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In the wiretap model of secure communication, Alice is connected to Bob over a noisy channel that is eavesdropped by Eve. The goal is to provide (asymptotic) reliability and perfect secrecy assuming that the Eve has unlimited computational power. The model has attracted considerable attention in recent years because it provides a natural model for passive eavesdropping in wireless communication. We consider a wiretap model with active adversaries, and define adversarial wiretap (AWTP) channels using a (\rho r , \rho w) wiretap adversary who can read a fraction \rho r , and modify a fraction \rho w of a sent codeword. The code components that are read and/or modified can be chosen adaptively, and the subsets of read and modified components could be different. AWTP codes provide secrecy and reliability for communication over AWTP channels. We define the security and reliability of AWTP channels and use these definitions to evaluate the security and reliability of codes for these channels. This paper has two main contributions. First, we prove an upper bound on the rate of AWTP codes for (\rho r , \rho w) -AWTP channels. Second, we give an explicit construction of a perfectly secure AWTP code family with efficient decoding that achieves the bound and, hence, obtain the secrecy capacity of AWTP channels for large alphabets. AWTP model is a natural extension of Wyner’s wiretap models, and somewhat surprisingly, it is also closely related to a seemingly unrelated cryptographic primitive, secure message transmission (SMT). This relation results in a new (and the only known) bound on the transmission rate of 1-round $(\epsilon , \delta )$ -SMT protocols. We discuss our results, give their relations to other works, and propose directions for future work. [ABSTRACT FROM AUTHOR]

The $K$ -user multiple-input multiple-output (MIMO) multi-way relay channel (Y-channel) consisting of $K$ users with $M$ antennas each and a common relay node with $N$ antennas is studied in this paper. Each user wants to exchange messages with all the other users through the relay. A transmission strategy is proposed for this channel. The proposed strategy is based on two steps: 1) channel diagonalization and 2) cyclic communication. The channel diagonalization is applied by using zero-forcing beam-forming. After the channel diagonalization, the channel is decomposed into parallel sub-channels. Cyclic communication is then applied, where signal-space alignment for network-coding is used over each sub-channel. The proposed strategy achieves the optimal degree-of-freedom (DoF) region of the channel if $N\leq M$ . To prove this, a new DoFs outer bound is derived. As a by-product, we conclude that the MIMO Y-channel is not separable, i.e., independent coding on separate sub-channels is not enough, and one has to code jointly over several sub-channels. [ABSTRACT FROM AUTHOR]

We consider the multi-user lossy source-coding problem for continuous alphabet sources. In a previous work, Ziv proposed a single-user universal coding scheme which uses uniform quantization with dither, followed by a lossless source encoder (entropy coder). In this paper, we generalize Ziv’s scheme to the multi-user setting. For this generalized universal scheme, upper bounds are derived on the redundancies, defined as the differences between the actual rates and the closest corresponding rates on the boundary of the rate region. For the mean-square error distortion measure, it is shown that this scheme can achieve redundancies of no more than 0.754 b per sample for each user. These bounds are obtained without the knowledge of the multi-user rate region, which is an open problem in general. As a direct consequence of these results, the inner and outer bounds on the rate-distortion achievable region are obtained. [ABSTRACT FROM PUBLISHER]

In this paper, we study the capacity of the diamond channel. We focus on the special case where the channel between the source node and the two relay nodes are two separate links with finite capacities and the link from the two relay nodes to the destination node is a Gaussian multiple access channel. We call this model the Gaussian multiple access diamond channel. We first propose an upper bound on the capacity. This upper bound is a single-letterization of an $n$ -letter upper bound proposed by Traskov and Kramer, and is tighter than the cut-set bound. As for the lower bound, we propose an achievability scheme based on sending correlated codes through the multiple access channel with superposition structure. We then specialize this achievable rate to the Gaussian multiple access diamond channel. Noting the similarity between the upper and lower bounds, we provide sufficient and necessary conditions that a Gaussian multiple access diamond channel has to satisfy such that the proposed upper and lower bounds meet. Thus, for a Gaussian multiple access diamond channel that satisfies these conditions, we have found its capacity. [ABSTRACT FROM PUBLISHER]

Communication systems are traditionally designed to have tight transmitter–receiver synchronization. This requirement has negligible overhead in the high-signal-to-noise ratio (SNR) regime. However, in many applications, such as wireless sensor networks, communication needs to happen primarily in the energy-efficient regime of low SNR, where requiring tight synchronization can be highly suboptimal. In this paper, we model the noisy channel with synchronization errors as a duplication/deletion/substitution channel. For this channel, we propose a new communication scheme that requires only loose transmitter–receiver synchronization. We show that the proposed scheme is asymptotically optimal for the Gaussian channel with synchronization errors in terms of energy efficiency as measured by the rate per unit energy. In the process, we also establish that the lack of synchronization causes negligible loss in energy efficiency. We further show that, for a general discrete memoryless channel with synchronization errors and a general input cost function admitting a zero-cost symbol, the rate per unit cost achieved by the proposed scheme is within a factor two of the information-theoretic optimum. [ABSTRACT FROM PUBLISHER]

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage nodes as possible. In this paper, we bound the minimum distance of a code in terms of its length, size, and locality. Unlike the previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence, the Simplex codes are the first example of an optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds. [ABSTRACT FROM PUBLISHER]

The well-known Huffman algorithm is an elegant approach to solve the basic problem of finding the optimal code among those with Kraft sums smaller than or equal to 1. In this paper, an extended problem is investigated, where the Kraft sum of the usable codes is limited to a given real number $\gamma $ , $0<\gamma <1$ . Unlike the case $\gamma =1$ (i.e., Huffman coding), for $\gamma <1$ , the Kraft sum of the optimal code is not necessarily equal to $\gamma $ . However, we show that the Kraft sum of the optimal code lies in a finite set of rational numbers which depend on the value of $\gamma $ . This motivates us to first devise a recursive algorithm to obtain the optimal codelengths with Kraft sum equal to $\gamma '$ , where $\gamma '$ has a finite binary representation and $0<\gamma '<1$ . To do so, the set of $N$ -tuple codelength vectors with a specific Kraft sum $\gamma '$ is characterized. The binary representation of $\gamma $ plays a fundamental role in all problems that arise for $\gamma <1$ . [ABSTRACT FROM PUBLISHER]

Polar codes are constructed based on kernels with polarizing properties. The performance of a polar code is characterized asymptotically in terms of the exponent of its kernel. The pioneering work of Arıkan on polar codes is based on a linear kernel of dimension two and exponent 0.5. In this paper, constructions of linear and nonlinear binary kernels of dimensions up to 16 are presented. The kernels are obtained using computer search or by shortening longer kernels obtained by computer search and a computer program is used to determine their exponents. It is proved that the constructed kernels have maximum exponents except in the case of nonlinear kernels of dimension 12 where it is demonstrated that the maximum exponent either equals that of the presented construction or assumes another specified value. The results show that the minimum dimension for which there exists a linear kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arıkan, is 15, while this minimum dimension is 14 for nonlinear kernels. Furthermore, it is shown that there is a linear kernel with maximum exponent up to dimension 11. For dimensions 13, 14, 15, and 16, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over \textrm GF(2) , are \mathbb Z4 -linear or \mathbb Z2\mathbb Z4 -linear. [ABSTRACT FROM PUBLISHER]

This paper studies the mismatched decoding problem for binary-input discrete memoryless channels. An example is provided for which an achievable rate based on superposition coding exceeds the Csiszár-Körner-Hui rate, thus providing a counter-example to a previously reported converse result. Both numerical evaluations and theoretical results are used in establishing this claim. [ABSTRACT FROM PUBLISHER]

A well-known technique in estimating the probabilities of rare events in general and in information theory in particular (used, for example, in the sphere–packing bound) is that of finding a reference probability measure under which the event of interest has the probability of order one and estimating the probability in question by means of the Kullback–Leibler divergence. A method has recently been proposed in [2] that can be viewed as an extension of this idea in which the probability under the reference measure may itself be decaying exponentially, and the Rényi divergence is used instead. The purpose of this paper is to demonstrate the usefulness of this approach in various information–theoretic settings. For the problem of channel coding, we provide a general methodology for obtaining matched, mismatched, and robust error exponent bounds, as well as new results in a variety of particular channel models. Other applications we address include rate-distortion coding and the problem of guessing. [ABSTRACT FROM PUBLISHER]

The artificial noise (AN) scheme, proposed by Goel and Negi, is being considered as one of the key enabling technology for secure communications over multiple-output multiple-input wiretap channels. However, the decrease in secrecy rate due to the increase in the number of Eve’s antennas is not well understood. In this paper, we develop an analytical framework to characterize the secrecy rate of the AN scheme as a function of Eve’s SNR, Bob’s SNR, the number of antennas in each terminal, and the power allocation scheme. We first derive a closed-form expression for the average secrecy rate. We then derive a closed-form expression for the asymptotic instantaneous secrecy rate with large number of antennas at all terminals. Finally, we derive simple lower and upper bounds on the average/instantaneous secrecy rate that provide a tool for the system design. [ABSTRACT FROM PUBLISHER]

The study of feedback has been mostly limited to single-hop communication settings. In this paper, we consider Gaussian networks where sources and destinations can communicate with the help of intermediate relays over multiple hops. We assume that links in the network can be bidirected providing opportunities for feedback. We ask the following question: can the information transfer in both directions of a link be critical to maximizing the end-to-end communication rates in the network? Equivalently, could one of the directions in each bidirected link (and more generally at least one of the links forming a cycle) be shut down and the capacity of the network still be approximately maintained? We show that in any arbitrary Gaussian network with bidirected edges and cycles and unicast traffic, we can always identify a directed acyclic subnetwork that approximately maintains the capacity of the original network. For Gaussian networks with multiple-access and broadcast traffic, an acyclic subnetwork is sufficient to achieve every rate point in the capacity region of the original network, however, there may not be a single acyclic subnetwork that maintains the whole capacity region. For networks with multicast and multiple unicast traffic, on the other hand, bidirected information flow across certain links can be critically needed to maximize the end-to-end capacity region. These results can be regarded as generalizations of the conclusions regarding the usefulness of feedback in various single-hop Gaussian settings and can provide opportunities for simplifying operation in Gaussian multihop networks. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

This paper shows that the Maddah-Ali–Tse (MAT) scheme, which achieves the symmetric capacity of two example broadcast channels with strictly causal state information at the transmitter, is a simple special case of the Shayevitz–Wigger (SW) scheme for the broadcast channel with generalized feedback, which involves block Markov coding, Gray–Wyner compression, superposition coding, and Marton coding. Focusing on the class of symmetric broadcast channels with state, we derive an expression for the maximum achievable symmetric rate using the SW scheme. We show that the MAT results for the two-receiver case can be recovered by evaluating this expression for the special case in which superposition coding and Marton coding are not used. We then introduce a new broadcast channel example that shares many features of the MAT examples. We show that another special case of our maximum symmetric rate expression in which superposition coding is also used attains a higher symmetric rate than the MAT scheme. The symmetric capacity of this new example is not known, however. [ABSTRACT FROM PUBLISHER]

We consider the problem of noisy Bayesian active learning where we are given a finite set of functions \mathcal H , a sample space \mathcal X , and a label set \mathcal L . One of the functions in \mathcal H assigns labels to samples in \mathcal X . The goal is to identify the function that generates the labels even though the result of a label query on a sample is corrupted by independent noise. More precisely, the objective is to declare one of the functions in \mathcal H as the true label generating function with high confidence using as a few label queries as possible, by selecting the queries adaptively and in a strategic manner. Previous work in Bayesian active learning considers generalized binary search and its variants for the noisy case, and analyzes the number of queries required by these sampling strategies. In this paper, we show that these schemes are, in general, suboptimal. Instead we propose and analyze an alternative strategy for sample collection. Our sampling strategy is motivated by a connection between Bayesian active learning and active hypothesis testing, and is based on querying the label of a sample, which maximizes the extrinsic Jensen–Shannon divergence at each step. We provide upper and lower bounds on the performance of this sampling strategy, and show that these bounds are better than the previous bounds in the literature. [ABSTRACT FROM PUBLISHER]

In the context of fix-free codes, the most important and immediate consequence of the 3/4-conjecture (if it is proven), is that the redundancy of an optimal binary fix-free code never exceeds 1 bit, as with the optimal prefix-free codes, i.e. Huffman codes. In this paper, this bound on the redundancy is proven without requiring the conjecture to be true. To do so, we use two known sufficient conditions for the existence of binary fix-free codes to derive an improved upper bound on the redundancy of an optimal fix-free code in terms of the largest symbol probability. [ABSTRACT FROM AUTHOR]

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a noncoherent multishot network, where the unknown and time-variant network is used several times. In order to create dependence between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding. [ABSTRACT FROM AUTHOR]

Let N=\n0, n1, \ldots , nk-1\ be a set of positive integers and M= \m0, m1, \ldots , mk-1\ be a multiset of positive integers. By an (N, M, w,1; \lambda ) -multilength optical orthogonal code (MLOOC), we mean an MLOOC of autocross correlation value and intracross correlation value one and intercross correlation value \lambda . The code contains mi codewords of weight w and length ni for $0\le i\le k-1$ . The study of MLOOCs is motivated by an application in optical networks requiring multiple signaling rates and quality-of-services. In this paper, we study $(N, M, w,1; \lambda )$ -MLOOCs with $\lambda =2$ (the least value among the nontrivial intercross correlations). Some new upper bounds on code size are derived under certain restrictions and a novel encoding approach is established. A number of series of new MLOOCs are then produced. These codes are of optimal sizes with respect to the new bounds. [ABSTRACT FROM AUTHOR]

This paper studies the achievable degrees of freedom (DoF) for multi-user multiple-input multiple-output (MIMO) two-way relay channels, where there are K source nodes, each equipped with M antennas, one relay node, equipped with N antennas, and each source node exchanges independent messages with an arbitrary set of other source nodes via the relay. By allowing an arbitrary information exchange pattern, the considered channel model is a unified one. It includes several existing channel models as special cases: 1) K -user MIMO Y channel; 2) multi-pair MIMO two-way relay channel; 3) generalized MIMO two-way X relay channel; and 4) L -cluster MIMO multiway relay channel. Previous studies mainly considered the achievability of the DoF cut-set bound 2N at the antenna configuration N<2M for the K -user MIMO Y channel; 2) \frac NM \in \big (0, 2+\frac 4K\big ] \cup \big [K-2, +\infty \big) for the multi-pair MIMO two-way relay channel; and 3) \frac NM \in \big (0, 2+\frac 8K^2\big ] \cup \big [K-2, +\infty \big) for the generalized MIMO two-way X relay channel. We also provide the antenna configuration regions for the general multi-user MIMO two-way relay channel to achieve the total DoF $KM$ . [ABSTRACT FROM AUTHOR]

In this paper, the difference between the redundancy of the optimal asymmetric/symmetric fix-free code, and that of the optimal prefix-free code is considered as the penalty of benefiting from the desired properties of fix-free codes. This penalty is studied from different perspectives. In particular, it is shown that the average penalty of asymmetric fix-free codes is less than 0.21 bit per symbol. Moreover, it is proved that when the source alphabet size is sufficiently large, for almost all sources, the penalty is less than or equal to 0.182 bit per symbol. Regarding symmetric fix-free codes, it is shown that the average penalty tends to infinity as the source alphabet size increases. [ABSTRACT FROM AUTHOR]

Motivated by the study of deletion channels, this paper presents improved bounds on the number of subsequences obtained from a binary string $X$ of length $n$ under $t$ deletions. It is known that the number of subsequences in this setting strongly depends on the number of runs in the string $X$ ; where a run is a maximal substring of the same character. Our improved bounds are obtained by a structural analysis of the family of $r$ -run strings $X$ , an analysis in which we identify the extremal strings with respect to the number of subsequences. Specifically, for every $r$ , we present $r$ -run strings with the minimum (respectively maximum) number of subsequences under any $t$ deletions; we perform an exact analysis of the number of subsequences of these extremal strings; and show that this number can be calculated in polynomial time. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies new bounds and code constructions that are applicable to the combinatorial granular channel model previously introduced by Sharov and Roth. We derive new bounds on the maximum cardinality of a grain-error-correcting code and propose constructions of codes that correct grain-errors. We demonstrate that a permutation of the classical group codes (e.g., Constantin–Rao codes) can correct a single grain-error. In many cases of interest, our results improve upon the currently best known bounds and constructions. Some of the approaches adopted in the context of grain-errors may have application to related channel models. [ABSTRACT FROM AUTHOR]

This paper studies adaptive sensing for estimating the nonzero amplitudes of a sparse signal with the aim of providing analytical guarantees on the performance gain due to adaptive resource allocation. We consider a previously proposed optimal two-stage policy for allocating sensing resources. For positive powers $q$ , we derive tight upper bounds on the mean $q$ th-power error resulting from the optimal two-stage policy and corresponding lower bounds on the improvement over nonadaptive uniform sensing. It is shown that the adaptation gain is related to the detectability of nonzero signal components as characterized by Chernoff coefficients, thus quantifying analytically the dependence on the sparsity level of the signal, the signal-to-noise ratio (SNR), and the sensing resource budget. For fixed sparsity levels and increasing SNR or sensing budget, we obtain the rate of convergence to oracle performance and the rate at which the fraction of resources spent on the first exploratory stage decreases to zero. For a vanishing fraction of nonzero components, the gain increases without bound as a function of SNR and sensing budget. Numerical simulations demonstrate that the bounds on adaptation gain are quite tight in nonasymptotic regimes as well. [ABSTRACT FROM AUTHOR]

The problem of detecting a single anomalous process among a finite number $M$ of processes is considered. At each time, a subset of the processes can be observed, and the observations from each chosen process follow two different distributions, depending on whether the process is normal or abnormal. The objective is a sequential search strategy that minimizes the expected detection time subject to an error probability constraint. This problem can be considered as a special case of active hypothesis testing first considered by Chernoff where a randomized strategy, referred to as the Chernoff test, was proposed and shown to be asymptotically (as the error probability approaches zero) optimal. For the special case considered in this paper, we show that a simple deterministic test achieves asymptotic optimality and offers better performance in the finite regime. We further extend the problem to the case where multiple anomalous processes are present. In particular, we examine the case where only an upper bound on the number of anomalous processes is known. [ABSTRACT FROM PUBLISHER]

Kitayama proposed a novel code-division multiple-access (CDMA) network for image transmission called spatial CDMA. Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial CDMA. An (m,n,k,\lambda ) -OOSPC is a set of m \times n~(0,1) -matrices with Hamming weight k and maximum correlation value \lambda . Let \Theta (m,n,k,\lambda ) be the largest possible number of codewords among all (m,n,k,\lambda ) -OOSPCs. In this paper, we concentrate on the calculation of the exact value of \Theta (m,n,3,1) and the construction of an when mn~\equiv ~14, 20~(\textrm {mod }24) , or mn~\equiv ~8, 16~(\textrm {mod }24) and \gcd (m,n,4)=2 , or mn~\equiv ~2~(\textrm {mod }6) and \gcd (m,n,4)=4$ , and \Theta (m,n,3,1)=\lfloor \frac {mn-1}{6}\rfloor otherwise. [ABSTRACT FROM AUTHOR]

The study of Armstrong codes is motivated by the problem of understanding complexities of dependences in relational database systems, where attributes have bounded domains. A $(q,k,n)$ -Armstrong code is a $q$ -ary code of length $n$ with minimum Hamming distance $n-k+1$ , and for any set of $k-1$ coordinates, there exist two codewords that agree exactly there. Let $f(q,k)$ be the maximum $n$ for which such a code exists. In this paper, $f(q,3)=3q-1$ is determined for all $q\geq 5$ with three possible exceptions. This disproves a conjecture of Sali. Furthermore, we introduce generalized Armstrong codes for branching, or $(s,t)$ -dependences, construct several classes of optimal Armstrong codes, and establish lower bounds for the maximum length $n$ in this more general setting. [ABSTRACT FROM AUTHOR]

In this paper, we study three-dimensional $(u\times v\times w,k,\lambda )$ optical orthogonal codes (OOCs) with at most one optical pulse per wavelength/time plane (AM-OPP) restriction, which is denoted by AM-OPP 3-D (u\times v\times w,k,\lambda )$ -OOC. We build an equivalence relation between such an OOC and a certain combinatorial subject, called a w$ -cyclic group divisible packing of type . By this link, the upper bound of the number of codewords is improved and some new combinatorial constructions are presented. As an application, the exact number of codewords of an optimal AM-OPP 3-D (u\times v\times w,3,1)$ -OOC is determined for any positive integers v,w$ , and with some possible exceptions. [ABSTRACT FROM PUBLISHER]

Vector space interference alignment (IA) is known to achieve high degrees of freedom (DoFs) with infinite-independent channel extensions, but its performance is largely unknown for a finite number of possibly dependent channel extensions. In this paper, we consider a K -user M_{t} \times M_{r} MIMO interference channel (IC) with an arbitrary number of channel extensions T and arbitrary channel diversity order L (i.e., each channel matrix is a generic linear combination of L fixed basis matrices). We study the maximum DoF achievable via vector space IA in the single beam case (i.e., each user sends one data stream). We prove that the total number of users K , where N = \min \MtT, MrT \ . An immediate consequence of this upper bound is that for a Single-Input Single-Output (SISO) IC the DoF in the single beam case is no more than \min \big \{\sqrt { {5}/{4}K}, L + {1}/{4}T\big \} . When the channel extensions are independent, i.e., L achieves the maximum M_{r} M_{t} T , we show that this maximum DoF lies in [M_{r} + Mt – 1, Mr + Mt] regardless of $T$ . Unlike the well-studied constant MIMO IC case, the main difficulty is how to deal with a hybrid system of equation (zero-forcing condition) and inequalities (full rank condition). Our approach combines algebraic tools that deal with equations with an induction analysis that indirectly considers the inequalities. [ABSTRACT FROM PUBLISHER]

The sliding window Lempel–Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios: 1) zero entropy sources that are characterized by strong long-term memory and 2) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of fixed shift variant of Lempel–Ziv (FSLZ) and SWLZ algorithms given in literature. Through an example of stationary and ergodic processes generated by an irrational rotation, we establish that for a window of size nw , a compression ratio given by O(\log nw/{{nw}^{a}}) , where $a$ depends on n_{w} \rightarrow \infty , is obtained under the application of FSLZ and SWLZ algorithms. In addition, we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy. Next, we extend the study of Ornstein and Weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a class of mixing processes. In addition, an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions. [ABSTRACT FROM PUBLISHER]

We consider the $N$ -relay Gaussian diamond network where a source node communicates to a destination node via $N$ parallel relays through a cascade of a Gaussian broadcast (BC) and a multiple access (MAC) channel. Introduced in 2000 by Schein and Gallager, the capacity of this relay network is unknown in general. The best currently available capacity approximation, independent of the coefficients and the SNRs of the constituent channels, is within an additive gap of $1.3 N$ bits, which follows from the recent capacity approximations for general Gaussian relay networks with arbitrary topology. In this paper, we approximate the capacity of this network within 2 log $N$ bits. We show that two strategies can be used to achieve the information-theoretic cut-set upper bound on the capacity of the network up to an additive gap of $O(\log N)$ bits, independent of the channel configurations and the SNRs. The first of these strategies is simple partial decode-and-forward. Here, the source node uses a superposition codebook to broadcast independent messages to the relays at appropriately chosen rates; each relay decodes its intended message and then forwards it to the destination over the MAC channel. A similar performance can be also achieved with compress-and-forward type strategies (such as quantize-map-and-forward and noisy network coding) that provide the $1.3 N$ -bit approximation for general Gaussian networks, but only if the relays quantize their observed signals at a resolution inversely proportional to the number of relay nodes $N$ . This suggests that the rule-of-thumb to quantize the received signals at the noise level in the current literature can be highly suboptimal. [ABSTRACT FROM AUTHOR]

This paper is devoted to the derivation of novel analytic expressions and bounds for a family of special functions that are useful in wireless communication theory. These functions are the well-known Nuttall Q- function, incomplete Toronto function, Rice Ie- function, and incomplete Lipschitz–Hankel integrals. Capitalizing on the offered results, useful identities are additionally derived between the above functions and Humbert, \Phi _{1} , function as well as for specific cases of the Kampéde Fériet function. These functions can be considered as useful mathematical tools that can be employed in applications relating to the analytic performance evaluation of modern wireless communication systems, such as cognitive radio, cooperative, and free-space optical communications as well as radar, diversity, and multiantenna systems. As an example, new closed-form expressions are derived for the outage probability over nonlinear generalized fading channels, namely, \alpha {-}\eta {-}\mu , \alpha -\lambda -\mu , and \alpha -\kappa -\mu as well as for specific cases of the \eta -\mu and \lambda -\mu fading channels. Furthermore, simple expressions are presented for the channel capacity for the truncated channel inversion with fixed rate and corresponding optimum cutoff signal-to-noise ratio for single-antenna and multiantenna communication systems over Rician fading channels. The accuracy and validity of the derived expressions is justified through extensive comparisons with respective numerical results. [ABSTRACT FROM AUTHOR]

Write-once memory (WOM) is a storage medium with memory elements, called cells, which can take on \(q\) levels. Each cell is initially in level 0 and can only increase its level. A \(t\) -write WOM code is a coding scheme, which allows one to store \(t\) messages to the WOM such that on consecutive writes every cell’s level does not decrease. The sum-rate of the WOM code, which is the ratio between the total amount of information written in the \(t\) writes and number of memory cells, is bounded by \(\log (t+1)\) . Our main contribution in this paper is a construction of binary three-write WOM codes with sum-rate approaching 1.885 for sufficiently large number of cells, whereas the upper bound is 2. This improves upon a recent construction of sum-rate 1.809. A key ingredient in our construction is a recent capacity achieving construction of two-write WOM codes, which uses the so-called Wozencraft ensemble of linear codes. In our construction, we encode information in the first and second write in a way that leaves a large number (roughly half) of the cells nonprogrammed. This allows us to use the above two-write construction in order to invoke a third write to the memory. We also give specific constructions of nonbinary two-write WOM codes and multiple writes, which give better sum-rate than the currently best known ones. In the construction of these codes, we build upon previous nonbinary constructions and show how tools such symbols relabeling can help in achieving high sum-rates. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the minimum energy of transmitting correlated sources over the Gaussian multiple-access channel. Compared to other works on joint source-channel coding, we consider the fundamental problem of the minimum transmission energy, where the source and channel bandwidths are not naturally matched. Different models of correlated sources are studied. We first treat lossy transmission of Gaussian sources, including multiterminal sources and CEO sources. We then consider lossless transmission of correlated binary sources. In all cases, we lower bound the minimum energy using a cut-set argument that couples transmission energy and the distortions for the Gaussian cases (or source entropy for the discrete case). For the achievable schemes, separate source and channel coding and uncoded transmission are studied as benchmarks. In addition, we show that hybrid digital/analog transmission achieves the best known energy efficiency. [ABSTRACT FROM AUTHOR]

We consider linear regression in the high-dimensional regime where the number of observations \(n\) is smaller than the number of parameters \(p\) . A very successful approach in this setting uses \(\ell _{1}\) -penalized least squares (also known as the Lasso) to search for a subset of \(s_{0}< n\) parameters that best explain the data, while setting the other parameters to zero. Considerable amount of work has been devoted to characterizing the estimation and model selection problems within this approach. In this paper, we consider instead the fundamental, but far less understood, question of statistical significance. More precisely, we address the problem of computing p-values for single regression coefficients. On one hand, we develop a general upper bound on the minimax power of tests with a given significance level. We show that rigorous guarantees for earlier methods do not allow to achieve this bound, except in special cases. On the other, we prove that this upper bound is (nearly) achievable through a practical procedure in the case of random design matrices with independent entries. Our approach is based on a debiasing of the Lasso estimator. The analysis builds on a rigorous characterization of the asymptotic distribution of the Lasso estimator and its debiased version. Our result holds for optimal sample size, i.e., when \(n\) is at least on the order of \(s_{0} \log (p/s_{0})\) . We generalize our approach to random design matrices with independent identically distributed Gaussian rows \( \boldsymbol {x}_{i}\sim {\sf N} (0, \boldsymbol {\Sigma })\) . In this case, we prove that a similar distributional characterization (termed standard distributional limit) holds for \(n\) much larger than \(s_{0}(\log p)^{2}\) . Our analysis assumes \( \boldsymbol {\Sigma }\) is known. To cope with unknown \( \boldsymbol {\Sigma }\) , we suggest a plug-in estimator for sparse covariances \( \boldsymbol {\Sigma }\) and validate the method through numerical simulations. Finally, we show that for optimal sample size, \(n\) being at least of order \(s_{0} \log (p/s_{0})\) , the standard distributional limit for general Gaussian designs can be derived from the replica heuristics in statistical physics. This derivation suggests a stronger conjecture than the result we prove, and near-optimality of the statistical power for a large class of Gaussian designs. [ABSTRACT FROM PUBLISHER]

Partial-period correlations of sequences are an important performance measure of communication systems employing them, but are known to be notoriously difficult to analyze. In this paper, we present a systematic approach to partial-period correlations as a generalization of the Paterson–Lothian approach. For a pair of sequences (not necessarily distinct), we introduce the linear phase-shifting sequences of one of them and analyze their full-period correlations, called the induced correlations, with the other one. By exploiting the induced correlations, we obtain the partial-period correlation properties of the given pair. We then investigate Zadoff–Chu sequences, and give some theoretical results on their partial-period autocorrelations, including closed-form expressions, and the first and second moments. We also derive an upper bound on the magnitudes of the partial-period cross-correlations of Zadoff–Chu sequences. Finally, we explore generalized chirp-like discrete Fourier transform sequences and show that their partial-period cross-correlations have the same magnitudes as the partial-period autocorrelations of the employed Zadoff–Chu sequence up to permutation. [ABSTRACT FROM AUTHOR]

A sufficient condition reported very recently for perfect recovery of a \(K\) -sparse vector via orthogonal matching pursuit (OMP) in \(K\) iterations (when there is no noise) is that the restricted isometry constant (RIC) of the sensing matrix satisfies \(\delta _{K+1} <({1}/{\sqrt {K} +1})\) . In the noisy case, this RIC upper bound along with a requirement on the minimal signal entry magnitude is known to guarantee exact support identification. In this paper, we show that, in the presence of noise, a relaxed RIC upper bound \(\delta _{K+1} <({\sqrt {4K+1} -1}/{2K})\) together with a relaxed requirement on the minimal signal entry magnitude suffices to achieve perfect support identification using OMP. In the noiseless case, our result asserts that such a relaxed RIC upper bound can ensure exact support recovery in \(K\) iterations: this narrows the gap between the so far best known bound \(\delta _{K+1} <({1}/{\sqrt {K} +1})\) and the ultimate performance guarantee \(\delta _{K+1} =({1}/{\sqrt {K}})\) . Our approach relies on a newly established near orthogonality condition, characterized via the achievable angles between two orthogonal sparse vectors upon compression, and, thus, better exploits the knowledge about the geometry of the compressed space. The proposed near orthogonality condition can be also exploited to derive less restricted sufficient conditions for signal reconstruction in two other compressive sensing problems, namely, compressive domain interference cancellation and support identification via the subspace pursuit algorithm. [ABSTRACT FROM PUBLISHER]

Bidirectional decoding improves the error resilience of the fix-free codes, which is important in some applications like video coding. However, in general, two separate decoders must be designed for forward and backward decoding and hence, the cost of decoding is almost doubled. For the symmetric fix-free codes, although the forward and backward decoders are the same, the redundancy could be much greater than the case of asymmetric fix-free codes. In this paper, a class of fix-free codes, called weakly symmetric, is studied. A code is weakly symmetric if the reverse of each codeword itself is a codeword. This imposes a much weaker constraint on the codewords than what is necessitated by the symmetric codes, i.e., each codeword must be equal to its reverse. We show that the forward and backward decoders of a weakly symmetric fix-free code are essentially the same. In addition, it is shown that two significant sufficient conditions for the existence of asymmetric fix-free codes are also applicable to the case of weakly symmetric fix-free codes. As a result, we conclude that for any memoryless source, the redundancy of the optimal weakly symmetric fix-free code is at most 1.0817 bits. Moreover, it can be only 0.8 bit greater than the redundancy of the Huffman code in the worst case. In summary, weakly symmetric fix-free codes can be regarded as an option by which one can to some extent achieve the low redundancy of asymmetric fix-free codes and the low-cost bidirectional decoding of symmetric fix-free codes, simultaneously. [ABSTRACT FROM AUTHOR]

Multimedia fingerprinting is an effective technique to trace the sources of pirate copies of copyrighted multimedia contents. Separable codes were introduced to detect colluders taking part in the averaging attack, which is the most feasible approach to perform a collusion attack. In this paper, we provide some improved bounds for the size of separable codes. Using a combinatorial technique named grouping coordinates, we have greatly reduced the upper bound. On the other hand, we also provide some lower bounds for the size of separable codes by both probabilistic and deterministic construction methods. In particular, \( \overline {2}\) -separable codes with asymptotically optimal rate are obtained by the deletion method. [ABSTRACT FROM AUTHOR]

Regenerating codes enable trading off repair bandwidth for storage in distributed storage systems (DSS). Due to their distributed nature, these systems are intrinsically susceptible to attacks, and they may also be subject to multiple simultaneous node failures. Cooperative regenerating codes allow bandwidth efficient repair of multiple simultaneous node failures. This paper analyzes storage systems that employ cooperative regenerating codes that are robust to (passive) eavesdroppers. The analysis is divided into two parts, studying both minimum bandwidth and minimum storage cooperative regenerating scenarios. First, the secrecy capacity for minimum bandwidth cooperative regenerating codes is characterized. Second, for minimum storage cooperative regenerating codes, a secure file size upper bound and achievability results are provided. These results establish the secrecy capacity for the minimum storage scenario for certain special cases. In all scenarios, the achievability results correspond to exact repair, and secure file size upper bounds are obtained using min-cut analyses over a suitable secrecy graph representation of DSS. The main achievability argument is based on an appropriate precoding of the data to eliminate the information leakage to the eavesdropper. [ABSTRACT FROM AUTHOR]

This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed entanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Christandl and Winter, establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed \(\log ( ( 1+\eta ) / ( 1-\eta ) ) \) for a pure-loss bosonic channel for which a fraction \(\eta \) of the input photons make it to the output on average. The best known lower bound on these capacities is equal to \(\log ( 1 / ( 1-\eta ) )\) . Thus, in the high-loss regime for which \(\eta \ll 1\) , this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal. [ABSTRACT FROM PUBLISHER]

In this paper, we revisit the combinatorial error model of Mazumdar et al. that models errors in high-density magnetic recording caused by lack of knowledge of grain boundaries in the recording medium. We present new upper bounds on the cardinality/rate of binary block codes that correct errors within this model. All our bounds, except for one, are obtained using combinatorial arguments based on hypergraph fractional coverings. The exception is a bound derived via an information-theoretic argument. Our bounds significantly improve upon existing bounds from the prior literature. [ABSTRACT FROM PUBLISHER]

This paper investigates the effect of sub-Nyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear time-invariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of right-invertible time-preserving sampling methods which includes irregular nonuniform sampling, and characterize in closed form the channel capacity achievable by this class of sampling methods, under a sampling rate and power constraint. Our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal-to-noise ratio among all spectral sets of measure equal to the sampling rate. This can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates, or through a single branch of modulation and filtering followed by uniform sampling. These results reveal that for a large class of channels, employing irregular nonuniform sampling sets, while are typically complicated to realize in practice, does not provide capacity gain over uniform sampling sets with appropriate preprocessing. Our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario, which is in contrast to the benefits obtained from random mixing in spectrum-blind compressive sampling schemes. [ABSTRACT FROM PUBLISHER]

We consider deletion correcting codes over a \(q\) -ary alphabet. It is well known that any code capable of correcting \(s\) deletions can also correct any combination of \(s\) total insertions and deletions. To obtain asymptotic upper bounds on code size, we apply a packing argument to channels that perform different mixtures of insertions and deletions. Even though the set of codes is identical for all of these channels, the bounds that we obtain vary. Prior to this paper, only the bounds corresponding to the all-insertion case and the all-deletion case were known. We recover these as special cases. The bound from the all-deletion case, due to Levenshtein, has been the best known for more than 45 years. Our generalized bound is better than Levenshtein's bound whenever the number of deletions to be corrected is larger than the alphabet size. [ABSTRACT FROM AUTHOR]

To the body of works on rewrite codes for constrained memories, we add a comprehensive study in a direction that is especially relevant to practical storage. The subject of this paper is codes for the \(q\) -ary extension of the write-once memories model, with input sizes that are fixed throughout the write sequence. Seven code constructions are given with guarantees on the number of writes they can support. For the parameters addressed by the constructions, we also prove upper bounds on the number of writes, which prove the optimality of three of the constructions. We concentrate on codes with short block lengths to keep the complexity of decoding and updates within the feasibility of practical implementation. Even with these short blocks the constructed codes are shown to be within a small additive constant from capacity for an arbitrarily large number of input bits. Part of the study addresses a new rewrite model where some of the input bits can be updated multiple times in a write sequence (hot bits), while other are updated at most once (cold bits). We refer to this new model as hot/cold rewrite codes. It is shown that adding cold bits to a rewrite code has a negligible effect on the total number of writes, while adding an important feature of leveling the physical wear of memory cells between hot and cold input data. [ABSTRACT FROM AUTHOR]

This paper corrects errors in the calculation of the normalized maximum likelihood (NML) code length for a Gaussian mixture model (GMM). It shows that the NML code length calculated in “efficient computation of the NML codes for the GMMs with its applications to clustering” is an upper bound on the NML code length strictly calculated for the GMM. In addition, we correct the NML code length for generalized logistic distribution. [ABSTRACT FROM AUTHOR]

We correct an intermediate step in the derivation of our main statements in [1]. Specifically, inequality (63) in the mentioned paper, intended to give an upper bound on the entropy of certain Gaussian states, is incorrect. In that paper, we used inequality (63) to derive the asymptotic (large-time) scaling of the entropy under the quantum version of the heat equation. We provide alternative derivations of this result, which sidestep the bound (63). The main conclusions of the paper therefore remain unaffected. We thank Giacomo de Palma, Andrea Mari and Vittorio Giovannetti for pointing out this issue, and simultaneously providing us with a resolution in a very detailed communication. We also thank them for their permission to include their discussion in this erratum. We note that one of the alternative derivations of the asymptotic scaling presented here has previously appeared in their publication. [ABSTRACT FROM PUBLISHER]