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

worst case to average case reductions based on gaussian measures we show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice the lattice problems we consider are the shortest vector problem the shortest independent vectors problem the covering radius problem and the guaranteed distance decoding problem a variant of the well known closest vector problem the approximation factor we obtain is n log o 1 n for all four problems this greatly improves on all previous work on the subject starting from ajtai s seminal paper generating hard instances of lattice problems in complexity of computations and proofs quad mat 13 dept math seconda univ napoli caserta italy 2004 pp 1 32 up to the strongest previously known results by micciancio siam j comput 34 2004 pp 118 169 our results also bring us closer to the limit where the problems are no longer known to be in np intersect conp our main tools are gaussian measures on lattices and the high dimensional fourier transform we start by defining a new lattice parameter which determines the amount of gaussian noise that one has to add to a lattice in order to get close to a uniform distribution in addition to yielding quantitatively much stronger results the use of this parameter allows us to simplify many of the complications in previous work our technical contributions are twofold first we show tight connections between this new parameter and existing lattice parameters one such important connection is between this parameter and the length of the shortest set of linearly independent vectors second we prove that the distribution that one obtains after adding gaussian noise to the lattice has the following interesting property the distribution of the noise vector when conditioning on the final value behaves in many respects like the original gaussian noise vector in particular its moments remain essentially unchanged

new bounds in some transference theorems in the geometry of numbers the aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in r to formulate the problem we have to introduce some notation and terminology we shall treat ir as an n dimensional euclidean space with the norm ii ii and metric d the inner product of vectors u v will be denoted by uv we shall usually write u 2 instead of uu the closed and open unit balls in f will be denoted by b and b respectively if a c ir then span a and a z denote the linear subspace spanned over a and the orthogonal complement of a in p a lattice in ir is an additive subgroup of ir generated by n linearly independent vectors the family of all lattices in ir will be denoted by a given a lattice l e a we define the polar dual reciprocal lattice l in the usual way l u u v z for each v l

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

exact post selection inference with application to the lasso we develop a general approach to valid inference after model selection in a nutshell our approach produces post selection inferences with the same frequency guarantees as those given by data splitting but are more powerful at the core of our framework is a result that characterizes the distribution of a post selection estimator conditioned on the selection event we specialize the approach to model selection by the lasso to form valid condence intervals for the selected coecients and test whether all relevant variables have been included in the model

hardness of approximating the shortest vector problem in lattices let p 1 be any fixed real we show that assuming np n rp there is no polynomial time algorithm that approximates the shortest vector problem svp in e p norm within a constant factor under the stronger assumption np n rtime 2poly log n we show that there is no polynomial time algorithm with approximation ratio 2 log n 1 2 e where n is the dimension of the lattice and e 0 is an arbitrarily small constant we first give a new randomized reduction from closest vector problem cvp to svp that achieves some constant factor hardness the reduction is based on bch codes its advantage is that the svp instances produced by the reduction behave well under the augmented tensor product a new variant of tensor product that we introduce this enables us to boost the hardness factor to 2 log n 1 2 e

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

a significance test for the lasso in the sparse linear regression setting we consider testing the significance of the predictor variable that enters the current lasso model in the sequence of models visited along the lasso solution path we propose a simple test statistic based on lasso fitted values called the covariance test statistic and show that when the true model is linear this statistic has an exp 1 asymptotic distribution under the null hypothesis the null being that all truly active variables are contained in the current lasso model our proof of this result for the special case of the first predictor to enter the model i e testing for a single significant predictor variable against the global null requires only weak assumptions on the predictor matrix x on the other hand our proof for a general step in the lasso path places further technical assumptions on x and the generative model but still allows for the important high dimensional case p n and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables of course for testing the significance of an additional variable between two nested linear models one typically uses the chi squared test comparing the drop in residual sum of squares rss to a formula see text distribution but when this additional variable is not fixed and has been chosen adaptively or greedily this test is no longer appropriate adaptivity makes the drop in rss stochastically much larger than formula see text under the null hypothesis our analysis explicitly accounts for adaptivity as it must since the lasso builds an adaptive sequence of linear models as the tuning parameter decreases in this analysis shrinkage plays a key role though additional variables are chosen adaptively the coefficients of lasso active variables are shrunken due to the formula see text penalty therefore the test statistic which is based on lasso fitted values is in a sense balanced by these two opposing properties adaptivity and shrinkage and its null distribution is tractable and asymptotically exp 1

on the number of zeros of abelian integrals a constructive solution of the infinitesimal hilbert sixteenth problem we prove that the number of limit cycles generated from nonsingular energy level ovals periodic trajectories in a small non conservative perturbation of a hamiltonian polynomial vector field on the plane is bounded by a double exponential of the degree of the fields this solves the long standing infinitesimal hilbert 16th problem

estimation of error variance via ridge regression summarywe propose a novel estimator of error variance and establish its asymptotic properties based on ridge regression and random matrix theory the proposed estimator is valid under both low and high dimensional models and performs well not only in nonsparse cases but also in sparse ones the finite sample performance of the proposed method is assessed through an intensive numerical study which indicates that the method is promising compared with its competitors in many interesting scenarios

a knockoff filter for high dimensional selective inference this paper develops a framework for testing for associations in a possibly high dimensional linear model where the number of features variables may far exceed the number of observational units in this framework the observations are split into two groups where the first group is used to screen for a set of potentially relevant variables whereas the second is used for inference over this reduced set of variables we also develop strategies for leveraging information from the first part of the data at the inference step for greater power in our work the inferential step is carried out by applying the recently introduced knockoff filter which creates a knockoff copy a fake variable serving as a control for each screened variable we prove that this procedure controls the directional false discovery rate fdr in the reduced model controlling for all screened variables this says that our high dimensional knockoff procedure discovers important variables as well as the directions signs of their effects in such a way that the expected proportion of wrongly chosen signs is below the user specified level thereby controlling a notion of type s error averaged over the selected set this result is nonasymptotic and holds for any distribution of the original features and any values of the unknown regression coefficients so that inference is not calibrated under hypothesized values of the effect sizes we demonstrate the performance of our general and flexible approach through numerical studies showing more power than existing alternatives finally we apply our method to a genome wide association study to find locations on the genome that are possibly associated with a continuous phenotype

valid post selection inference it is common practice in statistical data analysis to perform data driven variable selection and derive statistical inference from the resulting model such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori we propose to produce valid post selection inference by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals simultaneity is required for all linear functions that arise as coefficient estimates in all submodels by purchasing simultaneity insurance for all possible submodels the resulting post selection inference is rendered universally valid under all possible model selection procedures this inference is therefore generally conservative for particular selection procedures but it is always less conservative than full scheffe protection importantly it does not depend on the truth of the selected submodel and hence it produces valid inference even in wrong models we describe the structure of the simultaneous inference problem and give some asymptotic results

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

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 lattices learning with errors random linear codes and cryptography our main result is a reduction from worst case lattice problems such as gapsvp and sivp to a certain learning problem this learning problem is a natural extension of the learning from parity with error problem to higher moduli it can also be viewed as the problem of decoding from a random linear code this we believe gives a strong indication that these problems are hard our reduction however is quantum hence an efficient solution to the learning problem implies a quantum algorithm for gapsvp and sivp a main open question is whether this reduction can be made classical i e nonquantum we also present a classical public key cryptosystem whose security is based on the hardness of the learning problem by the main result its security is also based on the worst case quantum hardness of gapsvp and sivp the new cryptosystem is much more efficient than previous lattice based cryptosystems the public key is of size o n2 and encrypting a message increases its size by a factor of o n in previous cryptosystems these values are o n4 and o n2 respectively in fact under the assumption that all parties share a random bit string of length o n2 the size of the public key can be reduced to o n

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

explicit codes achieving list decoding capacity error correction with optimal redundancy in this paper we present error correcting codes that achieve the information theoretically best possible tradeoff between the rate and error correction radius specifically for every 0 r 1 and epsiv 0 we present an explicit construction of error correcting codes of rate that can be list decoded in polynomial time up to a fraction 1 r epsiv of worst case errors at least theoretically this meets one of the central challenges in algorithmic coding theory our codes are simple to describe they are folded reed solomon codes which are in fact exactly reed solomon rs codes but viewed as a code over a larger alphabet by careful bundling of codeword symbols given the ubiquity of rs codes this is an appealing feature of our result and in fact our methods directly yield better decoding algorithms for rs codes when errors occur in phased bursts the alphabet size of these folded rs codes is polynomial in the block length we are able to reduce this to a constant depending on epsiv using existing ideas concerning ldquolist recoveryrdquo and expander based codes concatenating the folded rs codes with suitable inner codes we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard gmd decoding

algebraic list decoding in projective space decoding with multiplicities and rank metric codes the problem of list decoding algebraic subspace codes and rank metric codes is considered we develop two separate methods via two different approaches for list decoding subspace codes and rank metric codes these methods provide for certain code parameters improved tradeoffs between rate and error correction capability than that of the koetter kschischang codes in the domain of subspace codes and than that of the gabidulin codes in the domain of rank metric codes and several other extensions thereof in the first approach we introduce the notion of root multiplicities for a certain sub ring of the ring of linearized polynomials in the list decoding algorithm multiple roots are enforced for the interpolation polynomial in order to achieve an improved error correction radius for an extended family of koetter kschischang subspace codes the normalized error correction radius for this approach is begin equation tau a frac 2 l 1 r 1 1 frac l l 1 l n r r 1 r end equation where l is the maximum list size n is the subspace code dimension r is the rate of the code and r is the multiplicity parameter in the second approach we construct a folded version of koetter kschischang codes a linear algebraic list decoding algorithm is proposed for these codes that achieves the error correction radius begin equation tau b s 1 sr end equation where s is the folding parameter as opposed to the first approach the size of output list in the second approach depends on the underlying field size and is at most q m s 1 where q m is the size of the field that message symbols are chosen from it is also shown that the output list size is 1 with high probability in a probabilistic setting we utilize the techniques of the second approach in the domain of rank metric codes to construct folded gabidulin codes to enable a linear algebraic list decoding algorithm for such codes our algorithm makes it possible to recover the message provided that the normalized rank of error is at most 1 r epsilon for any epsilon 0 notably this achieves the singleton bound on the error correction radius of a rank metric code

on the complexity of the rank syndrome decoding problem in this paper we propose two new generic attacks on the rank syndrome decoding rsd problem let c be a random n k rank code over gf q m and let y x e be a received word such that x in c and rank e r the first attack the support attack is combinatorial and permits to recover an error e of rank weight r in min o n k 3 m 3 q r lfloor km n rfloor o n k 3 m 3 q r 1 lfloor k 1 m n rfloor operations on gf q this new attack improves the exponent for the best generic attack for the rsd problem in the case n m by introducing the ratio m n in the exponential coefficient of the previously best known attacks the second attack the annulator polynomial attack is an algebraic attack based on the theory of q polynomials introduced by ore we propose a new algebraic setting for the rsd problem that permits to consider equations and unknowns in the extension field gf q m rather than in gf q as it is usually the case we consider two approaches to solve the problem in this new setting the linearization technique shows that if n ge k 1 r 1 1 the rsd problem can be solved in polynomial time more generally we prove that if lceil r 1 k 1 n 1 r rceil le k the rsd problem can be solved with an average complexity of o r 3 k 3 q r lceil r 1 k 1 n 1 r rceil operations in the base field gf q we also consider solving with grobner bases for which we discuss theoretical complexity we also consider hybrid solving with grobner bases on practical parameters as an example of application we use our new attacks on all recent cryptosystems parameters which repair the gpt cryptosystem we break all examples of published proposed parameters and some parameters are broken in less than 1 s in certain cases

efficient encryption from random quasi cyclic codes we propose a framework for constructing efficient code based encryption schemes that do not hide any structure in their public matrix the framework is in the spirit of the schemes first proposed by alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight we depart somewhat from alekhnovich s approach and propose an encryption scheme based on the difficulty of decoding random quasi cyclic codes we propose two new cryptosystems instantiated within our framework the hamming quasi cyclic cryptosystem hqc based on the hamming metric and the rank quasi cyclic cryptosystem rqc based on the rank metric we give a security proof which reduces the indistinguishability under chosen plaintext attack security of our systems to a decision version of the well known problem of decoding random families of quasi cyclic codes for the hamming and rank metrics the respective mathsf qcsd and mathsf rqcsd problems we also provide an analysis of the decryption failure probability of our scheme in the hamming metric case for the rank metric there is no decryption failure our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits the cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication asymptotically for lambda the security parameter the public key sizes are respectively in mathcal o lambda 2 for hqc and in mathcal o left lambda frac 4 3 right for rqc practical parameter compares well to the systems based on ring learning parity with noise or the recent moderate density parity check codes system

a deterministic reduction for the gap minimum distance problem determining the minimum distance of a linear code is one of the most important problems in algorithmic coding theory the exact version of the problem was shown to be np complete by vardy the gap version of the problem was shown to be np hard for any constant factor under a randomized reduction in an earlier work it was shown in the same paper that the minimum distance problem is not approximable in randomized polynomial time to the factor 2log1 n unless np rtime 2polylog n in this paper we derandomize the reduction and thus prove that there is no deterministic polynomial time algorithm to approximate the minimum distance to any constant factor unless p np we also prove that the minimum distance is not approximable in deterministic polynomial time to the factor 2log1 n unless np dtime 2polylog n as the main technical contribution for any constant 2 3 1 we present a deterministic algorithm that given a positive integer s runs in time poly s and constructs a code c of length poly s with an explicit hamming ball of radius d c such that the projection at the first s coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension s where d c denotes the minimum distance of c the codes are obtained by concatenating reed solomon codes with hadamard codes

on ideal lattices and learning with errors over rings the learning with errors lwe problem is to distinguish random linear equations which have been perturbed by a small amount of noise from truly uniform ones the problem has been shown to be as hard as worst case lattice problems and in recent years it has served as the foundation for a plethora of cryptographic applications unfortunately these applications are rather inefficient due to an inherent quadratic overhead in the use of lwe a main open question was whether lwe and its applications could be made truly efficient by exploiting extra algebraic structure as was done for lattice based hash functions and related primitives we resolve this question in the affirmative by introducing an algebraic variant of lwe called ring lwe and proving that it too enjoys very strong hardness guarantees specifically we show that the ring lwe distribution is pseudorandom assuming that worst case problems on ideal lattices are hard for polynomial time quantum algorithms applications include the first truly practical lattice based public key cryptosystem with an efficient security reduction moreover many of the other applications of lwe can be made much more efficient through the use of ring lwe

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

generators and irreducible polynomials over finite fields weil s character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field an application to the distribution of irreducible polynomials is given which confirms an asymptotic version of a conjecture of hansen mullen

on the hardness of the decoding and the minimum distance problems for rank codes we give a randomized reduction for the rank syndrome decoding problem and rank minimum distance problem for rank codes over extension fields our results are based on embedding linear codes in the hamming space into linear codes over an extension field equipped with the rank metric we prove that if any of the previous problems for the rank metric is in zpp rp cap corp then we would have np zpp we also give complexity results for the respective rank metric approximation problems

eigenprism inference for high dimensional signal to noise ratios consider the following three important problems in statistical inference namely constructing confidence intervals for 1 the error of a high dimensional p n regression estimator 2 the linear regression noise level and 3 the genetic signal to noise ratio of a continuous valued trait related to the heritability all three problems turn out to be closely related to the little studied problem of performing inference on the formula see text norm of the signal in high dimensional linear regression we derive a novel procedure for this which is asymptotically correct when the covariates are multivariate gaussian and produces valid confidence intervals in finite samples as well the procedure called eigenprism is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level we investigate the width of the eigenprism confidence intervals including a comparison with a bayesian setting in which our interval is just 5 wider than the bayes credible interval we are then able to unify the three aforementioned problems by showing that the eigenprism procedure with only minor modifications is able to make important contributions to all three we also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate gaussian covariates finally we apply eigenprism to a genetic dataset to estimate the genetic signal to noise ratio for a number of continuous phenotypes

maximal ferrers diagram codes constructions and genericity considerations this paper investigates the construction of rank metric codes with specified ferrers diagram shapes these codes play a role in the multilevel construction for subspace codes a conjecture from 2009 provides an upper bound for the dimension of a rank metric code with given specified ferrers diagram shape and rank distance while the conjecture in its generality is wide open several cases have been established in the literature this paper contributes further cases of ferrers diagrams and ranks for which the conjecture holds true in addition the proportion of maximal ferrers diagram codes within the space of all rank metric codes with the same shape and dimension is investigated special attention is being paid to mrd codes it is shown that for growing field size the limiting proportion depends highly on the ferrers diagram for instance for m times 2 mrd codes with rank 2 this limiting proportion is close to 1 e

communication over finite field matrix channels this paper is motivated by the problem of error control in network coding when errors are introduced in a random fashion rather than chosen by an adversary an additive multiplicative matrix channel is considered as a model for random network coding the model assumes that n packets of length m are transmitted over the network and up to t erroneous packets are randomly chosen and injected into the network upper and lower bounds on capacity are obtained for any channel parameters and asymptotic expressions are provided in the limit of large field or matrix size a simple coding scheme is presented that achieves capacity in both limiting cases the scheme has decoding complexity o n 2 m and a probability of error that decreases exponentially both in the packet length and in the field size in bits extensions of these results for coherent network coding are also presented

a flexible framework for hypothesis testing in high dimensions hypothesis testing in the linear regression model is a fundamental statistical problem we consider linear regression in the high dimensional regime where the number of parameters exceeds the number of samples p n to make informative inference we assume that the model is approximately sparse i e the effect of covariates on the response can be well approximated by conditioning on a relatively small number of covariates whose identities are unknown we develop a framework for testing very general hypotheses regarding the model parameters our framework encompasses testing whether the parameter lies in a convex cone testing the signal strength and testing arbitrary functionals of the parameter we show that the procedure proposed controls the type i error and we also analyse the power of the procedure our numerical experiments confirm our theoretical findings and demonstrate that we control the false positive rate type i error near the nominal level and have high power by duality between hypotheses testing and confidence intervals the framework proposed can be used to obtain valid confidence intervals for various functionals of the model parameters for linear functionals the length of confidence intervals is shown to be minimax rate optimal

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

some gabidulin codes cannot be list decoded efficiently at any radius gabidulin codes can be seen as the rank metric equivalent of reed solomon codes it was recently proved using subspace polynomials that gabidulin codes cannot be list decoded beyond the so called johnson radius in another result cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials in this paper these subspace codes are used to prove two bounds on the list size in decoding certain gabidulin codes the first bound is an existential one showing that exponentially sized lists exist for codes with specific parameters the second bound presents exponentially sized lists explicitly for a different set of parameters both bounds rule out the possibility of efficiently list decoding several families of gabidulin codes for any radius beyond half the minimum distance such a result was known so far only for non linear rank metric codes and not for gabidulin codes using a standard operation called lifting identical results also follow for an important class of constant dimension subspace codes

optimal estimation of genetic relatedness in high dimensional linear models estimating the genetic relatedness between two traits based on the genome wide association data is an important problem in genetics research in the framework of high dimensional linear models we

on asymptotics of eigenvectors of large sample covariance matrix let x ij i j be a double array of i i d complex random variables with ex 11 0 e x 11 2 1 and e x 11 4 and let an 1 n t 1 2 n x n x n t 1 2 n where t 1 2 n is the square root of a nonnegative definite matrix t n and x n is the n x n matrix of the upper left comer of the double array the matrix an can be considered as a sample covariance matrix of an i i d sample from a population with mean zero and covariance matrix t n or as a multivariate f matrix if t n is the inverse of another sample covariance matrix to investigate the limiting behavior of the eigenvectors of an a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights moreover if x ij and t n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of an are proved to have gaussian limits which suggests that the eigenvector matrix of an is nearly haar distributed when t n is a multiple of the identity matrix an easy consequence for a wishart matrix

complete abelian integrals for polynomials whose generic fiber is biholomorphic to c abstract let h be a polynomial of degree m 1 on c 2 such that its generic fiber is biholomorphic to c and let be an arbitrary polynomial 1 form of degree n on c 2 we give an upper bound depending only on m and n for the number of isolated zeros of the complete abelian integral defined by h and

variance estimation in high dimensional linear models the residual variance and the proportion of explained variation are important quantities in many statistical models and model fitting procedures they play an important role in regression diagnostics and model selection procedures as well as in determining the performance limits in many problems in this paper we propose new method of moments based estimators for the residual variance the proportion of explained variation and other related quantities such as the l2 signal strength the proposed estimators are consistent and asymptotically normal in high dimensional linear models with gaussian predictors and errors where the number of predictors d is proportional to the number of observations n in fact consistency holds even in settings where d n existing results on residual variance estimation in high dimensional linear models depend on sparsity in the underlying signal our results require no sparsity assumptions and imply that the residual variance and the proportion of explained variation can be consistently estimated even when d n and the underlying signal itself is nonestimable numerical work suggests that some of our distributional assumptions may be relaxed a real data analysis involving gene expression data and single nucleotide polymorphism data illustrates the performance of the proposed methods