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In the $q$ -ary online (or “causal”) channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf {x} =(x_{1},\ldots,x_{n}) \in \{0,1,\ldots,q-1\}^{n}$ symbol-by-symbol via a channel limited to at most $pn$ errors and $p^{*} n$ erasures. The channel is “online” in the sense that at the $i$ th step of communication the channel decides whether to corrupt the $i$ th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_{1},\ldots,x_{i})$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge of the sent codeword $\mathbf {x}$. In this paper, we study the capacity of $q$ -ary online channels for a combined corruption model, in which the channel may impose at most $pn$ errors and at most $p^{*} n$ erasures on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies, which present both upper and lower bounds on its capacity. In this paper, we give a full characterization of the capacity as a function of $q,p$ , and $p^{*}$. [ABSTRACT FROM AUTHOR]

The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function, of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The computing rate of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies. This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are not only able to enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non-achievable, but whether it is asymptotically achievable or not remains open. [ABSTRACT FROM AUTHOR]

Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$ -ary optimal locally repairable codes from subcodes of Reed-Solomon codes given by Tamo and Barg have length upper bounded by $q$. Recently, it was shown through extension of construction by Tamo and Barg that length of $q$ -ary optimal locally repairable codes can be $q+1$ by Jin et al.. Surprisingly it was shown by Barg et al. that, unlike classical MDS codes, $q$ -ary optimal locally repairable codes could have length bigger than $q+1$. Thus, it becomes an interesting and challenging problem to construct $q$ -ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we make use of rich algebraic structures of elliptic curves to construct a family of $q$ -ary optimal locally repairable codes of length up to $q+2\sqrt {q}$. It turns out that locality of our codes can be as big as 23 and distance can be linear in length. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of constructing maximum distance separable (MDS) codes that enable exact reconstruction (repair) of each code block by downloading a small amount of information from the remaining code blocks. The total amount of information flow from the remaining code blocks during this reconstruction process is referred to as repair bandwidth of the underlying code. Existing constructions of exact-repairable MDS codes with optimal repair bandwidth require working with large subpacketization levels, which restrict their applicability in practice. This paper presents two general approaches to construct exact-repairable MDS codes that aim at significantly reducing the required subpacketization level at the cost of slightly suboptimal repair bandwidth. The first approach provides MDS codes that have repair bandwidth at most twice the optimal repair bandwidth. In addition, these codes also have the smallest possible subpacketization level $O(r)$ , where $r$ denotes the number of parity blocks. This approach is then generalized to design codes that have their repair bandwidth approaching the optimal repair bandwidth at the cost of graceful increment in the required subpacketization level. The second approach transforms an MDS code with optimal repair bandwidth and large subpacketization level into a longer MDS code with small subpacketization level and near-optimal repair bandwidth. For a given $r$ , the codes constructed using this approach have their subpacketization level scaling logarithmically with the code length. In addition, the obtained codes require field size only linear in the code length and ensure load balancing among the intact code blocks in terms of the information downloaded from these blocks during the exact reconstruction of a code block. [ABSTRACT FROM AUTHOR]

A nonlinear feedback shift register (NFSR) of $n$ -stage is called irreducible if, the family of output sequences of any NFSR of stage less than $n$ is not included in that of the NFSR. Tian and Qi in this paper [IEEE-IT, 2013(6),4006–4012] gave a lower bound on the density of irreducible NFSRs. In this paper, we improve their lower bound and also give an upper bound on the density of irreducible NFSRs. Moreover, the gap between our upper and lower bounds is less than 0.04. [ABSTRACT FROM PUBLISHER]

This paper provides fundamental limits on the sample complexity of estimating dictionaries for tensor data. The specific focus of this work is on $K$ th-order tensor data and the case where the underlying dictionary can be expressed in terms of $K$ smaller dictionaries. It is assumed the data are generated by linear combinations of these structured dictionary atoms and observed through white Gaussian noise. This work first provides a general lower bound on the minimax risk of dictionary learning for such tensor data and then adapts the proof techniques for specialized results in the case of sparse and sparse-Gaussian linear combinations. The results suggest the sample complexity of dictionary learning for tensor data can be significantly lower than that for unstructured data: for unstructured data it scales linearly with the product of the dictionary dimensions, whereas for tensor-structured data the bound scales linearly with the sum of the product of the dimensions of the (smaller) component dictionaries. A partial converse is provided for the case of 2nd-order tensor data to show that the bounds in this paper can be tight. This involves developing an algorithm for learning highly-structured dictionaries from noisy tensor data. Finally, numerical experiments highlight the advantages associated with explicitly accounting for tensor data structure during dictionary learning. [ABSTRACT FROM AUTHOR]

This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n,m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Tr_{m}^{n} (\lambda x^{d}) from \mathbb {F}_{2^{n}} to \mathbb F2^{m} (where m is a divisor of n ) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratović-Ribić et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n,m) -functions. No tight upper bound is known when \frac n2

This paper considers the problem of lossy source coding with a specific distortion measure: logarithmic loss. The focus of this paper is on the single-shot approach, which exposes crisply the connection between lossless source coding with list decoding and lossy source coding with log-loss. Fixed-length and variable-length bounds are presented. Fixed-length bounds include the single-shot fundamental limit for average as well as excess distortion. Variable-length bounds include the single-shot fundamental limit for average as well as excess length. Two multi-terminal problems are addressed: coding with side information (Wyner–Ziv) and multiple descriptions coding. In both the cases, the application of the Shannon–McMillan theorem to the single-shot bounds yields the rate-distortion function and the rate distortion-region for stationary ergodic sources. [ABSTRACT FROM PUBLISHER]

Shannon in his 1956 seminal paper introduced the concept of the zero error capacity, C0 , of a noisy channel. This is defined as the least upper bound of rates, at which, it is possible to transmit information with zero probability of error. At present not many codes are known to achieve the zero error capacity. In this paper, some codes which achieve zero error capacities in limited magnitude error channels are described. The code lengths of these zero error capacity achieving codes can be of any finite length $n=1,2,\ldots$ , in contrast to the long lengths required for the known regular capacity achieving codes, such as turbo codes, LDPC codes, and polar codes. Both wrap around and non-wrap around limited magnitude error models are considered in this paper. For non-wrap around error model, the exact value of zero error capacities is derived, and optimal non-systematic and systematic codes are designed. The non-systematic codes achieve the zero error capacity with any finite length. The optimal systematic codes achieve the systematic zero error capacity of the channel, which is defined as the zero error capacity with the additional requirements that the communication must be carried out with a systematic code. It is also shown that the rates of the proposed systematic codes are equal to or approximately equal to the zero error capacity of the channel. For the wrap around model bounds are derived for the zero error capacity and in many cases the bounds give the exact value. In addition, optimal wrap around non-systematic and systematic codes are developed which either achieve or are close to achieving the zero error capacity with finite length. [ABSTRACT FROM PUBLISHER]

This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: a peak power constraint is imposed on every transmitted codeword, and the average error probabilities of decoding the transmitted message are non-vanishing as the blocklength increases. The main contribution of this paper is a self-contained proof of an upper bound on the first- and second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss’ theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the first- and second-order asymptotics. [ABSTRACT FROM PUBLISHER]

In practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing, and codebooks with small inner-product correlation are required. It is extremely difficult to construct codebooks achieving the Levenshtein bound. In this paper, two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented. These constructions produce complex codebooks asymptotically achieving the Levenshtein bound and codebooks asymptotically achieving the Welch bound. The codebooks presented in this paper have new parameters. [ABSTRACT FROM PUBLISHER]

This paper studies the noncausal relay channel, also known as the relay channel with unlimited look ahead, introduced by by El Gamal, Hassanpour, and Mammen. Unlike the standard relay channel model, where the relay encodes its signal based on the previous received output symbols, the relay in the noncausal relay channel encodes its signal as a function of the entire received sequence. In the existing coding schemes, the relay uses this noncausal information solely to recover the transmitted message or part of it and then cooperates with the sender to communicate this message to the receiver. However, it is shown in this paper that by applying the Gelfand–Pinsker coding scheme, the relay can take further advantage of the noncausally available information and achieve rates strictly higher than those of the existing coding schemes. This paper also provides a new upper bound on the capacity of the noncausal relay channel that strictly improves upon the existing cutset bound. These new lower and upper bounds on the capacity coincide for the class of degraded noncausal relay channels and establish the capacity for this class. [ABSTRACT FROM AUTHOR]

Sparse recovery from linear Gaussian measurements has been the subject of much investigation since the breaktrough papers Candès et al. and Donoho on compressed sensing. Application to sparse vectors and sparse matrices via least squares penalized with sparsity promoting norms is now well understood thanks to tools, such as Gaussian mean width, statistical dimension, and the notion of descent cones. Extention of these ideas to low rank tensor recovery is starting to enjoy considerable interest due to its many potential applications to independent component analysis, hidden Markov models, Gaussian mixture models, and hyperspectral image analysis, to name a few. In this paper, we demonstrate that the recent approach of Vershynin provides useful error bounds in the tensor setting with the nuclear norm or the Romera-Paredes–Pontil penalization. [ABSTRACT FROM AUTHOR]

In a recent work, S. Nitinawarat obtained a lower bound and an upper bound for the minimum list size in list decoding for an arbitrarily varying multiple access channel (AVMAC), for which the interior of the capacity region of deterministic list codes is nonempty. In the same paper, he proved that for a binary AVMAC, the minimum list size is finite, if and only if the interior of the capacity region of random correlated codes is nonempty. The goal of this paper is to close the gap between the two bounds for the minimum list size. We find a necessary and sufficient condition for the list codes for an AVMAC to have a capacity region with a nonempty interior in terms of bipartite graphs. Therefore, we determine the minimum list size. Moreover, we prove that for any AVMAC, the minimum list size is finite, if and only if the interior of the capacity region of random correlated codes is nonempty. [ABSTRACT FROM PUBLISHER]

Sparse representation over redundant dictionaries constitute a good model for many classes of signals (e.g., patches of natural images, segments of speech signals, and so on). However, despite its popularity, a very little are known about the representation capacity of this model. In this paper, we study how redundant a dictionary must be so as to allow any vector to admit a sparse approximation with a prescribed sparsity and a prescribed level of accuracy. We address this problem, both in a worst-case setting and in an average-case one. For each scenario, we derive lower and upper bounds on the minimal required overcompleteness. Our bounds have simple closed-form expressions that allow to easily deduce the asymptotic behavior in large dimensions. In particular, we find that the required overcompleteness grows exponentially with the sparsity level and polynomially with the allowed representation error. This implies that universal sparse representation is practical only at moderate sparsity levels, but can be achieved with a relatively high accuracy. As a side effect of our analysis, we obtain a tight lower bound on the regularized incomplete beta function, which may be interesting in its own right. We illustrate the validity of our results through numerical simulations, which support our findings. [ABSTRACT FROM AUTHOR]

Sampling from the lattice Gaussian distribution plays an important role in various research fields. In this paper, the Markov chain Monte Carlo (MCMC)-based sampling technique is advanced in several fronts. First, the spectral gap for the independent Metropolis-Hastings-Klein (MHK) algorithm is derived, which is then extended to Peikert’s algorithm and rejection sampling; we show that independent MHK exhibits faster convergence. Then, the performance of bounded distance decoding (BDD) using MCMC is analyzed, revealing a flexible trade-off between the decoding radius and complexity. MCMC is further applied to trapdoor sampling, again offering a trade-off between security and complexity. Finally, the independent multiple-try Metropolis-Klein (MTMK) algorithm is proposed to enhance the convergence rate. The proposed algorithms allow parallel implementation, which is beneficial for practical applications. [ABSTRACT FROM AUTHOR]

A binary matrix is called an ${s}$ -separable code for the disjunctive multiple-access channel (disj-MAC) if Boolean sums of sets of ${s}$ columns are all distinct. The well-known issue of the combinatorial coding theory is to obtain upper and lower bounds on the rate of ${s}$ -separable codes for the ${disj}$ -MAC. In our paper, we generalize the problem and discuss upper and lower bounds on the rate of ${q}$ -ary ${s}$ -separable codes for the models of noiseless symmetric MAC, i.e., at each time instant the output signal of MAC is a symmetric function of its ${s}$ input signals. [ABSTRACT FROM AUTHOR]

Mathematical tools that are widely employed for the distribution-free analysis of the finite-sample performance of empirical estimators—like Efron-Stein’s inequality, or Azuma’s martingale inequality and derivatives—rely on the provision of tight bounds for the Lipschitz constants of the estimators. Lipschitz constants can be easily derived for simple functionals of the empirical measures, such as the Shannon or the Tsallis entropy. However, obtaining tight bounds for more generalized entropy functionals which cannot be decomposed into sums of identical terms is much more involved. The goal in this paper is the derivation of (empirical) Lipschitz constants for the maximum likelihood estimator of the family of Renyi entropies of order $\lambda >0$ , $\lambda \neq 1$. Analytic solutions for the optimal constants are obtained for the most important special cases; namely, for $0 < \lambda < 1$ , for $1 < \lambda < 2$ as well as for $\lambda =2$ (collision entropy) and $\lambda = 3$. For the remaining cases where no analytic solution is obtained (i.e., $\lambda > 2$ , $\lambda \neq 3$), an efficient way to compute the optimal constants is derived by reducing the complexity of the underlying optimization problem from exponential ${\varOmega}(n^{\| \mathcal {A} \|-1})$ to linear $\mathcal {O}(n)$ , where $n$ the number of available samples and $\| \mathcal {A} \| \geq 2$ the number of symbols of the source (with the assumption $\| \mathcal {A} \|$ does not grow with $n$). The optimal constants are subsequently used for the distribution-free performance analysis of the maximum likelihood estimator of Renyi entropy; this includes variance and concentration bounds, both under the assumption of independence as well as under strong mixing conditions. [ABSTRACT FROM AUTHOR]

This paper contains two main contributions concerning the expurgation of hierarchical ensembles for the asymmetric broadcast channel. The first is an analysis of the optimal maximum likelihood (ML) decoders for the weak and strong user. Two different methods of code expurgation will be used, that will provide two competing error exponents. The second is the derivation of expurgated exponents under the generalized stochastic likelihood decoder (GLD). We prove that the expurgated exponents achieved for the hierarchical ensemble under GLD decoding are at least as good as the maximum between the random coding error exponents derived in an earlier work by Averbuch and Merhav (2018) and one of our ML–based expurgated exponents. [ABSTRACT FROM AUTHOR]

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a family of easily encodable rate-compatible low-density parity-check (LDPC) codes. PBRL codes have an excellent performance across all design rates. Quasi-cyclic (QC) PBRL code families permit high-speed decoder implementations. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the PBRL family at various design rates. This paper introduces a design method to obtain better QC PBRL code families at short block-lengths (of a few hundred bits) for low frame error rate (FER) requirements. We first select a protomatrix for the highest design rate. To add a new row to lower the rate, we keep all the previously obtained rows of the PBRL protomatrix fixed and select the new row that maximizes an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protomatrix. The new QC PBRL code families outperform the original PBRL codes at short block-lengths by providing a significantly better low-FER performance. The standard approach to computing the aforementioned upper bounds requires complexity that grows exponentially with the size of the protomatrix. However, we show that the structure of the PBRL protomatrix lets us obtain the upper bounds with complexity that grows only linearly with the size of the PBRL protomatrix. Using the complexity reduction results, we also establish an equivalence between the exhaustive search to design a new row for the PBRL protomatrix according to the new design method and an integer linear program. [ABSTRACT FROM AUTHOR]

We consider an $\ell _{2}$ -regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting $m,n \rightarrow \infty $ , $m/n \rightarrow \delta $ and obtain sharp performance bounds, where $m$ is the number of measurements and $n$ is the signal dimension. We show that for complex signals, the algorithm can perform accurate recovery with only $m = (({64}/{\pi ^{2}})-4)n \approx 2.5n$ measurements. Also, we provide a sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: 1) adding $\ell _{2}$ regularization to the non-convex loss function can be beneficial and 2) spectral initialization has a marginal impact on the performance of the algorithm. The sharp analyses, in this paper, not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non-convex optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a fractional version of Haemers’ bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers’ bound and the fractional chromatic number of a graph. We show that this fractional version outperforms any bound on the Shannon capacity that could be attained through Haemers’ bound. We show also that this bound is multiplicative, unlike Haemers’ bound. [ABSTRACT FROM AUTHOR]

BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes $\mathcal C_{(q, m, \delta _{2})}$ and $\mathcal C_{(q, m, \delta _{3})}$ and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes $\overline {\mathcal C}_{(q, m, \delta _{2})}$ have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code $\mathcal C_{(q, m, \delta _{2})}^\perp $ for $q \ge 5$. Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained. [ABSTRACT FROM AUTHOR]

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly noisy tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. One of the defining features of the group testing problem is the distinction between the non-adaptive and adaptive settings. In the non-adaptive case, all tests must be designed in advance, whereas in the adaptive case, each test can be designed based on the previous outcomes. While tight information-theoretic limits and near-optimal practical algorithms are known for the adaptive setting in the absence of noise, surprisingly little is known in the noisy adaptive setting. In this paper, we address this gap by providing information-theoretic achievability and converse bounds under various noise models, as well as a slightly weaker achievability bound for a computationally efficient variant. These bounds are shown to be tight or near-tight in a broad range of scaling regimes, particularly at low noise levels. The algorithms used for the achievability results have the notable feature of only using two or three stages of adaptivity. [ABSTRACT FROM AUTHOR]

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth. [ABSTRACT FROM AUTHOR]

Polar codes are a family of capacity-achieving codes that have explicit and low-complexity construction, encoding, and decoding algorithms. Decoding of polar codes is based on the successive-cancellation decoder, which decodes in a bit-wise manner. A decoding error occurs when at least one bit is erroneously decoded. The various codeword bits are correlated, yet performance analysis of polar codes ignores this dependence: the upper bound is based on the union bound, and the lower bound is based on the worst-performing bit. Improvement of the lower bound is afforded by considering error probabilities of two bits simultaneously. These are difficult to compute explicitly due to the large alphabet size inherent to polar codes. In this paper, we propose a method to lower-bound the error probabilities of bit pairs. We develop several transformations on pairs of synthetic channels that make the resultant synthetic channels amenable to alphabet reduction. Our method yields lower bounds that significantly improve upon currently known lower bounds for polar codes under successive-cancellation decoding. [ABSTRACT FROM AUTHOR]

This paper deals with lower bounds on aperiodic correlation of sequences. It intends to solve two open questions. The first one is on the validity of the Levenshtein bound for a set of sequences other than binary sequences or those over the roots of unity. Although this result could be a priori extended to polyphase sequences, a formal demonstration is presented here, proving that it does actually hold for these sequences. The second open question is on the possibility to find a bound tighter than Welch’s, in the case of a set consisting of two sequences $M = 2$. By including the specific structure of correlation sequences, a tighter lower bound is introduced for this case. Besides, this method also provides in the cases $M = 3$ and $M = 4$ a tighter bound than the up-to-now tightest bound provided by Liu et al. [ABSTRACT FROM AUTHOR]

The polarization process of polar codes over a prime $q$ -ary alphabet is studied. Recently, it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this paper, a different approach to computing the degree of this polynomial for any prime alphabet size is shown. This approach yields a lower degree polynomial for various values of the alphabet size that were examined. It is also shown that even lower degree polynomial can be computed with an additional numerical effort. [ABSTRACT FROM AUTHOR]

This paper investigates the phase retrieval problem, which aims to recover a signal from the magnitudes of its linear measurements. We develop statistically and computationally efficient algorithms for the situation when the measurements are corrupted by sparse outliers that can take arbitrary values. We propose a novel approach to robustify the gradient descent algorithm by using the sample median as a guide for pruning spurious samples in initialization and local search. Adopting a Poisson loss and a reshaped quadratic loss, respectively, we obtain two algorithms termed median-truncated Wirtinger flow and median-reshaped Wirtinger flow, both of which provably recover the signal from a near-optimal number of measurements when the measurement vectors are composed of independent and identically distributed Gaussian entries, up to a logarithmic factor, even when a constant fraction of the measurements is adversarially corrupted. We further show that both algorithms are stable in the presence of additional dense bounded noise. Our analysis is accomplished by developing non-trivial concentration results of median-related quantities, which may be of independent interest. We provide numerical experiments to demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we prove a unified achievability bound that generalizes and improves random coding bounds for any combination of source coding, channel coding, joint source–channel coding, and coding for computing problems assuming blockwise node operation. As a general network setup, we consider an acyclic discrete memoryless network, where the network demands and constraints are specified by a joint-typicality constraint on the whole channel input and output sequences. For achievability, a basic building block for node operation consists of simultaneous nonunique decoding, simultaneous compression, and symbol-by-symbol mapping. Our bound can be useful for deriving random coding bounds without error analysis, especially for large and complex networks. In particular, our bound can be used for unifying and generalizing many known relaying strategies. For example, a generalized decode-compress-amplify-and-forward bound is obtained as a simple corollary of our main theorem, and it is shown to strictly outperform the previously known relaying schemes. Furthermore, by exploiting the symmetry in our bound, we formally define and characterize three types of network duality based on channel input–output reversal and network flow reversal combined with packing–covering duality. [ABSTRACT FROM AUTHOR]

The zero-error capacity of a channel is defined as the maximum rate at which it is possible to transmit information with zero probability of error. In this paper, we settle all previously unsolved cases for the zero-error capacity of binary channels with one memory. [ABSTRACT FROM AUTHOR]

This paper investigates the joint source-channel coding problem of sending a memoryless source over a memoryless broadcast channel. An inner bound and several outer bounds on the admissible distortion region are derived, which, respectively, generalize and unify several existing bounds. As a consequence, we also obtain an inner bound and an outer bound for the degraded broadcast channel case. When specialized to the Gaussian or binary source broadcast, the inner bound and outer bound not only recover the best known inner bound and outer bound in the literature but also generate some new results. Besides, we also extend the inner bound and outer bounds to the Wyner–Ziv source broadcast problem, i.e., source broadcast with side information available at decoders. Some new bounds are obtained when specialized to the Wyner–Ziv Gaussian and Wyner–Ziv binary cases. [ABSTRACT FROM AUTHOR]

It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verdú, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver. [ABSTRACT FROM AUTHOR]

This paper starts by considering the minimization of the Rényi divergence subject to a constraint on the total variation distance. Based on the solution of this optimization problem, the exact locus of the points \bigl ( D(Q\|P1), D(Q\|P2) \bigr ) is determined when P1 , P2 , and $Q$ are arbitrary probability measures which are mutually absolutely continuous, and the total variation distance between P1 and P2 is not below a given value. It is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet. This characterization yields a geometric interpretation of the minimal Chernoff information subject to a constraint on the variational distance. This paper also derives an exponential upper bound on the performance of binary linear block codes (or code ensembles) under maximum-likelihood decoding. Its derivation relies on the Gallager bounding technique, and it reproduces the Shulman–Feder bound as a special case. The bound is expressed in terms of the Rényi divergence from the normalized distance spectrum of the code (or the average distance spectrum of the ensemble) to the binomially distributed distance spectrum of the capacity-achieving ensemble of random block codes. This exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes (or code ensembles) as a function of the deviation of their distance spectra from the binomial distribution. An efficient use of this bound is considered. [ABSTRACT FROM PUBLISHER]

In this paper, we provide a comprehensive survey on designs and analyses of various relay-destination transceiving schemes, such as zero-forcing (ZF), minimum mean squared error (MMSE), and maximum information rate (MIR) criteria, in multiple-input multiple-output (MIMO) amplify-and-forward (AF) relaying systems. In the first part of the paper, we suggest a new framework for the transceiver designs utilizing a decomposable property of the error covariance matrix to give a general insight on the system and make the analysis more tractable. Then, in the second part of the paper, we provide an in-depth analysis on their diversity performance. Our analysis embraces two different scenarios, namely, the diversity-multiplexing tradeoff (DMT) and the diversity-rate tradeoff (DRT). First, we derive compact closed-form expressions for the DMT through tight upper and lower bounds. Then, we observe that while our DMT analysis accurately predicts performance of the ZF and MIR schemes, the MMSE-based designs exhibit a complicated rate-dependent behavior and, thus, are very unpredictable via DMT for finite rate cases. Thus, second, we highlight this interesting observation and characterize the diversity of the MMSE schemes at all finite rates. This leads to closed-form expressions for the DRT which reveals relationship between diversity, spectral efficiency, and the number of antennas at each node. The DRT analysis compliments our work on DMT, and thus, the paper provides a complete understanding on the diversity of MIMO AF relaying systems. [ABSTRACT FROM PUBLISHER]

In this paper, a generic construction of codebooks based on binary codes is introduced. With this generic construction, a few previous constructions of optimal codebooks are extended, and a new class of codebooks almost meeting the Levenshtein bound is presented. Exponentially many codebooks meeting or almost meeting the Levenshtein bound from binary codes are obtained in this paper. The codebooks constructed in this paper have alphabet size 4. As a byproduct, three bounds on the parameters of binary codes are derived. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to handle a gap that was found in the proof of Theorem 2 in the paper “The generalized stochastic likelihood decoder: random coding and expurgated bounds.” [ABSTRACT FROM PUBLISHER]

The location awareness is useful in many wireless networking solutions, such as cellular, ad hoc, self-organizing, cognitive radio networks, and so on. The RSS-based, non-Bayesian source localization is of particular interest, due to the presence of the RSS measurements in all radio devices, as well as the instantaneous estimations without extensive training and learning phases. However, most RSS-based localization algorithms usually neglect problems arising from inherent network topology uncertainty. The positions of some measuring anchors, usually assumed to be precisely known in advance, are often a subject of previous estimation, which propagates errors in the source localization procedure and results in performance degradation. A joint localization framework, termed source position estimation for anchor position uncertainty reduction (SPEAR), has been recently proposed as a possible solution to the problem of RSS-based localization in the presence of network topology uncertainty. The framework simultaneously estimates the positions of the sources and the uncertain anchors and serves as robust tool that provides improved source localization performance and more reliable anchor position estimates. This paper focuses on the theoretical assessment of SPEAR’s performance by deriving its fundamental lower bounds using the Cramér–Rao information inequality. The main contribution of this paper is the detailed information-theoretic interpretation of the obtained results. The interpretation provides valuable insight into the essence of the SPEAR localization problem because it identifies different position information components in the observed data, discovers how they are mutually related, and determines how they contribute toward the resulting position information. This paper also provides geometric interpretation of the obtained results and illustrates the principles of distribution of position information in space. The theoretical results as well as their information-theoretic and geometric interpretations can be used as a benchmark for RSS-based location-aware systems that operate in the presence of network topology uncertainty. [ABSTRACT FROM AUTHOR]

The Haroutunian exponent arises in the study of channel reliability functions for both block coding with feedback and fixed-delay coding without feedback. For asymmetric channels, such as the $Z$ -channel, the Haroutunian exponent is strictly larger than the sphere-packing exponent. The sphere-packing exponent is believed to be an upper bound for the reliability function in the two aforementioned communication problems, but in attempting to prove this, one gets stuck at the Haroutunian exponent because of entanglements between the channel behavior and the input distribution. In this paper, we present a characteristic of the Haroutunian exponent that differentiates it from the random coding and sphere-packing exponents. We consider the parallel channel, the repeated use of the original discrete memoryless channel independently some $L$ number of times. It is well known that the capacity of the parallel channel is $L$ times the capacity of the original channel, and the random coding and sphere-packing exponents of the $L$ -use parallel channel decompose into $L$ times the exponents of the original channel. The main result of this paper is to show that the (appropriately normalized) Haroutunian exponent of the parallel channel asymptotically decomposes to the sphere-packing exponent of the original channel, as opposed to the Haroutunian exponent of the original channel. This fact is then used to prove two results. First, an upper bound for the reliability function for fixed blocklength coding with delayed feedback is proved. This upper bound converges to the sphere-packing exponent as the delay in the feedback path tends to infinity. Second, the reliability function for fixed delay coding without feedback is shown to be upper bounded by the sphere-packing exponent. [ABSTRACT FROM PUBLISHER]

The objective of this paper is the characterization of the Wyner–Ziv rate-distortion function for memoryless continuous sources, when the correlation between the sources is modeled via an additive noise channel. Modeling the distribution of the correlation noise via a Gaussian mixture, with discrete or continuous mixing variable, provides a unified signal model able to describe a wide class of distributions, useful in the context of practical applications. The Wyner–Ziv rate-distortion function associated with this signal model cannot, in general, be obtained in analytical form. This paper contributes a method for its analysis, by providing computable upper and lower bounds. [ABSTRACT FROM AUTHOR]

This paper considers a base station that delivers packets to multiple receivers through a sequence of coded transmissions. All receivers overhear the same transmissions. Each receiver may already have some of the packets as side information, and requests another subset of the packets. This problem is known as the index coding problem and can be represented by a bipartite digraph. An integer linear program is developed that provides a lower bound on the minimum number of transmissions required for any coding algorithm. Conversely, its linear programming relaxation is shown to provide an upper bound that is achievable by a simple form of vector linear coding. Thus, the information theoretic optimum is bounded by the integrality gap between the integer program and its linear relaxation. In the special case, when the digraph has a planar structure, the integrality gap is shown to be zero, so that exact optimality is achieved. Finally, for nonplanar problems, an enhanced integer program is constructed that provides a smaller integrality gap. The dual of this problem corresponds to a more sophisticated partial clique coding strategy that time-shares between maximum distance separable codes. This paper illuminates the relationship between index coding, duality, and integrality gaps between integer programs and their linear relaxations. [ABSTRACT FROM PUBLISHER]

In this paper, a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced. It includes some earlier constructions of codebooks meeting the Levenstein bound as special cases. With this construction, two new families of codebooks achieving the Levenstein bound are obtained. The codebooks constructed in this paper could have a very small alphabet size. [ABSTRACT FROM PUBLISHER]

In this paper, distributed storage systems with exact repair are studied. Constructions for exact-regenerating codes between the minimum storage regenerating (MSR) and the minimum bandwidth regenerating (MBR) points are given. To the best of our knowledge, no previous construction of exact-regenerating codes between MBR and MSR points is done except in the works by Tian et al. and Sasidharan et al. In contrast to their works, the methods used here are elementary. In this paper, it is shown that in the case that the parameters \(n\) , \(k\) , and \(d\) are close to each other, the given construction is close to optimal when comparing with the known functional repair capacity. This is done by showing that when the distances of the parameters \(n\) , \(k\) , and \(d\) are fixed but the actual values approach to infinity, the fraction of the performance of constructed codes with exact repair and the known capacity of codes with functional repair, approaches to one. Also, a simple variation of the constructed codes with almost the same performance is given. Also some bounds for the capacity of exact-repairing codes are given. These bounds illustrate the relationships between storage codes with different parameters. [ABSTRACT FROM PUBLISHER]

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate $R$ and minimum distance $\delta $ of a binary code. The best known tradeoff is the Gilbert–Varshamov bound, and says that for every $\delta \in (0, 1/2)$ , there are codes with minimum distance $\delta $ and rate $R = {R_{\mathsf {GV}}}(\delta) > 0$ (for a certain simple function ${R_{\mathsf {GV}}}(\cdot)$ ). In this paper, we show that the Gilbert–Varshamov bound can be achieved by codes, which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1) Local testing: for all $\delta \in (0,1/2)$ and all $R < {R_{\mathsf {GV}}}(\delta)$ , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally testable with $\mathsf {quasipolylog}(n)$ query complexity. 2) Local correction: for all $\epsilon > 0$ , for all $\delta < 1/2$ sufficiently large, and all $R < (1-\epsilon) {R_{\mathsf {GV}}}(\delta)$ , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally correctable from $({\delta }/{2}) - o(1)$ fraction errors with $O(n^{\epsilon })$ query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen’s random concatenation technique and the best known locally testable codes by Kopparty et al. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami–Sudan–Indyk list decoding strategy for concatenated codes, Alon–Edmonds–Luby distance amplification, and the local list-decodability, local list-recoverability, and local testability of Reed–Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code. [ABSTRACT FROM AUTHOR]

For non-binary codes the Elias bound is a good upper bound for the asymptotic information rate at low-relative minimum distance, whereas the Plotkin bound is better at high-relative minimum distance. In this paper, we obtain a hybrid of these bounds, which improves both. This in turn is based on the anticode bound, which is a hybrid of the Hamming and Singleton bounds and improves both bounds. The question of convexity of the asymptotic rate function is an important open question. We conjecture a much weaker form of the convexity, and we show that our bounds follow immediately if we assume the conjecture. [ABSTRACT FROM AUTHOR]

This paper examines fundamental error characteristics for a general class of matrix completion problems, where the matrix of interest is a product of two a priori unknown matrices, one of which is sparse, and the observations are noisy. Our main contributions come in the form of minimax lower bounds for the expected per-element squared error for this problem under several common noise models. Specifically, we analyze scenarios where the corruptions are characterized by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one bit) observations, as instances of our general result. Our results establish that the error bounds derived in (Soni et al., 2016) for complexity-regularized maximum likelihood estimators achieve, up to multiplicative constants and logarithmic factors, the minimax error rates in each of these noise scenarios, provided that the nominal number of observations is large enough, and the sparse factor has (on an average) at least one non-zero per column. [ABSTRACT FROM PUBLISHER]

Regular spatially-coupled low-density parity-check ensembles have gained significant interest, since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this paper, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per code word and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability ( P_ \mathrm \scriptscriptstyle B ) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random location in the code word and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability $p$ , potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations. [ABSTRACT FROM PUBLISHER]

Cooperation strategies allow communication devices to work together to improve network capacity. Consider a network consisting of $k$ encoders, a multiple access channel (MAC), a decoder, and a node, referred to as a “cooperation facilitator” (CF), that is connected to each encoder via a pair of rate-limited links, with one link going from the encoder to the CF and the other link going back. Let the “cooperation rate” be the total outgoing rate of the CF. This paper demonstrates the existence of a class of MACs where the ratio of the sum-capacity gain to cooperation rate tends to infinity as the cooperation rate tends to zero. For any $k\geq 2$ , examples of channels in this class include the $k$ -user binary adder MAC and the $k$ -user Gaussian MAC. [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]

A sum-network is an instance of a function computation problem over a directed acyclic network, in which each terminal node wants to compute the sum over a finite field of the information observed at all the source nodes. Many characteristics of the well-studied multiple unicast network communication problem also hold for sum-networks, due to a known reduction between the two problems. In this paper, we describe an algorithm to construct families of sum-network instances using incidence structures. The computation capacity of several of these sum-network families is evaluated. Unlike the coding capacity of a multiple unicast problem, the computation capacity of sum-networks depends on the characteristic of the finite field over which the sum is computed. This dependence is very strong; we show examples of sum-networks that have a rate-1 solution over one characteristic but a rate close to zero over a different characteristic. In addition, a sum-network can have arbitrarily different computation capacities for different alphabets. [ABSTRACT FROM PUBLISHER]