secret sharing and non shannon information inequalities the known secret sharing schemes for most access structures are not efficient even for a one bit secret the length of the shares in the schemes is 2o n where n is the number of participants in the access structure it is a long standing open problem to improve these schemes or prove that they cannot be improved the best known lower bound is by csirmaz who proved that there exist access structures with n participants such that the size of the share of at least one party is n logn times the secret size csirmaz s proof uses shannon information inequalities which were the only information inequalities known when csirmaz published his result on the negative side csirmaz proved that by only using shannon information inequalities one cannot prove a lower bound of n on the share size in the last decade a sequence of non shannon information inequalities were discovered in fact it was proved that there are infinity many independent information inequalities even in four variables this raises the hope that these inequalities can help in improving the lower bounds beyond n however we show that any information inequality with four or five variables cannot prove a lower bound of n on the share size in addition we show that the same negative result holds for all information inequalities with more than five variables that are known to date

eigenschemes and the jordan canonical form abstract we study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix the two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the jordan canonical form of the matrix we also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space this interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint

on network coding for sum networks a directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources we call such a network a sum network it is shown that there exists a solvably and linear solvably equivalent sum network for any multiple unicast network and thus for any directed acyclic communication network it is also shown that there exists a linear solvably equivalent multiple unicast network for every sum network it is shown that for any set of polynomials having integer coefficients there exists a sum network which is scalar linear solvable over a finite field f if and only if the polynomials have a common root in f for any finite or cofinite set of prime numbers a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set the insufficiency of linear net work coding and unachievability of the network coding capacity are proved for sum networks by using similar known results for communication networks under fractional vector linear network coding a sum network and its reverse network are shown to be equivalent however under nonlinear coding it is shown that there exists a solvable sum network whose reverse network is not solvable

computing linear functions by linear coding over networks we consider the scenario in which a set of sources generates messages in a network and a receiver node demands an arbitrary linear function of these messages we formulate an algebraic test to determine whether an arbitrary network can compute linear functions using linear codes we identify a class of linear functions that can be computed using linear codes in every network that satisfies a natural cut based condition conversely for another class of linear functions we show that the cut based condition does not guarantee the existence of a linear coding solution for linear functions over the binary field the two classes are complements of each other

linearity and solvability in multicast networks it is known that for every solvable multicast network there exists a large enough finite field alphabet such that a scalar linear solution exists we prove i every binary solvable multicast network with at most two messages has a binary scalar linear solution ii for more than two messages not every binary solvable multicast network has a binary scalar linear solution iii a multicast network that has a solution for a given alphabet might not have a solution for all larger alphabets

a bayesian approach to robust adaptive beamforming an adaptive beamformer that is robust to uncertainty in source direction of arrival doa is derived using a bayesian approach the doa is assumed to be a discrete random variable with a known a priori probability density function pdf that reflects the level of uncertainty in the source doa the resulting beamformer is a weighted sum of minimum variance distortionless response mvdr beamformers pointed at a set of candidate doas where the relative contribution of each mvdr beamformer is determined from the a posteriori pdf of the doa conditioned on previously observed data a simple approximation to the a posteriori pdf results in a straightforward implementation performance of the approximate bayesian beamformer is compared with linearly constrained minimum variance lcmv beamformers and data driven approaches that attempt to estimate signal characteristics or the steering vector from the data

topological interference management through index coding this paper studies linear interference networks both wired and wireless with no channel state information at the transmitters except a coarse knowledge of the end to end one hop topology of the network that only allows a distinction between weak zero and significant nonzero channels and no further knowledge of the channel coefficients realizations the network capacity wired and degrees of freedom dof wireless are found to be bounded above by the capacity of an index coding problem for which the antidote graph is the complement of the given interference graph the problems are shown to be equivalent under linear solutions an interference alignment perspective is then used to translate the existing index coding solutions into the wired network capacity and wireless network dof solutions as well as to find new and unified solutions to different classes of all three problems

characteristic dependent linear rank inequalities with applications to network coding two characteristic dependent linear rank inequalities are given for eight variables in particular the first inequality holds for all finite fields whose characteristic is not three and does not in general hold over characteristic three the second inequality holds for all finite fields whose characteristic is three and does not in general hold over characteristics other than three applications of these inequalities to the computation of capacity upper bounds in network coding are demonstrated

reversible and irreversible information networks it is shown that there exist information networks where messages can be sent utilizing network coding more easily in one direction than in the opposite direction this is valid even though each channel is assumed to have the same capacity in both directions it is shown that irreversible information networks only have solutions that use nonlinear network coding this correspondence argues that this result is more surprising than it might appear at first sight and that it follows using ideas resembling the path integral in quantum mechanics

on the hardness of approximating the network coding capacity this work addresses the computational complexity of achieving the capacity of a general network coding instance it has been shown lehman and lehman soda 2005 that determining the scalar linear capacity of a general network coding instance is np hard in this paper we address the notion of approximation in the context of both linear and nonlinear network coding loosely speaking we show that given an instance of the general network coding problem of capacity c constructing a code of rate c for any universal i e independent of the size of the instance constant 1 is hard specifically finding such network codes would solve a long standing open problem in the field of graph coloring our results refer to scalar linear vector linear and nonlinear encoding functions and are the first results that address the computational complexity of achieving the network coding capacity in both the vector linear and general network coding scenarios in addition we consider the problem of determining the scalar linear capacity of a planar network coding instance i e an instance in which the underlying graph is planar we show that even for planar networks this problem remains np hard

dualities between entropy functions and network codes in communications networks the capacity region of multisource network coding is given in terms of the set of entropy functions gamma more broadly determination of gamma would have an impact on converse theorems for multi terminal problems in information theory this paper provides several new dualities between entropy functions and network codes given a function g ges 0 defined on all subsets of n random variables we provide a construction for a network multicast problem which is ldquosolvablerdquo if and only if g is the entropy function of a set of quasi uniform random variables the underlying network topology is fixed and the multicast problem depends on g only through link capacities and source rates a corresponding duality is developed for linear network codes where the constructed multicast problem is linearly solvable if and only if g is linear group characterizable relaxing the requirement that the domain of g be subsets of random variables we obtain a similar duality between polymatroids and the linear programming bound these duality results provide an alternative proof of the insufficiency of linear and abelian network codes and demonstrate the utility of non shannon inequalities to tighten outer bounds on network coding capacity regions

truncation technique for characterizing linear polymatroids linear polymatroids have a strong connection to network coding the problem of finding the linear network coding capacity region is equivalent to the characterization of all linear polymatroids it is well known that linear polymatroids must satisfy the inequalities of ingleton combin math appln 1971 however it has been an open question for years as to whether these inequalities are sufficient it was until recently that new subspace rank inequalities have been discovered independently by kinser and dougherty in this paper we propose a new approach to investigate properties of linear polymatroids specifically we demonstrate how to construct a new polymatroid that satisfies not only the ingleton and dfz inequalities but also lies outside the minimal closed and convex cone containing all linear polymatroids using this polymatroid we prove that all truncation preserving inequalities including ingleton inequalities and dfz inequalities are insufficient to characterize linear polymatroids

achievable rate regions for network coding determining the achievable rate region for networks using routing linear coding or nonlinear coding is thought to be a difficult task in general and few are known we describe the achievable rate regions for four interesting networks completely for three and partially for the fourth in addition to the known matrix computation method for proving outer bounds for linear coding we present a new method that yields actual characteristic dependent linear rank inequalities from which the desired bounds follow immediately

a new set of linear constraints for broad band time domain element space processors a new set of linear constraints for designing broad band time domain element space antenna array processors is presented the set of linear constraints is used to ensure that a desired look direction response of the processor over a frequency band of interest can be closely approximated the design technique is posed in such a way that three types of presteering can be handled no presteering coarse presteering and exact presteering the elimination of presteering time delays or the possibility to use coarse presteering is an attractive feature in a digital implementation of antenna array processors the relationship that the new processor has to other broad band processors is also established furthermore the approach enables various types of errors and mismatches between signal model and actual scenario to be incorporated in the problem formulation

nonreversibility and equivalent constructions of multiple unicast networks we prove that for any finite directed acyclic network there exists a corresponding multiple unicast network such that for every alphabet each network is solvable if and only if the other is solvable and for every finite field alphabet each network is linearly solvable if and only if the other is linearly solvable the proof is constructive and creates an extension of the original network by adding exactly s 5m r 1 new nodes where in the original network m is the number of messages r is the average number of receiver nodes demanding each source message and s is the number of messages emitted by more than one source the construction is then used to create a solvable multiple unicast network which becomes unsolvable over every alphabet size if all of its edge directions are reversed and if the roles of source receiver pairs are reversed

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

beampattern synthesis via a matrix approach for signal power estimation we present new beampattern synthesis approaches based on semidefinite relaxation sdr for signal power estimation the conventional approaches use weight vectors at the array output for beampattern synthesis which we refer to as the vector approaches va instead of this we use weight matrices at the array output which leads to matrix approaches ma we consider several versions of ma including a data adaptive ma ama as well as several data independent ma designs for all of these ma designs globally optimal solutions can be determined efficiently due to the convex optimization formulations obtained by sdr numerical examples as well as theoretical evidence are presented to show that the optimal weight matrix obtained via sdr has few dominant eigenvalues and often only one when the number of dominant eigenvalues of the optimal weight matrix is equal to one ma reduces to va and the main advantage offered by sdr in this case is to determine the globally optimal solution efficiently moreover we show that the ama allows for strict control of main beam shape and peak sidelobe level while retaining the capability of adaptively nulling strong interferences and jammers numerical examples are also used to demonstrate that better beampattern designs can be achieved via the data independent ma than via its va counterpart

a robust adaptive beamformer based on worst case semi definite programming in this correspondence a novel robust adaptive beamformer is proposed based on the worst case semi definite programming sdp a recent paper has reported that a beamformer robust against large steering direction error can be constructed by using linear constraints on magnitude response in sdp formulation in practice however array system also suffers from many other array imperfections other than steering direction error in order to make the adaptive beamformer robust against all kinds of array imperfections the worst case optimization technique is proposed to reformulate the beamformer by minimizing the array output power with respect to the worst case array imperfections the resultant beamformer has the mathematical form of a regularized sdp problem and possesses superior robustness against arbitrary array imperfections although the formulation of robust beamformer uses weighting matrix with the help of spectral factorization approach the weighting vector can be obtained so that the beamformer can be used for both signal power and waveform estimation simple implementation flexible performance control as well as significant signal to interference plus noise ratio sinr enhancement support the practicability of the proposed method

average sinr calculation of a persymmetric sample matrix inversion beamformer based on the bartlett s decomposition of a wishart random matrix we study the normalized output signal to interference plus noise ratio sinr of a sample matrix inversion smi beamformer with exploiting a priori information on persymmetric structures in the received signal the persymmetric structure exists when a system is equipped with a symmetrically spaced linear array or symmetrically spaced pulse trains in the matched case we obtain an exact expression for the expectation of the normalized output sinr i e average sinr loss of the persymmetric smi beamformer considering the mismatch in the signal steering vector we derive an approximate expression for the average sinr loss of the persymmetric smi beamformer in a non homogeneous environment where the test and training data have different covariance matrices these theoretical results are all verified by using monte carlo techniques simulation results reveal that the exploitation of the persymmetric structure is equivalent to doubling the amount of training data and thus the sinr loss of the persymmetric smi beamformer can be significantly reduced in particular the persymmetric smi beamformer can work in the case of limited training data where the traditional beamformers cannot work

linear network codes and systems of polynomial equations if beta and gamma are nonnegative integers and f is a field then a polynomial collection p1 hellip pbeta sube z alpha1 hellip alphagamma is said to be solvable over f if there exist omega1hellip omegagamma isin f such that for all i 1 hellip beta we have pi omega1hellip omegagamma 0 we say that a network and a polynomial collection are solvably equivalent if for each field f the network has a scalar linear solution over f if and only if the polynomial collection is solvable over f koetter and medard s work implies that for any directed acyclic network there exists a solvably equivalent polynomial collection we provide the converse result namely that for any polynomial collection there exists a solvably equivalent directed acyclic network hence the problems of network scalar linear solvability and polynomial collection solvability have the same complexity the construction of the network is modeled on a matroid construction using finite projective planes due to maclane in 1936 a set psi of prime numbers is a set of characteristics of a network if for every q isin psi the network has a scalar linear solution over some finite field with characteristic q and does not have a scalar linear solution over any finite field whose characteristic lies outside of psi we show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or co finite two networks n and n are is equivalent if for any finite field f n is scalar linearly solvable over f if and only if n is scalar linearly solvable over f we further show that every network is ls equivalent to a multiple unicast matroidal network

capacity and achievable rate regions for linear network coding over ring alphabets the rate of a network code is the ratio of the block size of the network s messages to that of its edge codewords we compare the linear capacities and achievable rate regions of networks using finite field alphabets to the more general cases of arbitrary ring and module alphabets for non commutative rings two sided linearity is allowed specifically we prove the following for directed acyclic networks first the linear rate region and the linear capacity of any network over a finite field depend only on the characteristic of the field furthermore any two fields with different characteristics yield different linear capacities for at least one network second whenever the characteristic of a given finite field divides the size of a given finite ring each network s linear rate region over the ring is contained in its linear rate region over the field thus any network s linear capacity over a field is at least its linear capacity over any other ring of the same size an analogous result also holds for linear network codes over module alphabets third whenever the characteristic of a given finite field does not divide the size of a given finite ring there is some network whose linear capacity over the ring is strictly greater than its linear capacity over the field thus for any finite field there always exist rings over which some networks have higher linear capacities than over the field

networks matroids and non shannon information inequalities we define a class of networks called matroidal networks which includes as special cases all scalar linearly solvable networks and in particular solvable multicast networks we then present a method for constructing matroidal networks from known matroids we specifically construct networks that play an important role in proving results in the literature such as the insufficiency of linear network coding and the unachievability of network coding capacity we also construct a new network from the vamos matroid which we call the vamos network and use it to prove that shannon type information inequalities are in general not sufficient for computing network coding capacities to accomplish this we obtain a capacity upper bound for the vamos network using a non shannon type information inequality discovered in 1998 by zhang and yeung and then show that it is smaller than any such bound derived from shannon type information inequalities this is the first application of a non shannon type inequality to network coding we also compute the exact routing capacity and linear coding capacity of the vamos network finally using a variation of the vamos network we prove that shannon type information inequalities are insufficient even for computing network coding capacities of multiple unicast networks

network coding capacity with a constrained number of coding nodes we study network coding capacity under a constraint on the total number of network nodes that can perform coding that is only a certain number of network nodes can produce coded outputs whereas the remaining nodes are limited to performing routing we prove that every nonnegative monotonically nondecreasing eventually constant rational valued function on the nonnegative integers is equal to the capacity as a function of the number of allowable coding nodes of some directed acyclic network

on a relation between information inequalities and group theory we establish a one to one correspondence between information inequalities and group inequalities the major implication of our result is that we can prove information inequalities by proving the corresponding group inequalities and vice versa by giving a group theoretic proof for all shannon type inequalities we suggest that new inequalities could be discovered by making use of the rich set of tools in group theory on the other hand via a non shannon type information inequality discovered by zhang and yeung 1997 we obtain a new inequality in group theory whose meaning is yet to be understood

the minimal set of ingleton inequalities the ingleton inequalities are the inequality constraints known to be required of representable matroids for a matroid of n elements there are 16n ingleton inequalities in this paper we show that many of these inequalities are redundant we explicitly determine the unique minimal set of ingleton inequalities which number on the order of 6n 4 o 5n in information theory these inequalities are required of the entropy functions of certain random variables constructed from linear subspaces as a result these inequalities appear as constraints in linear programming outer bounds for the capacity region of multisource network coding where the codes are required to be linear ingleton inequalities have also played an instrumental role in demonstrating the insufficiency of linear codes for multisource network coding the reduction that we obtain for ingleton inequalities is sufficiently large to meaningfully reduce the complexity of these linear programming bounds

closure solvability for network coding and secret sharing network coding is a new technique to transmit data through a network by letting the intermediate nodes combine the packets they receive given a network the network coding solvability problem decides whether all the packets requested by the destinations can be transmitted in this paper we introduce a new approach to this problem we define a closure operator on a digraph closely related to the network coding instance and we show that the constraints for network coding can all be expressed according to that closure operator thus a solution for the network coding problem is equivalent to a so called solution of the closure operator we can then define the closure solvability problem in general which surprisingly reduces to finding secret sharing matroids when the closure operator is a matroid based on this reformulation we can easily prove that any multiple unicast where each node receives at least as many arcs as there are sources solvable by linear functions we also give an alternative proof that any nontrivial multiple unicast with two source receiver pairs is always solvable over all sufficiently large alphabets based on singular properties of the closure operator we are able to generalize the way in which networks can be split into two distinct parts we also provide a new way of identifying and removing useless nodes in a network we also introduce the concept of network sharing where one solvable network can be used to accommodate another solvable network coding instance finally the guessing graph approach to network coding solvability is generalized to any closure operator which yields bounds on the amount of information that can be transmitted through a network

a framework for linear information inequalities we present a framework for information inequalities namely inequalities involving only shannon s information measures for discrete random variables a region in ir 2 sup n 1 denoted by spl gamma is identified to be the origin of all information inequalities involving n random variables in the sense that all such inequalities are partial characterizations of spl gamma a product from this framework is a simple calculus for verifying all unconstrained and constrained linear information identities and inequalities which can be proved by conventional techniques these include all information identities and inequalities of such types in the literature as a consequence of this work most identities and inequalities involving a definite number of random variables can now be verified by a software called itip which is available on the world wide web our work suggests the possibility of the existence of information inequalities which cannot be proved by conventional techniques we also point out the relation between spl gamma and some important problems in probability theory and information theory

index coding an interference alignment perspective author s maleki h cadambe vr jafar sa abstract the index coding problem is studied from an interference alignment perspective providing new results as well as new insights into and generalizations of previously known results an equivalence is established between the capacity of multiple unicast index coding where each message is desired by exactly one receiver and groupcast index coding where a message can be desired by multiple receivers which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings where this question has previously been answered necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints and generalizations to feasibility of rate 1 l 1 per message when each destination desires at least l messages are similarly obtained finally capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks the solutions are based on vector linear coding 1963 2012 ieee

on ingleton violating finite groups given n discrete random variables its entropy vector is the 2 n 1 dimensional vector obtained from the joint entropies of all non empty subsets of the random variables it is well known that there is a close relation between such an entropy vector and a certain group characterizable vector obtained from a finite group and n of its subgroups indeed roughly speaking knowing the region of all such group characterizable vectors is equivalent to knowing the region of all entropy vectors this correspondence may be useful for characterizing the space of entropic vectors and for designing network codes if one restricts attention to abelian groups then not all entropy vectors can be obtained this is an explanation for the fact shown by dougherty et al that linear network codes cannot achieve capacity in general network coding problems since linear network codes come from abelian groups all abelian group characterizable vectors and by fiat all entropy vectors generated by linear network codes satisfy a linear inequality called the ingleton inequality general entropy vectors however do not necessarily have this property it is therefore of interest to identify groups that violate the ingleton inequality in this paper we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the ingleton inequality using a refined computer search we find the symmetric group s 5 to be the smallest group that violates the ingleton inequality careful study of the structure of this group and its subgroups reveals that it belongs to the ingleton violating family pgl 2 q with a prime power q geq 5 i e the projective group of 2 times 2 nonsingular matrices with entries in mathbb f q we further interpret this family of groups and their subgroups using the theory of group actions and identify the subgroups as certain stabilizers we also extend the construction to more general groups such as pgl n q and gl n q the families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes and we discuss some considerations for constructing such group network codes

precoding based network alignment using transform approach for acyclic networks with delay the algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known based on this formulation for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links the output symbols at the sink nodes at any given time instant is a f p m linear combination of the input symbols across different generations where f p m denotes the field over which the network operates p is prime and m is a positive integer we use finite field discrete fourier transform to convert the output symbols at the sink nodes at any given time instant into a f p m linear combination of the input symbols generated during the same generation without making use of memory at the intermediate nodes we call this as transforming the acyclic network with delay into n instantaneous networks n is sufficiently large we show that under certain conditions there exists a network code satisfying sink demands in the usual nontransform approach if and only if there exists a network code satisfying sink demands in the transform approach when the zero interference conditions are not satisfied we propose three precoding based network alignment pbna schemes for three source three destination multiple unicast network with delays 3 s 3 d mun d termed as pbna using transform approach and time invariant local encoding coefficients lecs pbna using time varying lecs and pbna using transform approach and block time varying lecs we derive sets of necessary and sufficient conditions under which throughputs close to n 1 2n 1 n 2n 1 and n 2n 1 are achieved for the three source destination pairs in a 3 s 3 d mun d employing pbna using transform approach and time invariant lecs and pbna using transform approach and block time varying lecs where n is a positive integer for pbna using time varying lecs we obtain a sufficient condition under which a throughput demand of n 1 n n 2 n and n 3 n can be met for the three source destination pairs in a 3 s 3 d mun d where n 1 n 2 and n 3 are positive integers less than or equal to the positive integer n this condition is also necessary when n 1 n 3 n 1 n 2 n where n 1 n 2 n 3

broadcasting with side information bounding and approximating the broadcast rate index coding has received considerable attention recently motivated in part by applications such as fast video on demand and efficient communication in wireless networks and in part by its connection to network coding optimal encoding schemes and efficient heuristics were studied in various settings while also leading to new results for network coding such as improved gaps between linear and non linear capacity as well as hardness of approximation the problem of broadcasting with side information a generalization of the index coding problem begins with a sender and sets of users and messages each user possesses a subset of the messages and desires an additional message from the set the sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message the fundamental parameter of interest is the broadcast rate the average communication cost for sufficiently long broadcasts though there have been many new nontrivial bounds on by bar yossef 2006 lubetzky and stav 2007 alon 2008 and blasiak 2011 there was no known polynomial time algorithm for approximating within a nontrivial factor and the exact value of remained unknown for all nontrivial instances using the information theoretic linear program introduced in blasiak 2011 we give a polynomial time algorithm for recognizing instances with 2 and pinpoint precisely for various classes of graphs e g various cayley graphs of cyclic groups further extending ideas from ramsey theory we give a polynomial time algorithm with a nontrivial approximation ratio for computing finally we provide insight into the quality of previous bounds by giving constructions showing separations between and the respective bounds in particular we construct graphs where is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse

multiround private information retrieval capacity and storage overhead private information retrieval pir is the problem of retrieving one message out of k messages from n non communicating replicated databases where each database stores all k messages in such a way that each database learns no information about which message is being retrieved the capacity of pir is the maximum number of bits of desired information per bit of downloaded information among all pir schemes the capacity has recently been characterized for pir as well as several of its variants in every case it is assumed that all the queries are generated by the user simultaneously here we consider multiround pir where the queries in each round are allowed to depend on the answers received in previous rounds we show that the capacity of multiround pir is the same as the capacity of single round pir the result is generalized to also include t privacy constraints combined with previous results this shows that there is no capacity advantage from multiround over single round schemes non linear over linear schemes or from epsilon error over zero error schemes however we show through an example that there is an advantage in terms of storage overhead we provide an example of a multiround non linear epsilon error pir scheme that requires a strictly smaller storage overhead than the best possible with single round linear zero error pir schemes

graph theoretical constructions for graph entropy and network coding based communications the guessing number of a directed graph digraph equivalent to the entropy of that digraph was introduced as a direct criterion on the solvability of a network coding instance this paper makes two contributions on the guessing number first we introduce an undirected graph on all possible configurations of the digraph referred to as the guessing graph which encapsulates the essence of dependence amongst configurations we prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph therefore network coding solvability is no more a problem on the operations made by each node but is simplified into a problem on the messages that can transit through the network by studying the guessing graph of a given digraph and how to combine digraphs or alphabets we are thus able to derive bounds on the guessing number of digraphs second we construct specific digraphs with high guessing numbers yielding network coding instances where a large amount of information can transit we first propose a construction of digraphs with finite parameters based on cyclic codes with guessing number equal to the degree of the generator polynomial we then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one despite these digraphs being arbitrarily sparse these constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear although these instances are sparse and the sources are arbitrarily far from their corresponding sinks

quasi uniform codes and their applications quasi uniform random vectors have probability distributions that are uniform over their projections they are of fundamental interest because a linear information inequality is valid if and only if it is satisfied by all quasi uniform random vectors in this paper we investigate properties of codes induced by quasi uniform random vectors we prove that quasi uniform codes which include linear and almost affine codes as special cases are distance invariant and that greene s theorem and the critical theorem of crapo and rota hold in the setting of quasi uniform codes we show that both theorems are essentially combinatorial but not algebraical in nature linear programming bounds proposed by delsarte are extended for quasi uniform codes

on information divergence measures and a unified typicality strong typicality which is more powerful for theorem proving than weak typicality can be applied to finite alphabets only while weak typicality can be applied to countable alphabets in this paper the relation between typicality and information divergence measures is discussed the new definition of information divergence measure in this paper leads to the definition of a unified typicality for finite or countably infinite alphabets which is stronger than both weak typicality and strong typicality unified typicality retains the asymptotic equipartition property and the structural properties of strong typicality and it can potentially be used to generalize those theorems which are previously established by strong typicality to countable alphabets the applications in rate distortion theory and multisource network coding problems are discussed

the resolution of the ideal of 2 2 minors of a 2 nmatrix of linear forms abstract let k be an algebraically closed field and let s k x 1 x m let m be a 2 n matrix of linear forms of s and let i 2 m denote the ideal generated by the determinants of the 2 2 minors of m we study in this paper the minimal finite free resolution of s i 2 m as an s module m corresponds to a certain 2 dimensional vector space l of m n matrices that is to a matrix pencil the kronecker weierstrass theory of such matrix pencils provides a normal form for l and we characterize the resolution of s i 2 m in terms of this normal form in particular if the general element of l is injective we explicitly construct the minimal resolution of s i 2 m by repeated application of the horseshoe lemma for any m we express the regularity of s i 2 m as a function of the invariants of l

computation in multicast networks function alignment and converse theorems the classical problem in a network coding theory considers communication over multicast networks multiple transmitters send independent messages to multiple receivers that decode the same set of messages in this paper computation over multicast networks is considered each receiver decodes an identical function of the original messages for a countably infinite class of two transmitter two receiver single hop linear deterministic networks the computation capacity is characterized for a linear function modulo 2 sum of bernoulli sources a new upper bound is derived that is tighter than cut set based and genie aided bounds a matching inner bound is established via the development of a network decomposition theorem which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality the decomposition theorem provides a conceptually simple proof of achievability that generalizes to l transmitter l receiver networks

entropy region and convolution the entropy region is constructed from vectors of random variables by collecting shannon entropies of all subvectors its shape is studied here by means of polymatroidal constructions notably by convolution the closure of the region is decomposed into the direct sum of tight and modular parts reducing the study to the tight part the relative interior of the reduction belongs to the entropy region behavior of the decomposition under self adhesivity is clarified results are specialized and extended to the region constructed from four tuples of random variables this and computer experiments help to visualize approximations of a symmetrized part of the entropy region the four atom conjecture on the minimal ingleton score is refuted

on base field of linear network coding for a single source multicast network the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field in this paper we design a new class mathcal n of multicast networks and obtain an explicit formula for the linear solvability of these networks which involves the associated coset numbers of a multiplicative subgroup in a base field the concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size it further facilitates us to unveil infinitely many new multicast networks linearly solvable over gf q but not over gf q with q based on a subgroup order criterion in particular 1 for every k geq 2 an instance in mathcal n can be found linearly solvable over gf 2 2k but not over gf 2 2k 1 and 2 for arbitrary distinct primes p and p there are infinitely many k and k such that an instance in mathcal n can be found linearly solvable over gf p k but not over gf p k with p k

performance of dmi and eigenspace based beamformers the performance of the direct matrix inversion dmi method for antenna arrays of arbitrary geometry is analyzed by asymptotic statistical techniques the effects of eigenspace disturbance caused by finite samples on the output interference and noise powers are examined under the unit gain constraint in the direction of the desired signal the results show that the performance of the dmi method is degraded mostly by the disturbed noise subspace that suggests the use of an eigenspace based beamformer in which the weight vector is computed by using the signal plus interference subspace component of the sample correlation matrix convergence properties of the eigenspace based beamformer are evaluated for the cases in which the source number is known and in which it is overestimated theoretical analyses validated by computer simulations indicate that the eigenspace based beamformer has faster convergence rate than the dmi method

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

a novel adaptive beamformer based on semidefinite programming sdp with magnitude response constraints a novel robust adaptive beamformer formulated as a semidefinite programming sdp problem is proposed in this paper with new constraints on the magnitude response the beamwidth and response ripple of the robust response region can be well controlled moreover only a small part of these inequality constraints on the magnitude response are active during optimization so that few degrees of freedom dofs of the adaptive beamformer are consumed consequently the resultant beamformer has significant improvement on signal to interference plus noise ratio sinr an important problem in the proposed beamformer is how to generate the array weight vector from the optimal semidefinite matrix in this paper a method utilizing the extended spectral factorization method is proposed to solve this problem simple implementation flexible performance control as well as significant sinr enhancement support the practicability of the proposed method

sensor networks consider the two related problems of sensor selection and sensor fusion in the first given a set of sensors one wishes to identify a subset of the sensors which while small in size captures the essence of the data gathered by the sensors in the second one wishes to construct a fused sensor which utilizes the data from the sensors possibly after discarding dependent ones in order to create a single sensor which is more reliable than each of the individual ones in this work we rigorously define the dependence among sensors in terms of joint empirical measures and incremental parsing we show that these measures adhere to a polymatroid structure which in turn facilitates the application of efficient algorithms for sensor selection we suggest both a random and a greedy algorithm for sensor selection given an independent set we then turn to the fusion problem and suggest a novel variant of the exponential weighting algorithm in the suggested algorithm one competes against an augmented set of sensors which allows it to converge to the best fused sensor in a family of sensors without having any prior data on the sensors performance

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

linear network coding over rings part i scalar codes and commutative alphabets linear network coding over finite fields is a well studied problem we consider the more general setting of linear coding for directed acyclic networks with finite commutative ring alphabets our results imply that for scalar linear network coding over commutative rings fields can always be used when the alphabet size is flexible but other rings may be needed when the alphabet size is fixed we prove that if a network has a scalar linear solution over some finite commutative ring then the unique smallest such commutative ring is a field we also show that fixed size commutative rings are quasi ordered such that all the scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher ordered ring we study commutative rings that are maximal with respect to this quasi order as they may be considered the best commutative rings of a given size we prove that a commutative ring is maximal if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size furthermore we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the ring s size finally we prove that there is a unique maximal commutative ring of size m if and only if each prime factor of m has multiplicity in 1 2 3 4 6 as consequences 1 every finite field is such a maximal ring and 2 for each prime p some network is scalar linearly solvable over a commutative ring of size p k but not over the field of the same size if and only if k not in 1 2 3 4 6

exact repair mds code construction using interference alignment the high repair cost of n k maximum distance separable mds erasure codes has recently motivated a new class of mds codes called repair mds codes that can significantly reduce repair bandwidth over conventional mds codes in this paper we describe n k d exact repair mds codes which allow for any failed node to be repaired exactly with access to d survivor nodes where k d n 1 we construct exact repair mds codes that are optimal in repair bandwidth for the cases of k n 1 2 and d 2k 11 b k 3 our codes are deterministic and require a finite field size of at most 2 n k our constructive codes are based on interference alignment techniques

aluffi torsion free ideals abstract a special class of algebras which are intermediate between the symmetric and the rees algebras of an ideal was introduced by p aluffi in 2004 to define characteristic cycle of a hypersurface parallel to conormal cycle in intersection theory these algebras are recently investigated by a nasrollah nejad and a simis who named them aluffi algebras for a pair of ideals j i of a commutative ring r the aluffi algebra of i j is called aluffi torsion free if it is isomorphic to the rees algebra of i j in this paper ideals generated by 2 minors of a 2 n matrix of linear forms and also edge ideals of graphs are considered and some conditions are presented which are equivalent to aluffi torsion free property of them

a new inequality for the von neumann entropy strong subadditivity of von neumann entropy proved in 1973 by lieb and ruskai is a cornerstone of quantum coding theory all other known inequalities for entropies of quantum systems may be derived from it here we prove a new inequality for the von neumann entropy which we prove is independent of strong subadditivity it is an inequality which is true for any four party quantum state provided that it satisfies three linear relations constraints on the entropies of certain reduced states

index coding capacity how far can one go with only shannon inequalities an interference alignment perspective is used to identify the simplest instances minimum possible number of edges in the alignment graph not more than 2 interfering messages at any destination of index coding problems where non shannon information inequalities are necessary for capacity characterization in particular this includes the first known example of a multiple unicast one destination per message index coding problem where non shannon information inequalities are shown to be necessary the simplest multiple unicast example has 7 edges in the alignment graph and 11 messages the simplest multiple groupcast multiple destinations per message example has 6 edges in the alignment graph 6 messages and 10 receivers for both the simplest multiple unicast and multiple groupcast instances the best outer bound based on only shannon inequalities is 2 5 which is tightened to 11 28 by the use of the zhang yeung non shannon type information inequality and the linear capacity is shown to be 5 13 using the ingleton inequality conversely identifying the minimal challenging aspects of the index coding problem allows an expansion of the class of solved index coding problems up to but not including these instances

a class of non linearly solvable networks for each positive composite integer m a network is constructed which is solvable over an alphabet of size m but is not solvable over any smaller alphabet these networks have no linear solutions over any module alphabets and are not asymptotically linearly solvable over any finite field alphabets the networks capacities are all shown to equal one and their linear capacities are all shown to be bounded away from one for all finite field alphabets in addition if m is a non power of prime composite number then such a network is not solvable over any prime power size alphabet

linear network coding linear index coding and representable discrete polymatroids discrete polymatroids are the multi set analogue of matroids in this paper we explore the connections among linear network coding linear index coding and representable discrete polymatroids we consider the vector linear solutions of networks over a field mathbb f q with possibly different message and edge vector dimensions which are referred to as linear fractional solutions it is well known that a scalar linear solution over mathbb f q exists for a network if and only if the network is matroidal with respect to a matroid representable over mathbb f q we define a discrete polymatroidal network and show that a linear fractional solution over a field mathbb f q exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over mathbb f q an algorithm to construct the networks starting from certain class of discrete polymatroids is provided every representation over mathbb f q for the discrete polymatroid results in a linear fractional solution over mathbb f q for the constructed network next we consider the index coding problem which involves a sender which generates a set of messages x x 1 x 2 dotso x k and a set of receivers mathcal r which demand messages a receiver r in mathcal r is specified by the tuple x h where x in x is the message demanded by r and h subseteq x setminus x is the side information possessed by r we first show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered rouayheb et al showed that the problem of finding a multi linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid the multi linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid we generalize the result of rouayheb et al by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid

cut set bounds on network information flow explicit characterization of the capacity region of communication networks is a long standing problem while it is known that network coding can outperform routing and replication the set of feasible rates is not known in general characterizing the network coding capacity region requires the determination of the set of all entropic vectors furthermore computing the explicitly known linear programming bound is infeasible in practice due to an exponential growth in complexity as a function of network size this paper focuses on the fundamental problems of characterization and computation of outer bounds for multi source multi sink networks starting from the known local functional dependence induced by the communication network we introduce the notion of irreducible sets which characterize implied functional dependence we provide recursions for the computation of all maximal irreducible sets these sets act as information theoretic bottlenecks and provide an easily computable outer bound for networks with correlated sources we extend the notion of irreducible sets and resulting outer bound for networks with independent sources we compare our bounds with existing bounds in the literature we find that our new bounds are the best among the known graph theoretic bounds for networks with correlated sources and for networks with independent sources

polynomial time algorithms for multicast network code construction the famous max flow min cut theorem states that a source node s can send information through a network v e to a sink node t at a rate determined by the min cut separating s and t recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re encode the information they receive we demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed we give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity we extend these algorithms to integer capacities and to codes that are tolerant to edge failures

linear network coding over rings part ii vector codes and non commutative alphabets in part i we studied linear network coding over finite commutative rings and made comparisons to the well studied case of linear network coding over finite fields here we consider the more general setting of linear network coding over finite possibly non commutative rings and modules we prove the following results regarding the linear solvability of directed acyclic networks over various finite alphabets for any network the following are equivalent i vector linear solvability over some field ii scalar linear solvability over some ring and iii linear solvability over some module analogously the following are equivalent a scalar linear solvability over some field b scalar linear solvability over some commutative ring and c linear solvability over some module whose ring is commutative whenever any network is linearly solvable over a module a smallest such module arises in a vector linear solution for that network over a field if a network is scalar linearly solvable over some non commutative ring but not over any commutative ring then such a non commutative ring must have size at least 16 and for some networks this bound is achieved an infinite family of networks is demonstrated each of which is scalar linearly solvable over some non commutative ring but not over any commutative ring whenever p is prime and 1 le k le 6 if a network is scalar linearly solvable over some ring of size p k then it is also k dimensional vector linearly solvable over the field mathrm gf p but the converse does not necessarily hold this result is extended to all k ge 1 when the ring is commutative

on the capacity of network coding for random networks we study the maximum flow possible between a single source and multiple terminals in a weighted random graph modeling a wired network and a weighted random geometric graph modeling an ad hoc wireless network using network coding for the weighted random graph model we show that the network coding capacity concentrates around the expected number of nearest neighbors of the source and the terminals specifically for a network with a single source l terminals and n relay nodes such that the link capacities between any two nodes is independent and identically distributed i i d spl sim x the maximum flow between the source and the terminals is approximately ne x with high probability for the weighted random geometric graph model where two nodes are connected if they are within a certain distance of each other we show that with high probability the network coding capacity is greater than or equal to the expected number of nearest neighbors of the node with the least coverage area

a matroidal framework for network error correcting codes matroidal networks were introduced by dougherty et al and have been well studied in the recent past it was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid a particularly interesting feature of this development is the ability to construct scalar and vector linearly solvable networks using certain classes of matroids furthermore it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios the current work attempts to establish a connection between matroid theory and network error correcting and detecting codes in a similar vein to the theory connecting matroids and network coding we abstract the essential aspects of linear network error detecting codes to arrive at the definition of a matroidal error detecting network and similarly a matroidal error correcting network abstracting from network error correcting codes an acyclic network with arbitrary sink demands is then shown to possess a scalar linear error detecting correcting network code if and only if it is a matroidal error detecting correcting network associated with a representable matroid therefore constructing such network error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions and vice versa we then present algorithms that enable the construction of matroidal error detecting and correcting networks with a specified capability of network error correction using these construction algorithms a large class of hitherto unknown scalar linearly solvable networks with multisource multicast and multiple unicast network error correcting codes is made available for theoretical use and practical implementation with parameters such as number of information symbols number of sinks number of coding nodes error correcting capability and so on being arbitrary but for computing power for the execution of the algorithms the complexity of the construction of these networks is shown to be comparable with the complexity of existing algorithms that design multicast scalar linear network error correcting codes finally we also show that linear network coding is not sufficient for the general network error correction detection problem with arbitrary demands in particular for the same number of network errors we show a network for which there is a nonlinear network error detecting code satisfying the demands at the sinks whereas there are no linear network error detecting codes that do the same

an implicit characterization of the achievable rate region for acyclic multisource multisink network coding the achievable information rate region problem of multisource multisink network coding for general acyclic networks with arbitrary transmission requirements has previously been studied where inner and outer bounds on the region were derived in terms of gamma ast the fundamental region of entropy functions in this paper we derive the exact characterization of the achievable rate region in terms of entropic functions thus closing the gap between the existing inner and outer bounds

line and lattice networks under deterministic interference models capacity bounds are compared for four different deterministic models of wireless networks representing four different ways of handling broadcast and superposition in the physical layer in particular the transport capacity under a multiple unicast traffic pattern is studied for a 1 d network of regularly spaced nodes on a line and for a 2 d network of nodes placed on a hexagonal lattice the considered deterministic models are i p p a model with exclusive transmission and reception ii p m a model with simultaneous reception of the sum of the signals transmitted by all nearby nodes iii b p a model with simultaneous transmission to all nearby nodes but exclusive reception and iv b m a model with both simultaneous transmission and simultaneous reception all four deterministic models are considered under half duplex constraints for the 1 d scenario it is found that the transport capacity under b m is twice that under p p for the 2 d scenario it is found that the transport capacity under b m is at least 2 5 times and no more than six times the transport capacity under p p the transport capacities under p m and b p fall between these bounds

robust adaptive beamforming for general rank signal models the performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment sources or sensor array this is especially true when the desired signal components are present in the beamformer training data snapshots because in this case the adaptive array performance is very sensitive to array and model imperfections the similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly but the training sample size is small we propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary type mismatches of the desired signal array response our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank one point source and higher rank scattered source fluctuating wavefront desired signal models the proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst case performance optimization simple closed form solutions to the considered robust adaptive beamforming problems are derived our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms while at the same time offer a significantly improved robustness and faster convergence rates

path gain algebraic formulation for the scalar linear network coding problem in the algebraic view the solution to a network coding problem is seen as a variety specified by a system of polynomial equations typically derived by using edge to edge gains as variables the output from each sink is equated to its demand to obtain polynomial equations in this paper we propose a method to derive the polynomial equations using source to sink path gains as the variables in the path gain formulation we show that linear and quadratic equations suffice therefore network coding becomes equivalent to a system of polynomial equations of maximum degree 2 we present algorithms for generating the equations in the path gains and for converting path gain solutions to edge to edge gain solutions because of the low degree simplification is readily possible for the system of equations obtained using path gains using small sized network coding problems we show that the path gain approach results in simpler equations and determines solvability of the problem in certain cases on a larger network with 87 nodes and 161 edges we show how the path gain approach continues to provide deterministic solutions to some network coding problems

pairwise intersession network coding on directed networks when there exists only a single multicast session in a directed acyclic cyclic network the existence of a network coding solution is characterized by the classic min cut max flow theorem for the case of more than one coexisting sessions network coding also demonstrates throughput improvement over noncoded solutions this paper proposes pairwise intersession network coding which allows for arbitrary directed networks but restricts the coding operations to being between two symbols for acyclic networks or between two strings of symbols for cyclic networks a graph theoretic characterization of pairwise intersession network coding is proven based on paths with controlled edge overlap this new characterization generalizes the edge disjoint path characterization of noncoded network communication and includes the well studied butterfly graph as a special case based on this new characterization various aspects of pairwise intersession network coding are studied including the sufficiency of linear codes the complexity of identifying coding opportunities its topological analysis and bandwidth and coding efficiency

abstract functional dependency structures abstract the functional dependency fd structures known mainly from database relationships theory appear to include other kinds of dependencies among objects like functions boolean or random variables etc we examine the representations of fd relations in the matroid theory manner a galois connection of the fd relations with closure system is elaborated

an equivalence between network coding and index coding we show that the network coding and index coding problems are equivalent this equivalence holds in the general setting which includes linear and nonlinear codes specifically we present a reduction that maps a network coding instance to an index coding instance while preserving feasibility i e the network coding instance has a feasible solution if and only if the corresponding index coding instance is feasible in addition we show that one can determine the capacity region of a given network coding instance with colocated sources by studying the capacity region of a corresponding index coding instance previous connections between network and index coding were restricted to the linear case

entropic inequalities and marginal problems a marginal problem asks whether a given family of marginal distributions for some set of random variables arises from some joint distribution of these variables here we point out that the existence of such a joint distribution imposes nontrivial conditions already on the level of shannon entropies of the given marginals these entropic inequalities are necessary but not sufficient criteria for the existence of a joint distribution for every marginal problem a list of such shannon type entropic inequalities can be calculated by fourier motzkin elimination and we offer a software interface to a fourier motzkin solver for doing so for the case that the hypergraph of given marginals is a cycle graph we provide a complete analytic solution to the problem of classifying all relevant entropic inequalities and use this result to bound the decay of correlations in stochastic processes furthermore we show that shannon type inequalities for differential entropies are not relevant for continuous variable marginal problems non shannon type inequalities are both in the discrete and in the continuous case in contrast to other approaches our general framework easily adapts to situations where one has additional conditional independence requirements on the joint distribution as in the case of graphical models we end with a list of open problems a complementary article discusses applications to quantum nonlocality and contextuality

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

a theory of network equivalence part i point to point channels a family of equivalence tools for bounding network capacities is introduced given a network n with node set v the capacity of n is a set of non negative vectors with elements corresponding to all possible multicast connections in n a vector is in the capacity region for n if and only if it is possible to simultaneously and reliably establish all multicast connections across n at the given rates any other demand type with independent messages is a special case of this multiple multicast problem and is therefore included in the given rate region in part i we show that the capacity of a network n is unchanged if any independent memoryless point to point channel in n is replaced by a noiseless bit pipe with throughput equal to the removed channel s capacity it follows that the capacity of a network comprised entirely of such point to point channels equals the capacity of an error free network that replaces each channel by a noiseless bit pipe of the corresponding capacity a related separation result was known previously for a single multicast connection over an acyclic network of independent memoryless point to point channels our result treats general connections e g a collection of simultaneous unicasts and allows cyclic or acyclic networks

a non shannon type conditional inequality of information quantities given n discrete random variables spl omega x sub 1 x sub n associated with any subset spl alpha of 1 2 n there is a joint entropy h x sub spl alpha where x sub spl alpha x sub i i spl isin spl alpha this can be viewed as a function defined on 2 sup 1 2 n taking values in 0 spl infin we call this function the entropy function of spl omega the nonnegativity of the joint entropies implies that this function is nonnegative the nonnegativity of the conditional joint entropies implies that this function is nondecreasing and the nonnegativity of the conditional mutual information implies that this function is two alternative these properties are the so called basic information inequalities of shannon s information measures an entropy function can be viewed as a 2 sup n 1 dimensional vector where the coordinates are indexed by the subsets of the ground set 1 2 n as introduced by yeng see ibid vol 43 no 6 p 1923 34 1997 spl gamma sub n stands for the cone in ir 2 sup n 1 consisting of all vectors which have all these properties let spl gamma sub n be the set of all 2 sup n 1 dimensional vectors which correspond to the entropy functions of some sets of n discrete random variables a fundamental information theoretic problem is whether or not spl gamma sub n spl gamma sub n here spl gamma sub n stands for the closure of the set spl gamma sub n we show that spl gamma sub n is a convex cone spl gamma sub 2 spl gamma sub 2 spl gamma sub 3 spl ne spl gamma sub 3 but spl gamma sub 3 spl gamma sub 3 for four random variables we have discovered a conditional inequality which is not implied by the basic information inequalities of the same set of random variables this lends an evidence to the plausible conjecture that spl gamma sub n spl ne spl gamma sub n for n 3

minimax robust discrete time matched filters the problem of designing finite length discrete time matched filters is considered for situations in which exact knowledge of the input signal and or noise characteristics is not available such situations arise in many applications due to channel distortion incoherencies nonlinear effects and other modeling uncertainties in such cases it is often of interest to design a minimax robust matched filter i e a nonadaptive filter with an optimum level of worst case performance for the expected uncertainty class this problem is investigated here for three types of uncertainty models for the input signal namely the mean absolute mean square and maximum absolute distortion classes and for a wide generality of norm deviation models for the noise covariance matrix some numerical examples illustrate the robustness properties of the proposed designs

on the index coding problem and its relation to network coding and matroid theory the index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks an instance of the index coding problem includes a sender that holds a set of information messages x x1 xk and a set of receivers r each receiver x h in r needs to obtain a message x x and has prior side information consisting of a subset h of x the sender uses a noiseless communication channel to broadcast encoding of messages in x to all clients the objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers in this paper we analyze the relation between the index coding problem the more general network coding problem and the problem of finding a linear representation of a matroid in particular we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem specifically we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions

robust adaptive beamformers based on worst case optimization and constraints on magnitude response in this paper novel robust adaptive beamformers are proposed with constraints on array magnitude response with the transformation from the array output power and the magnitude response to linear functions of the autocorrelation sequence of the array weight the optimization of an adaptive beamformer which is often described as a quadratic optimization problem in conventional beamforming methods is then reformulated as a linear programming lp problem unlike conventional robust beamformers the proposed method is able to flexibly control the robust response region with specified beamwidth and response ripple in practice an array has many imperfections besides steering direction error in order to make the adaptive beamformer robust against all kinds of imperfections worst case optimization is exploited to reconstruct the robust beamformer by minimizing array output power with the existence of the worst case array imperfections the robust beamforming can be expressed as a second order cone programming socp problem the resultant beamformer possesses superior robustness against arbitrary array imperfections with the proposed methods a large robust response region and a high signal to interference plus noise ratio sinr enhancement can be achieved readily simple implementation flexible performance control as well as significant sinr enhancement support the practicability of the proposed methods

response vector constrained robust lcmv beamforming based on semidefinite programming although linearly constrained minimum variance lcmv beamforming is robust against imprecise target information it usually leads to relatively high sidelobe and distorted mainlobe which would induce a high false alarm probability to circumvent this problem this work devises a novel robust lcmv beamforming approach by utilizing response vector optimization it intends to find the optimal response vector in lieu of the all one response vector in traditional lcmv beamformer the proposed robust beamformer is first formulated as a nonconvex quadratically constrained quadratic programming problem and then transformed into a semidefinite programming problem which can be efficiently and exactly solved the proposed beamformer not only improves the performance in terms of signal to interference plus noise ratio substantially but also possesses low sidelobe and well maintained mainlobe moreover since the response vector is quite small in size the complexity of calculating the optimal response vector is negligible additionally the proposed beamformer is also extended to two dimensional space time adaptive processing simulation results are presented to demonstrate the superiority of the proposed approach

piecewise linear conditional information inequality a new information inequality of non shannon type is proved for three discrete random variables under conditional independence constraints using the framework of entropy functions and polymatroids tightness of the inequality is described via quasi groups

on vector linear solvability of multicast networks vector linear network coding lnc is a generalization of the conventional scalar lnc such that the data unit transmitted on every edge is an l dimensional vector of data symbols over a base field gf q vector lnc enriches the choices of coding operations at intermediate nodes and there is a popular conjecture on the benefit of vector lnc over scalar lnc in terms of alphabet size of data units there exist single source multicast networks that are vector linearly solvable of dimension l over gf q but not scalar linearly solvable over any field of size q leq q l this paper introduces a systematic way to construct such multicast networks and subsequently establish explicit instances to affirm the positive answer of this conjecture for infinitely many alphabet sizes p l with respect to an arbitrary prime p on the other hand this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension l over gf 2 but have scalar linear solutions over gf q for some q where q can be odd or even this discovery also unveils that over a given base field a multicast network that has a vector linear solution of dimension l does not necessarily have a vector linear solution of dimension l l

capacity bounds for networks with correlated sources and characterisation of distributions by entropies characterising the capacity region for a network can be extremely difficult even with independent sources determining the capacity region can be as hard as the open problem of characterising all information inequalities the majority of computable outer bounds in the literature are relaxations of the linear programming bound which involves entropy functions of random variables related to the sources and link messages when sources are not independent the problem is even more complicated extension of linear programming bounds to networks with correlated sources is largely open source dependence is usually specified through a joint probability distribution and one of the main challenges in extending linear program bounds is the difficulty or impossibility of characterising arbitrary dependences via entropy functions this paper tackles the problem by answering the question of how well entropy functions can characterise correlation among sources we show that by using carefully chosen auxiliary random variables the characterisation can be fairly accurate using such auxiliary random variables we also give implicit and explicit outer bounds on the capacity of networks with correlated sources the characterisation of correlation or joint distribution via shannon entropy functions is also applicable to other information measures such as renyi entropy and tsallis entropy