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

generalized sphere packing bound kulkarni and kiyavash recently introduced a new method to establish upper bounds on the size of deletion correcting codes this method is based upon tools from hypergraph theory the deletion channel is represented by a hypergraph whose edges are the deletion balls or spheres so that a deletion correcting code becomes a matching in this hypergraph consequently a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear programming problem the problem of finding fractional transversals the method by kulkarni and kiyavash can be applied not only for the deletion channel but also for other error channels this paper studies this method in its most general setup first it is shown that if the error channel is regular and symmetric then the upper bound by this method coincides with the well known sphere packing bound and thus is called here the generalized sphere packing bound even though this bound is explicitly given by a linear programming problem finding its exact value may still be a challenging task the art of finding the exact upper bound or slightly weaker ones is the assignment of weights to the hypergraph s vertices in a way that they satisfy the constraints in the linear programming problem in order to simplify the complexity of the linear programming we present a technique based upon graph automorphisms that in many cases significantly reduces the number of variables and constraints in the problem we then apply this method on specific examples of error channels we start with the z channel and show how to exactly find the generalized sphere packing bound for this setup next studied is the nonbinary limited magnitude channel both for symmetric and asymmetric errors where we focus on the single error case we follow up on the deletion channel which was the original motivation of the work by kulkarni and kiyavash and show how to improve upon their upper bounds for single deletion correcting codes since the deletion and grain error channels have a similar structure for a single error we also improve upon the existing upper bounds on single grain error correcting codes finally we apply this method for projective spaces and find its generalized sphere packing bound for the single error case

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

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

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

skew feedback shift register synthesis and decoding interleaved gabidulin codes an efficient algorithm which synthesizes all shortest skew feedback shift registers defined in the paper generating l sequences of varying length over a field is derived and its correctness is proved it generalizes the berlekamp massey algorithm and some other algorithms and has time complexity o ln2 where n is the length of a longest sequence the proposed algorithm can be applied for efficiently solving the key equation when decoding interleaved or direct sum of gabidulin codes beyond half minimum distance those codes have many applications and as shown by kotter and kschischang can be used for random network coding

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

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

on the decoder error probability of bounded rank distance decoders for maximum rankdistance codes in this correspondence we first introduce the concept of elementary linear subspace which has similar properties to those of a set of coordinates we then use elementary linear subspaces to derive properties of maximum rank distance mrd codes that parallel those of maximum distance separable codes using these properties we show that for mrd codes with error correction capability the decoder error probability of bounded rank distance decoders decreases exponentially with based on the assumption that all errors with the same rank are equally likely

construction of network error correction codes in packet networks recently network error correction coding nec has been studied extensively several bounds in classical coding theory have been extended to nec especially the singleton bound in this paper following the research line using the extended global encoding kernels proposed by zhang in 2008 the refined singleton bound of nec can be proved more explicitly moreover we give a constructive proof of the attainability of this bound and indicate that the required field size for the existence of network maximum distance separable mds codes can become smaller further by this proof an algorithm is proposed to construct general linear network error correction codes including the linear network error correction mds codes finally we study the error correction capability of random linear nec motivated partly by the performance analysis of random linear network coding we evaluate the different failure probabilities defined in this paper in order to analyze the performance of random linear nec several upper bounds on these probabilities are obtained and they show that these probabilities will approach to zero as the size of the base field goes to infinity using these upper bounds we slightly improve on the probability mass function of the minimum distance of random linear network error correction codes in a paper by balli and colleagues as well as the upper bound on the field size required for the existence of linear network error correction codes with degradation at most d

iterative coding for network coding we consider communication over a noisy network under randomized linear network coding possible error mechanisms include node or link failures byzantine behavior of nodes or an overestimate of the network min cut building on the work of kotter and kschischang we introduce a systematic oblivious random channel model within this model codewords contain a header this is the systematic part the header effectively records the coefficients of the linear encoding functions thus simplifying the decoding task under this constraint errors are modeled as random low rank perturbations of the transmitted codeword we compute the capacity of this channel and we define an error correction scheme based on random sparse graphs and a low complexity decoding algorithm by optimizing over the code degree profile we show that this construction achieves the channel capacity in complexity which is jointly quadratic in the number of coded information bits and sublogarithmic in the error probability

on list decodability of random rank metric codes and subspace codes codes in rank metric have a wide range of applications to construct such codes with better list decoding performance explicitly it is of interest to investigate the listdecodability of random rank metric codes it is shown that if n m b is a constant then for every rank metric code in fm n q with rate r and list decoding radius must obey the gilbert varshamov bound that is r 1 1 b otherwise the list size can be exponential and hence no polynomial time list decoding is possible on the other hand for arbitrary 0 0 with e and being independent of each other with high probability a random rank metric code with rate r 1 1 b can be efficiently list decoded up to a fraction of rank errors with constant list size o 1 e we establish similar results for constant dimension subspace codes moreover we show that with high probability the list decoding radius of random fq linear rank metric codes also achieve the gilbert varshamov bound with constant list size o exp 1 e

conditions for ignoring the missing data mechanism in likelihood inferences for parameter subsets abstractfor likelihood based inferences from data with missing values models are generally needed for both the data and the missing data mechanism however modeling the mechanism can be challenging and parameters are often poorly identified rubin in 1976 showed that for likelihood and bayesian inference sufficient conditions for ignoring the missing data mechanism are a the missing data are missing at random mar in the sense that missingness does not depend on the missing values after conditioning on the observed data and b the parameters of the data model and the missingness mechanism are distinct that is there are no a priori ties via parameter space restrictions or prior distributions between these two sets of parameters these conditions are sufficient but not always necessary and they relate to the full vector of parameters of the data model we propose definitions of partially mar and ignorability for a subvector of the parameters of particular substantive interest for direct likel

linearity and complements in projective space abstract the projective space of order n over the finite field f q denoted here as p q n is the set of all subspaces of the vector space f q n the projective space can be endowed with distance function d s x y dim x dim y 2 dim x y which turns p q n into a metric space with this an n m d code c in projective space is a subset of p q 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 2 t 2 d this motivates new interest in such codes in this paper we examine the two fundamental concepts of complements and linear codes in the context of p q n these turn out to be considerably more involved than their classical counterparts these concepts are examined from two different points of view coding theory and lattice theory our results reveal a number of surprising phenomena pertaining to complements and linearity in p q n and gives rise to several interesting problems

natural finite element techniques for viscous fluid motion abstract the paper surveys recent work on fluid dynamics performed at the isd university of stuttgart it is in particular directed to a natural description of the flow phenomena and includes also a consideration of thermally coupled problems the derivation of the relevant finite element equations when referred to natural quantities is outlined and examples of application are given also presented is a discussion on the associated modern developments in numerical solutions techniques

on metrics for error correction in network coding the problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model for coherent network coding where knowledge of the network topology and network code is assumed at the source and destination nodes the error correction capability of an outer code is succinctly described by the rank metric as a consequence it is shown that universal network error correcting codes achieving the singleton bound can be easily constructed and efficiently decoded for noncoherent network coding where knowledge of the network topology and network code is not assumed the error correction capability of a subspace code is given exactly by a new metric called the injection metric which is closely related to but different than the subspace metric of koumltter and kschischang in particular in the case of a non constant dimension code the decoder associated with the injection metric is shown to correct more errors then a minimum subspace distance decoder all of these results are based on a general approach to adversarial error correction which could be useful for other adversarial channels beyond network coding

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

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

on the sparseness of certain linear mrd codes abstract we determine the proportion of linear 3 3 3 mrd codes over f q within the space of all 3 dimensional 3 3 rank metric codes over the same field this shows that for these parameters linear mrd codes are sparse in the sense that the proportion tends to 0 as q this is so far the only parameter case for which mrd codes are known to be sparse the computation is accomplished by reducing the space of all such rank metric codes to a space of specific bases and subsequently making use of a result by menichetti 1973 on 3 dimensional semifields

packing and covering properties of subspace codes for error control in random linear network coding codes in the projective space and codes in the grassmannian over a finite field referred to as subspace codes and constant dimension codes cdcs respectively have been proposed for error control in random linear network coding for subspace codes and cdcs a subspace metric was introduced to correct both errors and erasures and an injection metric was proposed to correct adversarial errors in this paper we investigate the packing and covering properties of subspace codes with both metrics we first determine some fundamental geometric properties of the projective space with both metrics using these properties we then derive bounds on the cardinalities of packing and covering subspace codes and determine the asymptotic rates of optimal packing and optimal covering subspace codes with both metrics our results not only provide guiding principles for the code design for error control in random linear network coding but also illustrate the difference between the two metrics from a geometric perspective in particular our results show that optimal packing cdcs are optimal packing subspace codes up to a scalar for both metrics if and only if their dimension is half of their length up to rounding in this case cdcs suffer from only limited rate loss as opposed to subspace codes with the same minimum distance we also show that optimal covering cdcs can be used to construct asymptotically optimal covering subspace codes with the injection metric only

a bayesian justification of cox s partial likelihood summary in this paper we establish both naive and formal bayesian justifications of cox s 1975 partial likelihood and its various modifications we extend the original work of kalbfieisch 1978 who showed that the partial likelihood is a limiting marginal posterior under noninformative priors for baseline hazards we extend the result to scenarios with timedependent covariates and time varying regression parameters we establish results for continuous time as well as grouped survival data in addition we present a bayesian justification of a modified partial likelihood for handling ties we also present tools for simplification of the gibbs sampling algorithm for implementing partial likelihood based bayesian inference in various practical applications

thermomechanical response of solids at high strains natural approach abstract the paper develops in pursuance of 4 the theoretical framework appertaining to coupled thermomechanical deformations of solids subject to large as well as inelastic deformations the essential feature of the analysis is a consistent natural formulation which encompasses also all thermodynamic aspects on the other hand it must be admitted that current concepts on mechanical constitutive assumptions 5 6 are not sufficiently sophisticated to permit a precise generalisation for the purpose of incorporating thermomechanical effects and delivering a fully compatible theory of the coupled problem uniqueness and bifurcation of the solutions as well as stable postcritical deformation paths are also considered in this account furthermore algorithms for the solution of the nonlinear algebraic equations representing the discretised problem and in particular quasi newton techniques are discussed the paper concludes with a number of applications of technical relevance

rank minimization over finite fields fundamental limits and coding theoretic interpretations this paper establishes information theoretic limits for estimating a finite field low rank matrix given random linear measurements of it these linear measurements are obtained by taking inner products of the low rank matrix with random sensing matrices necessary and sufficient conditions on the number of measurements required are provided it is shown that these conditions are sharp and the minimum rank decoder is asymptotically optimal the reliability function of this decoder is also derived by appealing to de caen s lower bound on the probability of a union the sufficient condition also holds when the sensing matrices are sparse a scenario that may be amenable to efficient decoding more precisely it is shown that if the n n sensing matrices contain on average nlog n entries the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field analogies are drawn between the aforementioned results and rank metric codes in the coding theory literature in fact we are also strongly motivated by understanding when minimum rank distance decoding of random rank metric codes succeeds to this end we derive minimum distance properties of equiprobable and sparse rank metric codes these distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one

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 randomized linear network codes and their error correction capabilities randomized linear network code for single source multicast was introduced and analyzed in ho et al ieee transactions on information theory october 2006 where the main results are upper bounds for the failure probability of the code in this paper these bounds are improved and tightness of the new bounds is studied by analyzing the limiting behavior of the failure probability as the field size goes to infinity in the linear random coding setting for single source multicast the minimum distance of the code defined in zhang ieee transactions on information theory january 2008 is a random variable taking nonnegative integer values that satisfy the inequality in the singleton bound recently established in yeung and cai communications in information and systems 2006 for network error correction codes we derive a bound on the probability mass function of the minimum distance of the random linear network code based on our improved upper bounds for the failure probability codes having the highest possible minimum distance in the singleton bound are called maximum distance separable mds the bound on the field size required for the existence of mds codes reported in zhang ieee transactions on information theory january 2008 and matsumoto arxiv cs it 0610121 oct 2006 suggests that such codes exist only when field size is large define the degradation of a code as the difference between the highest possible minimum distance in the singleton bound and the actual minimum distance of the code the bound for the probability mass function of the minimum distance leads to a bound on the field size required for the existence of network error correction codes with a given maximum degradation the result shows that allowing minor degradation reduces the field size required dramatically

error correcting regenerating and locally repairable codes via rank metric codes this paper presents and analyzes a novel concatenated coding scheme for enabling error resilience in two distributed storage settings one being storage using existing regenerating codes and the second being storage using locally repairable codes the concatenated coding scheme brings together a maximum rank distance code as an outer code and either a globally regenerating or a locally repairable code as an inner code in addition error resilience for combination of locally repairable codes with regenerating codes is considered this concatenated coding system is designed to handle two different types of adversarial errors the first type includes an adversary that can replace the content of an affected node only once while the second type studies an adversary that is capable of polluting data an unbounded number of times the paper establishes an upper bound on the resilience capacity for a locally repairable code this paper also proves that the proposed concatenated coding approach attains the upper bound on the resilience capacity in the presence of the first type of adversary for both minimum storage regenerating codes and locally repairable codes further this paper presents mechanisms that combine the presented concatenated coding scheme with subspace signatures to achieve error resilience for the second type of errors

convolutional codes with maximum column sum rank for network streaming the column hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels we introduce a metric known as the column sum rank that parallels the column hamming distance when streaming over a network with link failures we prove the rank analogues of several known column hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory our construction involves finding a class of super regular matrices that preserve this property after multiplication with non singular block diagonal matrices in the ground field

new lmrd code bounds for constant dimension codes and improved constructions we prove upper bounds for the cardinality of constant dimension codes cdc which contain a lifted maximum rank distance lmrd code as a subset thereby we cover all parameters fulfilling k lt 3d 2 where k is the codeword dimension and d is the minimum subspace distance the proofs of the bounds additionally show that an lmrd code l can be unioned with a cdc c of fitting parameters without violating the subspace distance condition iff each codeword of c intersects the special subspace of l in at least dimension d 2 this connection is used to find the new largest and sometimes bound achieving cdcs for small parameters

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

constructions for optimal ferrers diagram rank metric codes optimal rank metric codes in ferrers diagrams can be used to construct good subspace codes such codes consist of matrices having zeros at certain fixed positions this paper generalizes the known constructions for ferrers diagram rank metric fdrm codes via a criterion for linear maximum rank distance mrd codes an explicit construction for a class of systematic mrd codes is presented which is used to produce new optimal fdrm codes by exploring the subcodes of gabidulin codes if each of the rightmost delta 1 columns in the ferrers diagram cal f has at least n r dots where r is taken in a range then the conditions that an fdrm code in cal f is optimal are established the known combining constructions for fdrm code are generalized by introducing the concept of proper combinations of ferrers diagrams

list decodability of random subcodes of gabidulin codes efficient list decoding of rank metric codes seems more difficult compared with classical block codes although list decodability of random rank metric codes is completely determined by ding for example it was shown by raviv and wachter zeh that the list decoding radius of gabidulin codes is the same as the unique decoding radius i e half the minimum distance for some instances of parameters on the other hand guruswami and xing give an explicit construction of subcodes of gabidulin codes which can be list decoded up to the singleton bound this implies that subcodes of gabidulin codes are good candidates for list decoding in this paper we confirm that with overwhelming probability a random subcode of a gabidulin code can be list decoded with decoding radius far beyond half of the minimum distance

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

cyclic orbit codes a constant dimension code consists of a set of k dimensional subspaces of bbf qn orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group acting on the set of all subspaces of bbf qn if the acting group is cyclic the corresponding orbit codes are called cyclic orbit codes in this paper we show how orbit codes can be seen as an analog of linear codes in the block coding case we investigate how the structure of cyclic orbit codes can be utilized to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass of cyclic orbit codes

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

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

some considerations on the natural approach abstract in this note we present some pertinent comments on the natural mode technique in the first part we demonstrate that for arbitrary finite elements the choice of urelements may exclusively be based on simplex elements in fact the corresponding infinitesimal building blocks the so called subelements may be used also for quadrilateral and hexahedronal elements or for that matter for any one two or three dimensional element the second part develops some new considerations on unconventional models associated with the patch test

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

pattern mixture models for multivariate incomplete data consider a random sample on variables x1 xv with some values of xv missing selection models specify the distribution of x1 xv over respondents and nonrespondents to xv and the conditional distribution that xv is missing given x1 xv in contrast pattern mixture models specify the conditional distribution of x 1 xv given that xv is observed or missing respectively and the marginal distribution of the binary indicator for whether or not xv is missing for multivariate data with a general pattern of missing values the literature has tended to adopt the selection modeling approach see for example little and rubin here pattern mixture models are proposed for this more general problem pattern mixture models are chronically underidentified in particular for the case of univariate nonresponse mentioned above there are no data on the distribution of xv given x1 xv 1 in the stratum with xv missing thus the models require restrictions or prior information to identify the paramet

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

refined coding bounds and code constructions for coherent network error correction coherent network error correction is the error control problem in network coding with the knowledge of the network codes at the source and sink nodes with respect to a given set of local encoding kernels defining a linear network code we obtain refined versions of the hamming bound the singleton bound and the gilbert varshamov bound for coherent network error correction similar to its classical counterpart this refined singleton bound is tight for linear network codes the tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound these two algorithms illustrate different design methods one makes use of existing network coding algorithms for error free transmission and the other makes use of classical error correcting codes the implication of the tightness of the refined singleton bound is that the sink nodes with higher maximum flow values can have higher error correction capabilities

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

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

continuum aspects of the natural approach abstract the natural finite element approach introduced by argyris in the early sixties is marked essentially by the distinction between rigid body motion and deformation on the one hand and by the description of the latter in compliance with the element purpose and geometry on the other hand for triangular and tetrahedral elements the approach suggests strain and stress measures defined along the sides respectively the edges as homogeneous normal quantities free of shear in the mechanics of continua the respective infinitesimal elements represent minimum configurations to define local deformation in two and three dimensions the paper presents developments of the natural approach on the continuum level within a consistent theoretical framework it is proposed to begin with a reference system of supernumerary coordinates associated with the elementary tetrahedron in the space and with the triangle in the plane vectorial quantities are defined the operations of gradient and divergence are interpreted in this system the natural deformation rate is deduced from the velocity field the stress is introduced as work conjugate measure the condition for local equilibrium is presented in natural quantities as well as the stress definition in association with the resultant forces the set up of material constitutive relations is exemplified for the viscous solid beyond the description of the momentary kinematics as from the velocity field the appearance of finite deformation is considered basing on displacements illustration of the methodology for a plane elastic case is appended

linearized polynomial interpolation and its applications in this paper we first propose an interpolation algorithm in a well ordered free module of a linearized polynomial ring and then use this algorithm to decode several important families of codes gabidulin codes kotter and kschischang kk codes and mahdavifar and vardy mv codes our decoding algorithm for gabidulin codes is different from the polynomial reconstruction algorithm by loidreau when applied to decode kk codes our interpolation algorithm is equivalent to the sudan style list 1 decoding algorithm proposed by kotter and kschischang for kk codes the interpolation approach is also capable of solving the interpolation problem for the list decoding of mv codes proposed by mahdavifar and vardy and has a lower complexity than gaussian elimination an interpolator for list decoding of mv codes has also been implemented in hardware and the synthesis results show that it leads to better throughput and efficiency than gaussian elimination

clarifying missing at random and related definitions and implications when coupled with exchangeability we clarify the key concept of missingness at random in incomplete data analysis we first distinguish between data being missing at random and the missingness mechanism being a missing at random one which we call missing always at random and which is more restrictive we further discuss how in general neither of these conditions is a statement about conditional independence we then consider the implication of the more restrictive missing always at random assumption when coupled with full unit exchangeability for the matrix of the variables of interest and the missingness indicators the conditional distribution of the missingness indicators for any variable that can have a missing value can depend only on variables that are always fully observed we discuss implications of this for modelling missingness mechanisms

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

optimal rate delay tradeoffs and delay mitigating codes for multipath routed and network coded networks via multiterminal information theory a framework is presented for deriving fundamental rate delay tradeoffs that delay mitigating codes must have when utilized over multipath routed and random linear network coded networks the rate delay tradeoff is formulated as a calculus problem on a capacity region of a related abstracted broadcast channel given this general framework for studying such rate delay tradeoffs the extreme case of uniform networks in which each possible received packet arrival order is equally likely is considered for these networks the rate delay calculus problem is simplified to an integer programming problem which for small numbers of packets may be solved explicitly or for larger numbers of packets may be accurately approximated through the calculus of variations by appropriate relaxation of an integer constraint explicit expressions for the rate delay tradeoff in uniform networks are presented in the special cases of i constant packet inter arrival times and ii exponential independent and identically distributed i i d packet arrival times finally the delay mitigating codes achieving these rate delay tradeoffs are discussed

prior distributions from pseudo likelihoods in the presence of nuisance parameters consider a model parameterized by where is the parameter of interest the problem of eliminating the nuisance parameter can be tackled by resorting to a pseudo likelihood function l for namely a function of only and the data y with properties similar to those of a likelihood function if one treats l as a true likelihood the posterior distribution y l for can be considered where is a prior distribution on the goal of this article is to construct probability matching priors for a scalar parameter of interest only i e priors for which bayesian and frequentist inference agree to some order of approximation to be used in y when l is a marginal a conditional or a modification of the profile likelihood we show that is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo maximum likelihood estimator the proposed priors are compared with the reference or jeffreys priors in four

on the bounds of certain maximal linear codes in a projective space the set of all subspaces of mathbb f q n is denoted by mathbb p q n the subspace distance d s x y dim x dim y 2 dim x cap y defined on mathbb p q n turns it into a natural coding space for error correction in random network coding a subset of mathbb p vphantom r r r q n is called a code and the subspaces that belong to the code are called codewords motivated by classical coding theory a linear coding structure can be imposed on a subset of mathbb p q n braun et al conjectured that the largest cardinality of a linear code that contains mathbb f q n is 2 n in this paper we prove this conjecture and characterize the maximal linear codes that contain mathbb f q n

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

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 decoding rank metric codes over large fields a decoding algorithm is presented for a rank metric array codes that are based on diagonal interleaving of maximum distance separable codes with respect to this metric such array codes are known to be optimal when the underlying field is algebraically closed it is also shown that for any list decoding radius that is smaller than the minimum rank distance the list size can be bounded from above by an expression that is independent of the field

subspace polynomials and cyclic subspace codes subspace codes have received an increasing interest recently due to their application in error correction for random network coding in particular cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms in this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits we further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials we present some constructions of such codes which are cyclic and analyze their parameters

linearized shift register synthesis an efficient algorithm synthesizing all shortest q linearized feedback shift registers generating a given sequence of length n over a finite field bbfqm is derived and its correctness is proved this algorithm which is a generalization of the berlekamp massey algorithm has time complexity o ln o n2 operations in bbfqm where l is the linearized complexity of the sequence the algorithm can be applied for efficiently solving the key equation when decoding gabidulin codes