subspace polynomials and limits to list decoding of reed solomon codes we show combinatorial limitations on efficient list decoding of reed solomon codes beyond the johnson guraswami sudan bounds in particular we show that for arbitrarily large fields fn fn n for any 0 1 and k n 1 existence there exists a received word wn fn fn that agrees with a super polynomial number of distinct degree k polynomials on n points each 2 explicit there exists a polynomial time constructible received word w n fn fn that agrees with a superpolynomial number of distinct degree k polynomials on 2 log n k points each in both cases our results improve upon the previous state of the art which was n points of agreement for the existence case proved by justesen and hoholdt and 2n points of agreement for the explicit case proved by guruswami and rudra furthermore for close to 1 our bound approaches the guruswami sudan bound which is n k and implies limitations on extending their efficient reed solomon list decoding algorithm to larger decoding radius our proof is based on some remarkable properties of sub space polynomials using similar ideas we then present a family of low rate codes that are efficiently list decodable beyond the johnson bound this leads to an optimal list decoding algorithm for the family of matrix codes

optimal matching with minimal deviation from fine balance in a study of obesity and surgical outcomes in multivariate matching fine balance constrains the marginal distributions of a nominal variable in treated and matched control groups to be identical without constraining who is matched to whom in this way a fine balance constraint can balance a nominal variable with many levels while focusing efforts on other more important variables when pairing individuals to minimize the total covariate distance within pairs fine balance is not always possible that is it is a constraint on an optimization problem but the constraint is not always feasible we propose a new algorithm which returns a minimum distance finely balanced match when one is feasible and otherwise minimizes the total distance among all matched samples that minimize the deviation from fine balance perhaps we can come very close to fine balance when fine balance is not attainable moreover in any event because our algorithm is guaranteed to come as close as possible to fine balance the investigator may perform one match and on that basis judge whether the best attainable balance is adequate or not we also show how to incorporate an additional constraint the algorithm is implemented in two similar ways first as an optimal assignment problem with an augmented distance matrix second as a minimum cost flow problem in a network the case of knee surgery in the obesity and surgical outcomes study motivated the development of this algorithm and is used as an illustration in that example two of 47 hospitals had too few nonobese patients to permit fine balance for the nominal variable with 47 levels representing the hospital but our new algorithm came very close to fine balance moreover in that example there was a shortage of nonobese diabetic patients and incorporation of an additional constraint forced the match to include all of these nonobese diabetic patients thereby coming as close as possible to balance for this important but recalcitrant covariate

additive rank metric codes we give an infinite family of maximum rank distance mrd codes which covers properly the largest known linear mrd code family our family contains infinite families of non linear mrd codes which are the first non linear examples for most of the parameters we also give explicit examples and a table that demonstrates the proportion of linear and non linear families for some small parameters

galois theory and linear algebra abstract let k be a field admitting a galois extension l of degree n with galois group g artin s lemma on the independence of characters implies that the algebra of k linear endomorphisms of l is identical with the set of l linear combinations of the elements of g this paper examines some consequences of this description of endomorphisms we provide a characterization of the rank 1 endomorphisms and describe the matrix theoretic trace of an endomorphism in terms of the field theoretic trace we also investigate in greater detail those endomorphisms annihilating a k subspace in the case when g is cyclic

exact nonparametric inference when doses are measured with random errors studies that estimate the effects of exposure to a possibly harmful agent often compare exposed subjects who received varied doses with matched controls who received zero dose if the doses are measured with error then one may wish to use the fallible doses to estimate a linear relationship between the unobserved true dose and the observed response if one is willing to assume that the dose errors for exposed subjects are symmetrically distributed about 0 that is the dose errors are pure errors and not say systematic underreporting of exposure then the presence of zero dose controls is all that is needed to obtain exact distribution free confidence intervals and tests and consistent point estimates the method is simpler for matched pairs than for matched sets with two or more matched subjects and it is illustrated using two studies one of each kind with matched pairs as in this first example the method uses wilcoxon s signed rank test as the basis for inference when there are several zero dos

on the classification of hyperovals abstract a hyperoval in the projective plane p 2 f q is a set of q 2 points no three of which are collinear hyperovals have been studied extensively since the 1950s with the ultimate goal of establishing a complete classification it is well known that hyperovals in p 2 f q are in one to one correspondence to polynomials with certain properties called o polynomials of f q we classify o polynomials of f q of degree less than 1 2 q 1 4 as a corollary we obtain a complete classification of exceptional o polynomials namely polynomials over f q that are o polynomials of infinitely many extensions of f q

commutative semifields and symplectic spreads abstract commutative semifields are constructed by using their relationship with symplectic spreads the number of pairwise nonisomorphic commutative semifield planes of even order n obtained in this manner is not bounded above by any polynomial in n the number previously known for any n was less than log n

several classes of optimal ferrers diagram rank metric codes abstract four constructions for ferrers diagram rank metric fdrm codes are presented the first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes by introducing restricted gabidulin codes the second construction is presented which unifies many known constructions for fdrm codes the third and fourth constructions are based on two different ways to represent elements of a finite field f q m vector representation and matrix representation each of these constructions produces optimal codes with different diagrams and parameters

exceptional scatteredness in prime degree abstract let q be an odd prime power and n an integer let l f q n x be a q linearized t scattered polynomial of linearized degree r let d max t r be an odd prime number in this paper we show that under these assumptions it follows that l x our technique involves a galois theoretical characterization of t scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field f q

class groups of number fields numerical heuristics extending previous work of h w lenstra jr and the first author we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four given the signature and the galois group of the galois closure the theoretical justifications for these conjectures will appear elsewhere but the agreement with the existing tables is quite good 1 introduction and notations in 3 h w lenstra jr and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree in a forthcoming paper 4 we shall show that this technique can be extended to a much wider class of number fields and also to relative extensions the aim of the present paper is to rapidly make available the numerical conjec tures obtained for people not really interested in our heuristic reasoning or not wanting to wait for 4 to appear hence apart from a total lack of justifications for the conjectures that we present this paper is essentially self contained the plan is as follows in the rest of this section we present the notations used in the sequel some of them being nonstandard and in general differing from the notations of 3 we urge the reader to read the notations carefully before applying the conjectures in the next section we present templates for the subsequent conjectures and then the conjectures themselves illustrated by numerical examples first for their own sake and second as a double check for the reader to understand the templates these conjectures are given for all types of fields of degree less than or equal to four in the final section we comment on the consistency of the conjectures with existing tables which is quite good combinatorial notations if x is a set xi denotes its cardinality

globally efficient non parametric inference of average treatment effects by empirical balancing calibration weighting the estimation of average treatment effects based on observational data is extremely important in practice and has been studied by generations of statisticians under different frameworks existing globally efficient estimators require non parametric estimation of a propensity score function an outcome regression function or both but their performance can be poor in practical sample sizes without explicitly estimating either functions we consider a wide class calibration weights constructed to attain an exact three way balance of the moments of observed covariates among the treated the control and the combined group the wide class includes exponential tilting empirical likelihood and generalized regression as important special cases and extends survey calibration estimators to different statistical problems and with important distinctions global semiparametric efficiency for the estimation of average treatment effects is established for this general class of calibration estimators the results show that efficiency can be achieved by solely balancing the covariate distributions without resorting to direct estimation of propensity score or outcome regression function we also propose a consistent estimator for the efficient asymptotic variance which does not involve additional functional estimation of either the propensity score or the outcome regression functions the proposed variance estimator outperforms existing estimators that require a direct approximation of the efficient influence function

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

on the use of propensity scores in principal causal effect estimation we examine the practicality of propensity score methods for estimating causal treatment effects conditional on intermediate posttreatment outcomes principal effects in the context of randomized experiments in particular we focus on the sensitivity of principal causal effect estimates to violation of principal ignorability which is the primary assumption that underlies the use of propensity score methods to estimate principal effects under principal ignorability pi principal strata membership is conditionally independent of the potential outcome under control given the pre treatment covariates i e there are no differences in the potential outcomes under control across principal strata given the observed pretreatment covariates under this assumption principal scores modeling principal strata membership can be estimated based solely on the observed covariates and used to predict strata membership and estimate principal effects while this assumption underlies the use of propensity scores in this setting sensitivity to violations of it has not been studied rigorously in this paper we explicitly define pi using the outcome model although we do not actually use this outcome model in estimating principal scores and systematically examine how deviations from the assumption affect estimates including how the strength of association between principal stratum membership and covariates modifies the performance we find that when pi is violated very strong covariate predictors of stratum membership are needed to yield accurate estimates of principal effects

isometries for rank distance and permutation group of gabidulin codes the rank distance was introduced by e m gabidulin see probl pered inform vol 21 p 1 12 1985 he determined an upper bound for the minimum rank distance of a code moreover he constructed a class of codes which meet this bound the so called gabidulin codes we first characterize the linear and semilinear isometries for the rank distance then we determine the isometry group and the permutation group of gabidulin codes of any length we give a characterization of equivalent gabidulin codes finally we prove that the number of equivalence classes of gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in gf p sup m under some particular relations

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

systematic encoders for generalized gabidulin codes and the q analogue of cauchy matrices abstract we characterize the generator matrix in standard form of generalized gabidulin codes the parametrization we get for the non systematic part of this matrix coincides with the q analogue of generalized cauchy matrices leading to the definition of q cauchy matrices these matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized gabidulin codes whose generator matrix is a structured matrix in particular as an application we construct gabidulin codes whose generator matrix is the concatenation of an identity block and a toeplitz hankel matrix in addition our results allow to give a new efficient criterion to verify whether a rank metric code of dimension k and length n is a generalized gabidulin code this criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires o m f k n field operations where f k n is the cost of computing the reduced row echelon form of a k n matrix moreover we also provide a characterization of the generator matrix in standard form of general mrd codes

galois extensions and subspaces of alternating bilinear forms with special rank properties abstract let k be a field admitting a cyclic galois extension of degree n the main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension n over k we show that this space of forms is the direct sum of n 1 2 subspaces each of dimension n and the non zero elements in each subspace have constant rank defined in terms of the orders of the galois automorphisms furthermore if ordered correctly for each integer k lying between 1 and n 1 2 the rank of any non zero element in the sum of the first k subspaces is at most n 2 k 1 slightly less sharp similar results hold for cyclic extensions of even degree

estimating propensity scores with missing covariate data using general location mixture models in many observational studies analysts estimate causal effects using propensity scores e g by matching sub classifying or inverse probability weighting based on the scores estimation of propensity scores is complicated when some values of the covariates are missing analysts can use multiple imputation to create completed data sets from which propensity scores can be estimated we propose a general location mixture model for imputations that assumes that the control units are a latent mixture of i units whose covariates are drawn from the same distributions as the treated units covariates and ii units whose covariates are drawn from different distributions this formulation reduces the influence of control units outside the treated units region of the covariate space on the estimation of parameters in the imputation model which can result in more plausible imputations in turn this can result in more reliable estimates of propensity scores and better balance in the true covariate distributions when matching or sub classifying we illustrate the benefits of the latent class modeling approach with simulations and with an observational study of the effect of breast feeding on children s cognitive abilities

vector network coding based on subspace codes outperforms scalar linear network coding this paper considers vector network coding solutions based on rank metric codes and subspace codes the main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same multicast network the multicast networks considered in this paper have one source with h messages and the vector solution is over a field of size q with vectors of length t for a given network let the smallest field size for which the network has a scalar linear solution be q s then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be q s q t in this contribution the achieved gap is q h 2 t 2 h o t for any q geq 2 and any even h geq 4 if h geq 5 is odd then the achieved gap of the alphabet size is q h 3 t 2 h 1 o t previously only a gap of size one had been shown for networks with a very large number of messages these results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear network coding solution for multicast networks for three messages we also show an advantage of vector network coding while for two messages the problem remains open several networks are considered all of them are generalizations and modifications of the well known combination networks the vector network codes that are used as solutions for those networks are based on subspace codes particularly subspace codes obtained from rank metric codes some of these codes form a new family of subspace codes which poses a new research problem

substantial gains in bias reduction from matching with a variable number of controls summary in observational studies that match several controls to each treated subject substantially greater bias reduction is possible if the number of controls is not fixed but rather is allowed to vary from one matched set to another in certain cases matching with a fixed number of controls may remove only 50 of the bias in a covariate whereas matching with a variable number of controls may remove 90 of the bias even though both control groups have the same number of controls in total an example of matching in a study of surgical mortality is discussed in detail

the 4 class ranks of quadratic fields let k be a quadratic extension field of the rational numbers q let c k be the 2 class group of k in the narrow sense it is a classical result that rank c k t l where t is the number of primes that ramify in k q now let c ai a e ck and let r k denote the 4 class rank of k in the narrow sense i e rk rank kc z dimv2 ttc z t 41 k kj here f 2 is the finite field with two elements and 2 4 ck c k is an elementary abelian 2 group which we are viewing as a vector space o v e r f 2 given a quadratic field k one can compute r k by computing the rank over f2 of a certain matrix of legendre symbols cf 11 now assume k is imaginary quadratic so k q 1 mm where m is a square free positive integer for each positive integer t each nonnegative integer e and each positive real number x we define

on the existence of fields governing the 2 invariants of the classgroup of as varies this paper formulates general conjectures relating the structure of the 2 classgroup c2 dp associated to q fdp to the splitting of the ideal p in certain algebraic number fields here d i 2 mod 4 is a fixed integer and p varies over primes the conjectures assert that there exists an algebraic number field qj d such that the artin symbol q2 d q p determines the first j 2 invariants of the group c2 dp i e it determines c2 dp c2 dp 2j these conjectures imply that the set of primes p for which c2 dp has a given set of 2 invariants has a natural density which is a rational number existing results prove the conjectures wheneverj i or 2 and also for an infinite set of d withj 3 the smallest open case isj 3 d 21 this paper presents evidence concerning these conjectures for d 4 8 and 21 numerical evidence is given that q23 21 exists and that natural densities which are rational numbers exist for the sets of primes with 2 h dp for d 4 and 8 for i j 7 a search for the hypothetical field 04 4 ruled out the simplest candidate fields 04 4 is not a normal extension of q of degree 16 ramifying only at 2

combining propensity score matching with additional adjustments for prognostic covariates abstract propensity score matching refers to a class of multivariate methods used in comparative studies to construct treated and matched control samples that have similar distributions on many covariates this matching is the observational study analog of randomization in ideal experiments but is far less complete as it can only balance the distribution of observed covariates whereas randomization balances the distribution of all covariates both observed and unobserved an important feature of propensity score matching is that it can be easily combined with model based regression adjustments or with matching on a subset of special prognostic covariates or combinations of prognostic covariates that have been identified as being especially predictive of the outcome variables we extend earlier results by developing approximations for the distributions of covariates in matched samples created with linear propensity score methods for the practically important situation where matching uses both the estimat

on the 4 rank of class groups of quadratic number fields we prove that the 4 rank of class groups of quadratic number fields behaves as predicted in an extension due to gerth of the cohen lenstra heuristics

maximum rank distance codes as space time codes the critical design criterion for space time codes in asymptotically good channels is the minimum rank between codeword pairs rank codes are a two dimensional matrix code construction where by the rank is the metric of merit we look at the application of rank codes to space time code design in particular we provide construction methods of full rank codes over different complex signal constellations for arbitrary numbers of antennas and codeword periods we also derive a singleton type bound on the rate of a code for the rank metric and we show that rank codes satisfy this bound with equality

on the number of roots of some linearized polynomials abstract linearized polynomials appear in many different contexts such as rank metric codes cryptography and linear sets and the main issue regards the characterization of the number of roots from their coefficients results of this type have been already proved in 7 10 24 in this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form a x b 0 x q s b 1 x q s n b 2 x q s 2 n b t 1 x q s n t 1 f q n t x with gcd s n 1 also we characterize the number of roots of such polynomials directly from their coefficients dealing with matrices which are much smaller than the relative dickson matrices and the companion matrices used in the previous papers furthermore we develop a method to find explicitly the roots of a such polynomial by finding the roots of a q n polynomial finally as an application of the above results we present a family of linear sets of the projective line whose points have a small spectrum of possible weights containing most of the known families of scattered linear sets in particular we carefully study the linear sets in pg 1 q 6 presented in 9

double error correcting cyclic codes and absolutely irreducible polynomials over gf 2 codewords of weight 4 in certain cyclic codes of length n 2s 1 parameterized by an odd integer t can be related to zeros of certain projective plane curves gt x y z some families of these codes have been shown to have no codewords of weight 4 i e they are 2 error correcting codes but if the polynomials gt x y z are absolutely irreducible weil s theorem shows that the codes do have codewords of weight 4 for all integers s that are sufficiently large with respect to t here we prove that gt x y z is absolutely irreducible for all integers t 3 such that t 3 mod 4 and also for some values t 1 mod 4 these cases provide us with evidence for a conjecture that would classify all such codes in terms of their minimum distance the methods of proving absolute irreducibility involve bezout s theorem and may be of independent interest

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

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

impact of multiple matched controls on design sensitivity in observational studies summary in an observational study one treated subject may be matched for observed covariates to either one or several untreated controls the common motivation for using several controls rather than one is to increase the power of a test of no effect under the doubtful assumption that matching for observed covariates suffices to remove bias from nonrandom treatment assignment does the choice between one or several matched controls affect the sensitivity of conclusions to violations of this doubtful assumption with continuous responses it is known that reducing the heterogeneity of matched pair differences reduces sensitivity to unmeasured biases but increasing the sample size has a highly circumscribed effect on sensitivity to bias is the use of several controls rather than one analogous to a reduction in heterogeneity or to an increase in sample size the issue is examined for huber s m statistics including the t test the examination having three components an example asymptotic calculations using design sensitivity and a simulation use of multiple controls with continuous responses yields a nontrivial reduction in sensitivity to unmeasured biases an example looks at lead and cadmium in the blood of smokers from the 2008 national health and nutrition examination survey a by product of the discussion is a new result giving the design sensitivity for the permutation distribution of m statistics

a new family of mrd codes abstract we introduce a family of linear sets of pg 1 q 2 n arising from maximum scattered linear sets of pseudoregulus type of pg 3 q n for n 3 4 and for certain values of the parameters we show that these linear sets of pg 1 q 2 n are maximum scattered and they yield new mrd codes with parameters 6 6 q 5 for q 2 and with parameters 8 8 q 7 for q odd

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

subspaces of matrices with special rank properties abstract let k be a field and let m m n k denote the space of m n matrices over k we investigate properties of a subspace m of m m n k of dimension n m r 1 in which each non zero element of m has rank at least r and enumerate the number of elements of a given rank in m when k is finite we also provide an upper bound for the dimension of a constant rank r subspace of m m n k when k is finite and give non trivial examples to show that our bound is optimal in some cases we include a similar a bound for the maximum dimension of a constant rank subspace of skew symmetric matrices over a finite field

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

a characterization of optimal designs for observational studies summary an empirical investigation of the effects of a treatment is an observational study if it involves the comparison of treated and control groups that were not formed by randomization in such studies treated and control groups may differ systematically with respect to pretreatment measures or covariates and addressing these pre treatment differences is a central concern matching and subclassification are two standard methods of adjusting for observed pretreatment differences they may be used alone e g cochran 1968 or in conjunction with analytical or model based adjustments e g rubin 1973 1979 holford 1978 rosenbaum and rubin 1984 section 3 3 and rosenbaum 1987 1988a in particular rubin s simulation studies suggest that model based adjustments applied to matched or subclassified samples are more robust than model based adjustments applied to unmatched samples the purpose of this paper is to identify the structure of an optimal subclassification i e one in which the treated and control subjects in the same subclass are on average as similar as possible with respect to observed covariates it turns out that this structure is simple and intuitive while adjustments may control imbalances in observed covariates they cannot

error correcting codes in projective spaces via rank metric codes and ferrers diagrams coding in the projective space has received recently a lot of attention due to its application in network coding reduced row echelon form of the linear subspaces and ferrers diagram can play a key role for solving coding problems in the projective space in this paper we propose a method to design error correcting codes in the projective space we use a multilevel approach to design our codes first we select a constant weight code each codeword defines a skeleton of a basis for a subspace in reduced row echelon form this skeleton contains a ferrers diagram on which we design a rank metric code each such rank metric code is lifted to a constant dimension code the union of these codes is our final constant dimension code in particular the codes constructed recently by koetter and kschischang are a subset of our codes the rank metric codes used for this construction form a new class of rank metric codes we present a decoding algorithm to the constructed codes in the projective space the efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank metric codes finally we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension

counting dihedral and quaternionic extensions we give asymptotic formulas for the number of biquadratic extensions of that admit a quadratic extension which is a galois extension of with a prescribed galois group for example with a galois group isomorphic to the quaternionic group our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the siegel walfisz theorem and the double oscillations theorem

asymptotic separability in sensitivity analysis summary in an observational study in which each treated subject is matched to several untreated controls by using observed pretreatment covariates a sensitivity analysis asks how hidden biases due to unobserved covariates might alter the conclusions the bounds required for a sensitivity analysis are the solution to an optimization problem in general this optimization problem is not separable in the sense that one cannot find the needed optimum by performing a separate optimization in each matched set and combining the results we show however that this optimization problem is asymptotically separable so that when there are many matched sets a separate optimization may be performed in each matched set and the results combined to yield the correct optimum with negligible error this is true when the wilcoxon rank sum test or the hodgeslehmann aligned rank test is applied in matching with multiple controls numerical calculations show that the asymptotic approximation performs well with as few as 10 matched sets in the case of the rank sum test a table is given containing the separable solution with this table only simple arithmetic is required to conduct the sensitivity analysis the method also supplies estimates such as the hodges lehmann estimate and confidence intervals associated with rank tests the method is illustrated in a study of dropping out of us high schools and the effects on cognitive test scores

interval estimation for treatment effects using propensity score matching in causal studies without random assignment of treatment causal effects can be estimated using matched treated and control samples where matches are obtained using estimated propensity scores propensity score matching can reduce bias in treatment effect estimators in cases where the matched samples have overlapping covariate distributions despite its application in many applied problems there is no universally employed approach to interval estimation when using propensity score matching in this article we present and evaluate approaches to interval estimation when using propensity score matching

reducing bias in observational studies using subclassification on the propensity score abstract the propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates previous theoretical arguments have shown that subclassification on the propensity score will balance all observed covariates subclassification on an estimated propensity score is illustrated using observational data on treatments for coronary artery disease five subclasses defined by the estimated propensity score are constructed that balance 74 covariates and thereby provide estimates of treatment effects using direct adjustment these subclasses are applied within sub populations and model based adjustments are then used to provide estimates of treatment effects within these sub populations two appendixes address theoretical issues related to the application the effectiveness of subclassification on the propensity score in removing bias and balancing properties of propensity scores with incomplete data

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

exceptional scattered polynomials abstract let f be an f q linear function over f q n if the f q subspace u x q t f x x f q n defines a maximum scattered linear set then we call f a scattered polynomial of index t as these polynomials appear to be very rare it is natural to look for some classification of them we say a function f is an exceptional scattered polynomial of index t if the subspace u associated with f defines a maximum scattered linear set in pg 1 q m n for infinitely many m our main results are the classifications of exceptional scattered monic polynomials of index 0 for q 5 and of index 1 the strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the hasse weil theorem to derive contradictions

identifiers for mrd codes abstract for any admissible value of the parameters n and k there exist n k maximum rank distance f q linear codes indeed it can be shown that if field extensions large enough are considered almost all rank metric codes are mrd on the other hand very few families up to equivalence of such codes are currently known in the present paper we study some invariants of mrd codes and evaluate their value for the known families providing a new characterization of generalized twisted gabidulin codes

on the list decodability of rank metric codes let k n m in mathbb z be integers such that k leq n leq m let mathrm g n k in mathbb f q m n be a delsarte gabidulin code recently wachter zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius tau providing tau is large enough this achievement essentially relies on proving a lower bound for the list size of some specific words in mathbb f q m n some years later raviv and wachter zeh improved this bound in a special case i e when n mid m as a consequence they were able to detect infinite families of delsarte gabidulin codes that cannot be efficiently list decoded at all in this article we determine similar lower bounds for maximum rank distance codes belonging to a wider class of examples containing generalized gabidulin codes generalized twisted gabidulin codes and examples recently described by trombetti and zhou by exploiting arguments such as those used by raviv and wachter zeh when n mid m we also show infinite families of generalized gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to left lfloor frac d 1 2 right rfloor 1 where d is its minimum distance nonetheless in all the examples belonging to above mentioned class we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to left lfloor frac d 1 2 right rfloor 2 where d is its minimum distance in particular this leads to show infinite families of gabidulin codes with underlying parameters not already covered by the result of raviv and wachter zeh having this decodability defect finally relying on the properties of a set of subspace trinomials recently presented by mcguire and mueller we are able to prove our main result that is any rank metric code of mathbb f q m n of order q kn with n dividing m such that 4n 3 is a square in mathbb z and containing mathrm g n 2 is not efficiently list decodable at some values of the radius tau

discrete optimization for interpretable study populations and randomization inference in an observational study of severe sepsis mortality abstractmotivated by an observational study of the effect of hospital ward versus intensive care unit admission on severe sepsis mortality we develop methods to address two common problems in observational studies 1 when there is a lack of covariate overlap between the treated and control groups how to define an interpretable study population wherein inference can be conducted without extrapolating with respect to important variables and 2 how to use randomization inference to form confidence intervals for the average treatment effect with binary outcomes our solution to problem 1 incorporates existing suggestions in the literature while yielding a study population that is easily understood in terms of the covariates themselves and can be solved using an efficient branch and bound algorithm we address problem 2 by solving a linear integer program to use the worst case variance of the average treatment effect among values for unobserved potential outcomes that are compatible with the null hy

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

nuclei and automorphism groups of generalized twisted gabidulin codes abstract generalized twisted gabidulin codes are one of the few known families of maximum rank metric codes over finite fields as a subset of m n matrices when m n the automorphism group of any generalized twisted gabidulin code has been completely determined by the authors in 20 in this paper we consider the same problem for m n under certain conditions on their parameters we determine their middle nuclei and right nuclei which are important invariants with respect to the equivalence for rank metric codes furthermore we also use them to derive necessary conditions on the automorphisms of generalized twisted gabidulin codes

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

proof of a conjecture on the sequence of exceptional numbers classifying cyclic codes and apn functions abstract we prove a conjecture that classifies exceptional numbers this conjecture arises in two different ways from cryptography and from coding theory an odd integer t 3 is said to be exceptional if f x x t is apn almost perfect nonlinear over f 2 n for infinitely many values of n equivalently t is exceptional if the binary cyclic code of length 2 n 1 with two zeros t has minimum distance 5 for infinitely many values of n the conjecture we prove states that every exceptional number has the form 2 i 1 or 4 i 2 i 1

optimal matching for observational studies abstract matching is a common method of adjustment in observational studies currently matched samples are constructed using greedy heuristics or stepwise procedures that produce in general suboptimal matchings with respect to a particular criterion a matched sample is suboptimal if it could be improved by changing the controls assigned to specific treated units that is if it could be improved with the data at hand here optimal matched samples are obtained using network flow theory in addition to providing optimal matched pair samples this approach yields optimal constructions for several statistical matching problems that have not been studied previously including the construction of matched samples with multiple controls with a variable number of controls and the construction of balanced matched samples that combine features of pair matching and frequency matching computational efficiency is discussed extensive use is made of ideas from two essentially disjoint literatures namely st

equivalence and characterizations of linear rank metric codes based on invariants abstract we show that the sequence of dimensions of the linear spaces generated by a given rank metric code together with itself under several applications of a field automorphism is an invariant for the whole equivalence class of the code the same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism these invariants give rise to easily computable criteria to check if two codes are inequivalent we derive some concrete values and bounds for these dimension sequences for some known families of rank metric codes namely gabidulin and generalized twisted gabidulin codes we then derive conditions on the length of the codes with respect to the field extension degree such that codes from different families cannot be equivalent furthermore we derive upper and lower bounds on the number of equivalence classes of gabidulin codes and twisted gabidulin codes improving a result of schmidt and zhou for a wider range of parameters in the end we use the aforementioned sequences to determine a characterization result for gabidulin codes

model based direct adjustment abstract direct adjustment or standardization applies population weights to subclass means in an effort to estimate population quantities from a sample that is not representative of the population direct adjustment has several attractive features but when there are many subclasses it can attach large weights to small quantities of data often in a fairly erratic manner in the extreme direct adjustment can attach infinite weight to nonexistent data a noticeable inconvenience in practice this article proposes a method of model based direct adjustment that preserves the attractive features of conventional direct adjustment while stabilizing the weights attached to small subclasses the sample mean and conventional direct adjustment are both special cases of model based direct adjustment under two different extreme models for the subclass specific selection probabilities the discussion of this method provides some insights into the behavior of true and estimated propensity scores the estimated scores

the parity of the period of the continued fraction of d we call a positive square free integer d special if d is not divisible by primes congruent to 3 mod 4 we show that the period of the expansion of ad in continued fractions is asymptotically more often odd than even when we restrict to special integers we note that this period is always even for a non special square free integer d it is well known that the above period is odd if and only if the negative pell equation x2 dy2 1 is solvable the latter problem is solvable if and only if the narrow and the ordinary class groups of q ad are equal in a prior work we fully described the asymptotics of the 4 ranks of those class groups here we get the first non trivial results about the asymptotic behavior of the 8 rank of the narrow class group for example we show that more than 76 of the quadratic fields q ad where d is special have the property that the 8 rank of the narrow class group is zero

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

vandermonde and wronskian matrices over division rings in l the first author initiated the study of vandermonde matrices with entries over a division ring k even in the case when k is a field the definition of a vandermonde matrix in l is already more general than that in the classical sense as it takes into account a given endomorphism s of k under the epilogue of l it was further pointed out that the results of that paper remain valid if in the definition of a vandermonde matrix one allows for a given s derivation d and uses the appropriate definition of the power functions in the s d setting such a generalization is worthwhile because in this setting the study of vandermonde matrices over k also encompasses the study of wronskian matrices with respect to d so results on vandermonde matrices may be used to study linear differential equations arising from the operator d more generally vandermonde matrices seem to be an important tool in studying the skew polynomial ring k t s d so a deeper understanding of vandermonde matrices should be important to the study of the arithmetic of a division ring the purpose of this paper is threefold first we shall develop under sec tion 2 the basic facts on skew polynomials in the s d setting define the evaluation of such polynomials and prove the all important product theorem 2 7 second we shall give a general computation of the rank of a vandermonde matrix with respect to s d as in l the computation is first reduced to the case when the elements a a used to build the vandermonde matrix are pairwise s d conjugate in this case the rank of the vandermonde matrix is computed by the dimension of a vector space over a certain division subring of k cf theorem 4 5 last under

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

effect modification and design sensitivity in observational studies in an observational study of treatment effects subjects are not randomly assigned to treatment or control so differing outcomes in treated and control groups may reflect a bias from nonrandom assignment rather than a treatment effect after adjusting for measured pretreatment covariates perhaps by matching a sensitivity analysis determines the magnitude of bias from an unmeasured covariate that would need to be present to alter the conclusions of the naive analysis that presumes adjustments eliminated all bias other things being equal larger effects tend to be less sensitive to bias than smaller effects effect modification is an interaction between a treatment and a pretreatment covariate controlled by matching so that the treatment effect is larger at some values of the covariate than at others in the presence of effect modification it is possible that results are less sensitive to bias in subgroups experiencing larger effects two cases are considered i an a priori grouping into a few categ

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

on the negative pell equation we give asymptotic upper and lower bounds for the number of squarefree d 0 d x such that the equation x 2 dy 2 1 is solvable these estimates as usual can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm 1 and give strong evidence in the direction of a conjecture due to p stevenhagen

size of regulators and consecutive square free numbers we prove that for almost all d the fundamental solution x 0 y 0 sqrt d of the associated pell equation x2 dy2 1 is greater than d1 749 we also show a strong link between this question and the error term in the asymptotic formula of the number of pairs of consecutive square free numbers

mrd codes arising from the trinomial xq xq3 cxq5 fq6 x abstract in 10 the existence of f q linear mrd codes of f q 6 6 with dimension 12 minimum distance 5 and left idealiser isomorphic to f q 6 defined by a trinomial of f q 6 x when q is odd and q 0 1 mod 5 has been proved in this paper we show that this family produces f q linear mrd codes of f q 6 6 with the same properties also in the remaining q odd cases but not in the q even case these mrd codes are not equivalent to the previously known mrd codes we also prove that the corresponding maximum scattered f q linear sets of pg 1 q 6 are not p l 2 q 6 equivalent to any previously known linear set

low degree planar polynomials over finite fields of characteristic two abstract planar functions are mappings from a finite field f q to itself with an extremal differential property such functions give rise to finite projective planes and other combinatorial objects there is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even we classify polynomials of degree at most q 1 4 that induce planar functions on f q by showing that such polynomials are precisely those in which the degree of every monomial is a power of two as a corollary we obtain a complete classification of exceptional planar polynomials namely polynomials over f q that induce planar functions on infinitely many extensions of f q the proof strategy is to study the number of f q rational points of an algebraic curve attached to a putative planar function our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions