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Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$ , i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this paper, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$ , dimension $k$ , locality $r$ , availability $t$ , and field size $q$ . Then, we study the binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed–Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared with other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, the availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code. [ABSTRACT FROM PUBLISHER]

The restricted isometry property (RIP) is a fundamental property of a matrix, which enables sparse recovery. Informally, an m \times n matrix satisfies RIP of order k for the \ell p norm, if \Ax\p \approx \x\p for every vector x with at most k non-zero coordinates. For every 1 \leq p < \infty , we obtain almost tight bounds on the minimum number of rows m necessary for the RIP property to hold. Prior to this paper, only the cases p = 1 , 1 + 1 / \log n , and 2 were studied. Interestingly, our results show that the case p = 2 for all p\in [1,\infty )\setminus \2\ , as opposed to \widetilde \Theta (k) for $p=2$ . We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss the implications of our results for the stable sparse recovery problem as defined by Candès et al. [ABSTRACT FROM AUTHOR]

We study the generalized degrees of freedom (gDoF) of the noncoherent diamond (parallel relay) wireless network with asymmetric distributions of link strengths. We use the noncoherent block-fading model introduced by Marzetta and Hochwald, where no channel state information is available at the transmitters or at the receivers and the channels remain constant for a coherence time of $T$ symbol durations. We first derive an upper bound for the capacity of this channel and then derive the optimal structure for the solution of the upper bound optimization problem. Using the optimal structure, we solve the upper bound optimization problem in terms of its gDoF. Using insights from our upper bound signaling solution, we devise an achievability strategy based on a novel scheme that we call train-scale quantize-map-forward (TS-QMF). This scheme uses training in the links from the source to the relays, scaling and quantizing at the relays combined with nontraining-based schemes. We show the optimality of this scheme by comparing it to the upper bound in terms of the gDoF. In noncoherent point-to-point multiple-input-multiple-output (MIMO) channels, where the fading realization is unknown to the transmitter and the receiver, an important tradeoff between communication and channel learning was revealed by Zheng and Tse, by demonstrating that not all the available antennas might be used, as it is suboptimal to learn all their channel parameters. Our results in this paper for the diamond network demonstrate that in certain regimes of relative channel strengths, the gDoF-optimal scheme uses a subnetwork, demonstrating a similar tradeoff between channel learning and communication. In some regimes, it is gDoF-optimal to do relay selection, i.e., use a part of the network. In the other regimes, even when it is essential to use the entire network, it is suboptimal to learn the channel states for all the links in the network, i.e., traditional training-based schemes are suboptimal in these regimes. [ABSTRACT FROM AUTHOR]

In this paper, locally repairable codes which have optimal minimum Hamming distance with respect to the bound presented by Prakash et al. are considered. New upper bounds on the length of such optimal codes are derived. The new bounds apply to more general cases, and have weaker requirements compared with the known ones. In this sense, they both improve and generalize previously known bounds. Further, optimal codes are constructed, whose length is order-optimal with respect to the new upper bounds. Notably, the length of the codes is super-linear in the alphabet size. [ABSTRACT FROM AUTHOR]

Recently, two extensions of Wyner’s common information—exact and Rényi common informations—were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution. For the exact common information problem, exact generation of the target distribution is required, while for Wyner’s and $\alpha $ -Rényi common informations, the relative entropy and Rényi divergence with order $\alpha $ were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner’s common information. However, it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions. In this paper, we first establish the equivalence between the exact and $\infty $ -Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and $\infty $ -Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner’s. This answers an open problem posed by KLE. Furthermore, we extend Wyner’s, $\infty $ -Rényi, and exact common informations to sources with countably infinite or continuous alphabets, including Gaussian sources. [ABSTRACT FROM AUTHOR]

Write-once memory (WOM) is a storage device consisting of binary cells that can only increase their levels. A $t$ -write WOM code is a coding scheme that makes it possible to write $t$ times to a WOM without decreasing the levels of any of the cells. The sum-rate of a WOM code is the ratio between the total number of bits written to the memory during the $t$ writes and the number of cells. It is known that the maximum possible sum-rate of a $t$ -write WOM code is $\log (t+1)$. This is also an achievable upper bound, both by information-theoretic arguments and through explicit constructions. While existing constructions of WOM codes are targeted at the sum-rate, we consider here two more figures of merit. The first one is the complexity of the encoding and decoding maps. The second figure of merit is the convergence rate, defined as the minimum code length $n(\delta)$ required to reach a point that is $\delta $ -close to the capacity region. One of our main results in this paper is a capacity-achieving construction of two-write WOM codes which has polynomial encoding/decoding complexity while the block length $n(\delta)$ required to be $\delta $ -close to capacity is significantly smaller than existing constructions. Using these two-write WOM codes, we then obtain three-write WOM codes that approach a sum-rate of 1.809 at relatively short block lengths. We also provide several explicit constructions of finite length three-write WOM codes; in particular, we achieve a sum-rate of 1.716 by using only 93 cells. Finally, we modify our two-write WOM codes to construct $\epsilon $ -error WOM codes of high rates and small probability of failure. [ABSTRACT FROM AUTHOR]

Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation ball sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing upper and lower bounds on multipermutation Ulam ball sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric. [ABSTRACT FROM AUTHOR]

In this paper, we consider the additive white Gaussian noise (AWGN) channel with a power constraint called the (\sigma , \rho ) -power constraint, which is motivated by energy harvesting communication systems. Given a codeword, the constraint imposes a limit of \sigma + k \rho on the total power of any k\geq 1 consecutive transmitted symbols. Such a channel has infinite memory and evaluating its exact capacity is a difficult task. Consequently, we establish an n , which is the set of all length n sequences satisfying the (\sigma , \rho ) -power constraints. For a noise power of \nu , which is the Minkowski sum of {\mathcal{ S}}_{n}(\sigma , \rho ) and the n -dimensional Euclidean ball of radius \sqrt n\nu . We analyze this bound using a result from convex geometry known as Steiner’s formula, which gives the volume of this Minkowski sum in terms of the intrinsic volumes of \mathcal Sn(\sigma , \rho ) . We show that as the dimension n increases, the logarithm of the sequence of intrinsic volumes of \{ {\mathcal{ S}}_{n}(\sigma , \rho )\} converges to a limit function under an appropriate scaling. The upper bound on capacity is then expressed in terms of this limit function. We derive the asymptotic capacity in the low- and high-noise regime for the (\sigma , \rho )$ -power constrained AWGN channel, with strengthened results for the special case of $\sigma = 0$ , which is the amplitude constrained AWGN channel. [ABSTRACT FROM PUBLISHER]

In secure multiparty computation (MPC), mutually distrusting users collaborate to compute a function of their private data without revealing any additional information about their data to the other users. While it is known that information theoretically secure MPC is possible among $n$ users having access to private randomness and are pairwise connected by secure, noiseless, and bidirectional links against the collusion of less than $n/2$ users (in the honest-but-curious model; the threshold is $n/3$ in the malicious model), relatively little is known about the communication and randomness complexity of secure computation, i.e., the amount of communication and randomness required to compute securely. In this paper, we employ information theoretic techniques to obtain lower bounds on communication and randomness complexity of secure MPC. We restrict ourselves to a concrete interactive setting involving three users under which all functions are securely computable against corruption of individual users in the honest-but-curious model. We derive lower bounds for both the perfect security case (i.e., zero-error and no leakage of information) and asymptotic security (where the probability of error and information leakage vanish as block-length goes to $\infty $ ). Our techniques include the use of a data processing inequality for residual information (i.e., the gap between mutual information and Gács–Körner common information), a new information inequality for three-user protocols, and the idea of distribution switching by which lower bounds computed under certain worst case scenarios can be shown to apply for the general case. Our lower bounds are shown to be tight for various functions of interest. In particular, we show concrete functions which have communication-ideal protocols, i.e., which achieve the minimum communication simultaneously on all links in the network. Also, we obtain the first explicit example of a function that incurs a higher communication cost than the input length, in the secure computation model of Feige et al. (26th Annual ACM Symposium on Theory of Computing, 1994), who had shown that such functions exist. We also show that our communication bounds imply tight lower bounds on the amount of randomness required by MPC protocols for many interesting functions. [ABSTRACT FROM PUBLISHER]

The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csiszár and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, R \text {SK} , defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by R \text {CO} , is an upper bound on R \text {SK} . For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain Type \mathcal S condition, which is verifiable in polynomial time, guarantees that our lower bound on R \text {SK} meets the R \text {CO} upper bound. Thus, the PIN models satisfying our condition are R \text {SK} -maximal, indicating that the upper bound R \text {SK} \le R \text {CO} holds with equality. This allows us to explicitly evaluate R \text {SK} for such PIN models. We also give several examples of PIN models that satisfy our Type \mathcal S condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type \mathcal S condition implies that communication from all terminals (omnivocality) is needed for establishing an SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type \mathcal S condition is satisfied. However, for the source models with four or more terminals, counterexamples exist showing that the converse does not hold in general. [ABSTRACT FROM PUBLISHER]

The squashed entanglement of a quantum channel is an additive function of quantum channels, which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication. This quantity has led to an upper bound of $\log ((1+\eta )/(1-\eta ))$ on the capacity of a pure-loss bosonic channel for such a task, where $\eta $ is the average fraction of photons that make it from the input to the output of the channel. The purpose of this paper is to extend these results beyond the single-sender single-receiver setting to the more general case of a single sender and multiple receivers (a quantum broadcast channel). We employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel, along the way developing several new properties of these measures. We apply our results to the case of a pure-loss broadcast channel with one sender and two receivers. [ABSTRACT FROM AUTHOR]

This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid and, thus, generalizes invariants that lie at the heart of several fundamental problems in a coding theory. An upper bound on the covering dimension is proved, improving Kung’s classical bound for the critical exponent. In addition, a construction is given for linear codes that attain equality in this covering dimension bound. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In this paper, we consider linear secret sharing schemes over a finite field $\mathbb {F}_{q}$ , where the secret is a vector in $\mathbb {F}_{q}^\ell $ and each of the $n$ shares is a single element of $\mathbb {F}_{q}$. We obtain lower bounds on the so-called threshold gap $g$ of such schemes, defined as the quantity $r-t$ where $r$ is the smallest number such that any subset of $r$ shares uniquely determines the secret and $t$ is the largest number such that any subset of $t$ shares provides no information about the secret. Our main result establishes a family of bounds which are tighter than previously known bounds for $\ell \geq 2$. Furthermore, we also provide bounds, in terms of $n$ and $q$ , on the partial reconstruction and privacy thresholds, a more fine-grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size. Finally, we compare our lower bounds with known upper bounds in the asymptotic setting. [ABSTRACT FROM AUTHOR]

We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability. New upper and lower bounds on the minimum distance of such codes are derived. The upper bound is based on the shortening method and generalized Hamming weights that are fundamental parameters of any linear codes with many useful applications. This bound improves existing upper bounds. To reduce the gap in between upper and lower bounds, we do not restrict the alphabet size and propose explicit constructions of codes with locality and availability via rank-metric codes. The first construction relies on expander graphs and is better in low rate region. The second construction utilizes the LRC codes developed by Wang et al. as inner codes and is better in high rate region. We also suggest one possible generalization of LRC codes in which the recovering sets can intersect in a small number of coordinates. This feature allows us to increase the achievable code rate and still meet load balancing requirements. We derive upper and lower bounds on the parameters of such codes and present explicit constructions of codes with such a property. [ABSTRACT FROM AUTHOR]

In this paper, we study data-dependent generalization error bounds that exhibit a mild dependency on the number of classes, making them suitable for multi-class learning with a large number of label classes. The bounds generally hold for empirical multi-class risk minimization algorithms using an arbitrary norm as the regularizer. Key to our analysis is new structural results for multi-class Gaussian complexities and empirical $\ell _\infty $ -norm covering numbers, which exploit the Lipschitz continuity of the loss function with respect to the $\ell _{2}$ - and $\ell _\infty $ -norm, respectively. We establish data-dependent error bounds in terms of the complexities of a linear function class defined on a finite set induced by training examples, for which we show tight lower and upper bounds. We apply the results to several prominent multi-class learning machines and show a tighter dependency on the number of classes than the state of the art. For instance, for the multi-class support vector machine of Crammer and Singer (2002), we obtain a data-dependent bound with a logarithmic dependency, which is a significant improvement of the previous square-root dependency. The experimental results are reported to verify the effectiveness of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the problem of finding the longest systematic-length k for a linear minimum storage regenerating (MSR) code with optimal repair of only systematic part, for a given per-node storage capacity l and an arbitrary number of parity nodes r. We study the problem by following a geometric analysis of linear subspaces and operators. First, a simple quadratic bound is given, which implies that k = r + 2 is the largest number of systematic nodes in the scalar scenario. Second, an r-based-log bound is derived, which is superior to the upper bound on log-base 2 in the prior work. Finally, an explicit upper bound depending on the value of r 2 /l is introduced, which further extends the corresponding result in the literature. [ABSTRACT FROM AUTHOR]

A combinatorial framework for adversarial network coding is presented. Channels are described by specifying the possible actions that one or more (possibly coordinated) adversaries may take. Upper bounds on three notions of capacity—the one-shot capacity, the zero-error capacity, and the compound zero-error capacity—are obtained for point-to-point channels, and generalized to corresponding capacity regions appropriate for multi-source networks. A key result of this paper is a general method by which bounds on these capacities in point-to-point channels may be ported to networks. This technique is illustrated in detail for Hamming-type channels with multiple adversaries operating on specific coordinates, which correspond, in the context of networks, to multiple adversaries acting on specific network edges. Capacity-achieving coding schemes are described for some of the considered adversarial models. [ABSTRACT FROM AUTHOR]

For nonnegative integers $n$ , $d$ , and $w$ , let $A(n,d,w)$ be the maximum size of a code $C \subseteq \mathbb {F}_{2}^{n}$ with a constant weight $w$ and minimum distance at least $d$. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on $A(n,d,w)$. The new upper bounds imply that $A(22,8,10)=616$ and $A(22,8,11)=672$. Lower bounds on $A(22,8,10)$ and $A(22,8,11)$ are obtained from the $(n,d)=(22,7)$ shortened Golay code of size 2048. It can be concluded that the shortened Golay code is a union of constant-weight $w$ codes of sizes $A(22,8,w)$. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

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]

We study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop-outs or symbol insertions with particular emphasis on the binary and non-binary deletion channels. We offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels. The resulting output symbols are combined appropriately to come up with an equivalent input–output representation of the original channel which allows for derivation of new upper bounds on the channel capacity. We consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels. With two specific applications of this idea, a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel, we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels. The resulting inequalities prove useful in deriving tighter capacity upper bounds for: 1) independent identically distributed (i.i.d.) deletion channels when the deletion probability exceeds 0.65 and 2) nonbinary deletion channels. Some extensions of these results, for instance, to the case of deletion/substitution channels are also explored. [ABSTRACT FROM PUBLISHER]

An erasure broadcast network is considered with two disjoint sets of receivers: a set of weak receivers with all-equal erasure probabilities and equal cache sizes and a set of strong receivers with all-equal erasure probabilities and no cache memories. Lower and upper bounds are presented on the capacity-memory tradeoff of this network (the largest rate at which messages can be reliably communicated for given cache sizes). The lower bound is achieved by means of a joint cache-channel coding scheme and significantly improves over traditional schemes based on the separate cache-channel coding. In particular, it is shown that the joint cache-channel coding offers new global caching gains that scale with the number of strong receivers in the network. The upper bound uses bounding techniques from degraded broadcast channels and introduces an averaging argument to capture the fact that the contents of the cache memories are designed before knowing users’ demands. The derived upper bound is valid for all stochastically degraded broadcast channels. The lower and upper bounds match for a single weak receiver (and any number of strong receivers) when the cache size does not exceed a certain threshold. Improved bounds are presented for the special case of a single weak and a single strong receiver with two files and the bounds are shown to match over a large range of cache sizes. [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]

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]

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]

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]

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]

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. These lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality. [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]

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 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]

The Stinson–Wei traceability scheme (known as traceability scheme) was proposed for broadcast encryption as a generalization of the Chor–Fiat–Naor traceability scheme (known as traceability code). Cover-free family was introduced by Kautz and Singleton in the context of binary superimposed code. In this paper, we find a new relationship between a traceability scheme and a cover-free family, which strengthens the anti-collusion strength from t to t^{2} , i.e., a t$ -traceability scheme is a t^{2} -cover-free family. Based on this interesting discovery, we derive new upper bounds for traceability schemes. By using combinatorial structures, we construct several infinite families of optimal traceability schemes, which attain our new upper bounds. We also provide a constructive lower bound for traceability schemes, the size of which has the same order of magnitude as our general upper bound. Meanwhile, we consider parent-identifying set systems, an anti-collusion key-distributing scheme requiring weaker conditions than traceability scheme but stronger conditions than cover-free family. A new upper bound is also given for parent-identifying set systems. [ABSTRACT FROM PUBLISHER]

Maximal repetition of a string is the maximal length of a repeated substring. This paper investigates maximal repetition of strings drawn from stochastic processes. Strengthening previous results, two new bounds for the almost sure growth rate of maximal repetition are identified: an upper bound in terms of conditional Rényi entropy of order $\gamma >1$ given a sufficiently long past and a lower bound in terms of unconditional Shannon entropy ( $\gamma =1$ ). Both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time. We also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity. In particular, it is shown that a power-law logarithmic growth of maximal repetition with respect to the string length, recently observed for texts in natural language, may hold only if the conditional Rényi entropy rate given a sufficiently long past equals zero. According to this observation, natural language cannot be faithfully modeled by a typical hidden Markov process, which is a class of basic language models used in computational linguistics. [ABSTRACT FROM AUTHOR]

Cutting a cake is a metaphor for the problem of dividing a resource (cake) among several agents. The problem becomes non-trivial when the agents have different valuations for different parts of the cake (i.e., one agent may like chocolate, while the other may like cream). A fair division of the cake is one that takes into account the individual valuations of agents and partitions of the cake based on some fairness criterion. Fair division may be accomplished in a distributed or centralized way. Due to its natural and practical appeal, it has been a subject of study in economics. To the best of our knowledge, the role of partial information in fair division has not been studied so far from an information theoretic perspective. Given the diversity of problems in fair division, we consider certain specific (yet important) problems that capture different aspects of information exchange in a fair division setting. From the class of distributed algorithms, we consider the classical divide and choose (DC) problem between two parties. Here, we study the effect of partial spying and voluntarily sharing of information in both one-shot and asymptotic scenarios. Furthermore, we consider implicit information transmission through actions for the repeated version of the problem. While identifying subgame perfect Nash equilibrium in repeated games with incomplete information on both sides is very difficult in general, for the special case of division of two items, we find a more stringent trembling hand perfect equilibrium. Next, from the class of centralized algorithms, we consider the Adjusted Winner (AW) algorithm between two players Alice and Bob. Brams and Taylor showed that if Alice can fully spy on Bob, she can trick the algorithm. We consider the same setup when partial spying is allowed and study the growth rate of Alice’s utility per spying bit. Via a transformation from AW to DC, it is shown that the problem reduces to the one studied earlier for DC. However, if Alice is forced to only spy certain simple structured functions of Bob’s valuation, an upper bound on the growth rate of utility per spying bit is derived. This bound is shown to be tight in some cases. We also consider a centralized algorithm for maximizing the overall welfare of the agents under the Nash collective utility function (CUF). This corresponds to a clustering problem. By observing a link between this problem and the portfolio selection problem in stock markets, we provide an upper bound on the increase of the Nash CUF for a clustering refinement. [ABSTRACT FROM PUBLISHER]

Binary-input memoryless channels with a run length constrained input are considered. Upper bounds to the capacity of such noisy run length constrained channels are derived using the dual capacity method with Markov test distributions satisfying the Karush-Kuhn-Tucker conditions for the capacity-achieving output distribution. Simplified algebraic characterizations of the bounds are presented for the binary erasure channel and the binary symmetric channel. These upper bounds are very close to achievable rates, and improve upon previously known feedback-based bounds for a large range of channel parameters. For the binary-input additive white Gaussian noise channel, the upper bound is simplified to a small-scale numerical optimization problem. These results provide some of the simplest upper bounds for an open capacity problem that has theoretical and practical relevance. [ABSTRACT FROM PUBLISHER]

We establish the first known upper bound on the exact and Wyner’s common information of n continuous random variables in terms of the dual total correlation between them (which is a generalization of mutual information). In particular, we show that when the pdf of the random variables is log-concave, there is a constant gap of n^{2}\log e+9n\log n between this upper bound and the dual total correlation lower bound that does not depend on the distribution. The upper bound is obtained using a computationally efficient dyadic decomposition scheme for constructing a discrete common randomness variable W$ from which the $n$ random variables can be simulated in a distributed manner. We then bound the entropy of $W$ using a new measure, which we refer to as the erosion entropy. [ABSTRACT FROM PUBLISHER]

We study the size (or volume) of balls in the metric space of permutations, Sn , under the infinity metric. We focus on the regime of balls with radius $r = \rho \cdot (n\!-\!1)$ , \rho \in [{0,1}] , i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0.029 bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes. [ABSTRACT FROM PUBLISHER]

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]