomagnetic directional data we propose a new distribution for analyzing paleomagnetic directional data that is a novel transformation of the von mises fisher distribution the new distribution has ellipse like symmetry as

a directional mixed effects model for compositional expenditure data abstractcompositional data are vectors of proportions defined on the unit simplex and this type of constrained data occur frequently in government surveys it is also possible for the compositional data to be correlated due to the clustering or grouping of the observations within small domains or areas we propose a new class of the mixed model for compositional data based on the kent distribution for directional data where the random effects also have kent distributions one useful property of the new directional mixed model is that the marginal mean direction has a closed form and is interpretable the random effects enter the model in a multiplicative way via the product of a set of rotation matrices and the conditional mean direction is a random rotation of the marginal mean direction in small area estimation settings the mean proportions are usually of primary interest and these are shown to be simple functions of the marginal mean direction for estimation we apply a quasi likelihood method whic

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

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools