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We consider the concatenation of a convolutional code (CC) with an optimized cyclic redundancy check (CRC) code as a promising paradigm for good short blocklength codes. The resulting CRC-aided convolutional code naturally permits the use of serial list Viterbi decoding (SLVD) to achieve maximum-likelihood decoding. The convolutional encoder of interest is of rate- $1/\omega $ and the convolutional code is either zero-terminated (ZT) or tail-biting (TB). The resulting CRC-aided convolutional code is called a CRC-ZTCC or a CRC-TBCC. To design a good CRC-aided convolutional code, we propose the distance-spectrum optimal (DSO) CRC polynomial. A DSO CRC search algorithm for the TBCC is provided. Our analysis reveals that the complexity of SLVD is governed by the expected list rank which converges to 1 at high SNR. This allows a good performance to be achieved with a small increase in complexity. In this paper, we focus on transmitting 64 information bits with a rate-1/2 convolutional encoder. For a target error probability $10^{-4}$ , simulations show that the best CRC-ZTCC approaches the random-coding union (RCU) bound within 0.4 dB. Several CRC-TBCCs outperform the RCU bound at moderate SNR values. [ABSTRACT FROM AUTHOR]

Previous algorithms for decoding AG codes up to the designed distance all assumed existence of an extra rational place on the base algebraic curve. The place is used to solve the decoding problem by linear algebra over the base field of the curve. The rationality of the place is essential, and therefore AG codes supported by all rational places on the curve are excluded from the domain of applicability of the decoding algorithms. This paper presents a decoding algorithm for those AG codes using an extra place of higher degree. Hence finally all AG codes, as Goppa defined 40 years ago, are equipped with a fast decoding algorithm. [ABSTRACT FROM AUTHOR]

This paper considers a real-time status update system in which an energy harvesting (EH)-powered transmitter node observes some physical process, and sends its sensed measurements in the form of status updates to a destination node. The status update and harvested energy packets are assumed to arrive at the transmitter according to independent Poisson processes, and the service time of each status update is assumed to be exponentially distributed. We quantify the freshness of status updates when they reach the destination using the concept of Age of Information (AoI). Unlike most of the existing analyses of AoI focusing on the evaluation of its average value when the transmitter is not subject to energy constraints, our analysis is focused on understanding the distributional properties of AoI through the characterization of its moment generating function (MGF). In particular, we use the stochastic hybrid systems (SHS) framework to derive closed-form expressions of the MGF of AoI under several queueing disciplines at the transmitter, including non-preemptive and preemptive in service/waiting strategies. Using these MGF results, we further obtain closed-form expressions for the first and second moments of AoI in each queueing discipline. We demonstrate the generality of this analysis by recovering several existing results for the corresponding system with no energy constraints as special cases of the new results. Our numerical results verify the analytical findings, and demonstrate the necessity of incorporating the higher moments of AoI in the implementation/optimization of real-time status update systems rather than just relying on its average value. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

This paper studies the continuous-time Poisson channel whose dark current is random and may change for every $\tau $ -second time interval, where the actual values of the dark current are known to the transmitter as channel-state information (CSI). In the limit where $\tau $ tends to zero, the capacity gain provided by both causal and noncausal CSI is shown to vanish linearly with $\tau $ , so CSI at the transmitter provides almost no capacity improvement. The paper also considers a related problem of the state-dependent very noisy channel. In the very noisy limit, the capacity gain provided by noncausal CSI is shown to be the same as that provided by causal CSI. [ABSTRACT FROM AUTHOR]

Modern communication and computation systems often consist of large networks of unreliable nodes. Still, it is well known that such systems can provide aggregate reliability via redundancy. While duplication may increase the load on a system, it can lead to significant performance improvement when combined with the judicious management of extra system resources. Prime examples of this abstract paradigm include multi-path routing across communication networks, content access from multiple caches in delivery networks, and master/slave computations on compute clusters. Several recent contributions in the area establish bounds on the performance of redundant systems, characterizing the latency-redundancy tradeoff under specific load profiles. Following a similar line of research, this paper introduces new analytical bounds and approximation techniques for the latency-redundancy tradeoff for a range of system loads and a class of symmetric redundancy schemes, under the assumption of Poisson arrivals, exponential service-rates, and fork-join scheduling policy. The proposed approach can be employed to efficiently approximate the latency distribution of a queueing system at equilibrium. Various metrics can subsequently be derived for this system, including the mean and variance of the sojourn time, and the tail decay rate of the stationary distribution. This paper also establishes the stability region in terms of arrival rates for redundant systems with certain symmetries. Finally, it offers selection guidelines for design parameters to provide latency guarantees based on the proposed approximations. Findings are substantiated by numerical results. [ABSTRACT FROM AUTHOR]

We investigate the list decodability of symbol-pair codes in this paper. First, we show that the list decodability of every symbol-pair code does not exceed the Gilbert–Varshamov bound. On the other hand, we are able to prove that with high probability, a random symbol-pair code can be list decoded up to the Gilbert–Varshamov bound. Our second result of this paper is to derive the Johnson-type bound, i.e., a lower bound on list decoding radius in terms of minimum distance. Finally, we present a list decoding algorithm of Reed–Solomon codes beyond the Johnson-type bound in the pair metric. 1 A symbol-pair code is referred to a code in the pair metric. [ABSTRACT FROM AUTHOR]

This paper derives upper and lower bounds on the secrecy capacity-memory tradeoff of a wiretap erasure broadcast channel (BC) with ${\mathsf{K}}_{w} $ weak receivers and ${\mathsf {K}}_{s} $ strong receivers, where weak receivers and strong receivers have the same erasure probabilities and cache sizes, respectively. The lower bounds are achieved by the schemes that meticulously combine joint cache-channel coding with wiretap coding and key-aided one-time pads. The presented upper bound holds more generally for arbitrary degraded BCs and arbitrary cache sizes. When only weak receivers have cache memories, upper and lower bounds coincide for small and large cache memories, thus providing the exact secrecy capacity-memory tradeoff for this setup. The derived bounds further allow us to conclude that the secrecy capacity is positive even when the eavesdropper is stronger than all the legitimate receivers with cache memories. Moreover, they show that the secrecy capacity-memory tradeoff can be significantly smaller than its non-secure counterpart, but it grows much faster when cache memories are small. This paper also presents a lower bound on the global secrecy capacity-memory tradeoff where one is allowed to optimize the cache assignment subject to a total cache budget. It is close to the best known lower bound without secrecy constraint. For small total cache budget, the global secrecy capacity-memory tradeoff is achieved by assigning all the available cache memory uniformly over all the receivers if the eavesdropper is stronger than all the legitimate receivers, and it is achieved by assigning the cache memory uniformly only over the weak receivers if the eavesdropper is weaker than the strong receivers. [ABSTRACT FROM AUTHOR]

This paper considers the problem of channel coding over Gaussian intersymbol interference (ISI) channels with a given decoding rule. Specifically, it is assumed that the mismatched decoder has an incorrect assumption on the channel impulse response. The mismatch capacity is the highest achievable rate for a given decoding rule. The existing achievable rates for channels and decoding metrics with memory (as in our model) are currently available only in the form of multi-letter expressions that cannot be calculated. Consequently, they provide little insight on the mismatch problem. In this paper, we derive the computable formulas of achievable rates and discuss some implications of our results. Our achievable rates are based on two ensembles: the ensemble of codewords generated by an autoregressive process and the ensemble of codewords drawn uniformly over a “type class” of real-valued sequences. We provide a few numerical results of our achievable rates, as functions of the mismatched ISI parameters. Finally, we compare our results with universal decoders which are designed outside the true class of channels that we consider in this paper. [ABSTRACT FROM AUTHOR]

Given two discrete random variables $X$ and $Y$ , with probability distributions $p=(p_{1}, \ldots , p_{n})$ and $q=(q_{1}, \ldots , q_{m})$ , respectively, let us denote by ${\mathcal{ C}}(p, q)$ the set of all couplings of $p$ and $q$ , that is, the set of all bivariate probability distributions that have $p$ and $q$ as marginals. In this paper, we study the problem of finding a joint probability distribution in ${\mathcal{ C}}(p, q)$ of minimum entropy (equivalently, a coupling that maximizes the mutual information between $X$ and $Y$), and we discuss several situations where the need for this kind of optimization naturally arises. Since the optimization problem is known to be NP-hard, we give an efficient algorithm to find a joint probability distribution in ${\mathcal{ C}}(p, q)$ with entropy exceeding the minimum possible at most by 1 bit, thus providing an approximation algorithm with an additive gap of at most 1 bit. Leveraging on this algorithm, we extend our result to the problem of finding a minimum-entropy joint distribution of arbitrary $k\geq {2}$ discrete random variables $X_{1}, \ldots , X_{k}$ , consistent with the known $k$ marginal distributions of the individual random variables $X_{1}, \ldots , X_{k}$. In this case, our algorithm has an additive gap of at most $\log k$ from optimum. We also discuss several related applications of our findings and extensions of our results to entropies different from the Shannon entropy. [ABSTRACT FROM AUTHOR]

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]

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]

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]

Partial MDS (PMDS) codes are a class of erasure-correcting array codes that combine local correction of the rows with global correction of the array. An $\boldsymbol {m}\times \boldsymbol {n}$ array code is called an $(\boldsymbol {r};\boldsymbol {s})$ PMDS code if each row belongs to an ${[}\boldsymbol {n},\boldsymbol {n}-\boldsymbol {r}, \boldsymbol {r}+\textbf {1}{]}$ MDS code and the code can correct erasure patterns consisting of $\boldsymbol {r}$ erasures in each row together with $\boldsymbol {s}$ more erasures anywhere in the array. While a recent construction by Calis and Koyluoglu generates $(\boldsymbol {r};\boldsymbol {s})$ PMDS codes for all $\boldsymbol {r}$ and $\boldsymbol {s}$ , its field size is exponentially large. In this paper, a family of PMDS codes with field size ${\mathcal{ O}}\left ({\max \{\boldsymbol {m},\boldsymbol {n}^{\boldsymbol {r}+\boldsymbol {s}}\}^{\boldsymbol {s}} }\right)$ is presented for the case where $\boldsymbol {r}= {\mathcal{ O}}(1), \boldsymbol {s}= {\mathcal{ O}}(1)$. [ABSTRACT FROM AUTHOR]

Sparse superposition codes, or sparse regression codes, are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic result by deriving a large deviation bound on the probability of an AMP decoding error. This bound gives insight into the error performance of the AMP decoder for large but finite problem sizes, giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths. For an appropriate choice of code parameters, we show that for any fixed rate less than the channel capacity, the decoding error probability decays exponentially in $n/(\log n)^{2T}$ , where $T$ , the number of AMP iterations required for successful decoding, is bounded in terms of the gap from capacity. [ABSTRACT FROM AUTHOR]

This paper considers the fundamental limit of random linear estimation for i.i.d. signal distributions and i.i.d. Gaussian measurement matrices. Its main contribution is a rigorous characterization of the asymptotic mutual information (MI) and minimum mean-square error (MMSE) in this setting. Under mild technical conditions, our results show that the limiting MI and MMSE are equal to the values predicted by the replica method from statistical physics. This resolves a well-known problem that has remained open for over a decade. [ABSTRACT FROM AUTHOR]

This paper generalizes the method proposed by Poulliat et al. for the determination of the optimal Galois field coefficients of a non-binary LDPC parity check constraint based on the binary image of the code. Optimal, or almost-optimal, parity check coefficients are given for check degree varying from 4 to 20 and Galois field varying from GF (64) up to GF (1024). For all given sets of coefficients, no codeword of Hamming weight two exists. A reduced complexity algorithm to compute the binary Hamming weight 3 of a parity check is proposed. When the number of sets of coefficients is too high for an exhaustive search and evaluation, a local greedy search is performed. Explicit tables of coefficients are given. The proposed sets of coefficients can effectively replace the random selection of coefficients often used in NB-LDPC construction. [ABSTRACT FROM AUTHOR]

This paper concerns the problem of recovering an unknown but structured signal ${x}\in \mathbb {R} ^{n}$ from $m$ -quadratic measurements of the form ${y}_{r}= \left |{\langle {a}_{r}, {x}\rangle }\right |^{{2}}$ for ${r}={1},{2},\ldots, {m}$. We focus on the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal (${m} < < {n}$). We formulate the recovery problem as a nonconvex optimization problem where prior structural information about the signal is enforced through constrains on the optimization variables. We prove the projected gradient descent, when initialized in a neighborhood of the desired signal, converges to the unknown signal at a linear rate. These results hold for any closed constraint set (convex or non-convex) providing convergence guarantees to the global optimum even when the objective function and constraint set are nonconvex. Furthermore, these results hold with a number of measurements that are only a constant factor away from the minimal number of measurements required to uniquely identify the unknown signal. Our results provide the first provably tractable algorithm for this data-poor regime, breaking local sample complexity barriers that have emerged in this paper. In this paper, we utilize and further develop powerful tools for uniform convergence of empirical processes that may have broader implications for rigorous understanding of constrained nonconvex optimization heuristics. [ABSTRACT FROM AUTHOR]

Vector autoregressive models characterize a variety of time series in which linear combinations of current and past observations can be used to accurately predict future observations. For instance, each element of an observation vector could correspond to a different node in a network, and the parameters of an autoregressive model would correspond to the impact of the network structure on the time series evolution. Often, these models are used successfully in practice to learn the structure of social, epidemiological, financial, or biological neural networks. However, little is known about statistical guarantees on the estimates of such models in non-Gaussian settings. This paper addresses the inference of the autoregressive parameters and associated network structure within a generalized linear model framework that includes Poisson and Bernoulli autoregressive processes. At the heart of this analysis is a sparsity-regularized maximum likelihood estimator. While sparsity-regularization is well-studied in the statistics and machine learning communities, those analysis methods cannot be applied to autoregressive generalized linear models because of the correlations and potential heteroscedasticity inherent in the observations. Sample complexity bounds are derived using a combination of martingale concentration inequalities and modern empirical process techniques for dependent random variables. These bounds, which are supported by several simulation studies, characterize the impact of various network parameters on the estimator performance. [ABSTRACT FROM AUTHOR]

We study the first- and second-order asymptotics of covert communication over binary-input discrete memoryless channels for three different covertness metrics and under maximum probability of error constraint. When covertness is measured in terms of the relative entropy between the channel output distributions induced with and without communication, we characterize the exact first- and second-order asymptotics of the number of bits that can be reliably transmitted with a maximum probability of error less than $\epsilon $ and a relative entropy less than $\delta $. When covertness is measured in terms of the variational distance between the channel output distributions or in terms of the probability of missed detection for fixed probability of false alarm, we establish the exact first-order asymptotics and bound the second-order asymptotics. Pulse position modulation achieves the optimal first-order asymptotics for all three metrics, as well as the optimal second-order asymptotics for relative entropy. The main conceptual contribution of this paper is to clarify how the choice of a covertness metric impacts the information-theoretic limits of covert communications. The main technical contribution underlying our results is a detailed expurgation argument to show the existence of a code satisfying the reliability and covertness criteria. [ABSTRACT FROM AUTHOR]

Over the past decade, the bulk of wireless traffic has shifted from speech to content. This shift creates the opportunity to cache part of the content in memories closer to the end users, for example in base stations. Most previous paper focuses on the reduction of load in the backhaul and core networks due to caching, i.e., on the benefits caching offers for the wireline communication link between the origin servers and the caches. In this paper, we are instead interested in the benefits caching can offer for the wireless communication link between the caches and the end users. To quantify the gains of caching for this wireless link, we consider an interference channel in which each transmitter is equipped with an isolated cache memory. Communication takes place in two phases, a content placement phase followed by a content delivery phase. The objective is to design both the placement and the delivery phases to maximize the rate in the delivery phase in response to any possible user demands. Focusing on the three-user case, we show that through careful joint design of these phases, we can reap three distinct benefits from caching: a load balancing gain, an interference cancellation gain, and an interference alignment gain. In our proposed scheme, load balancing is achieved through a specific file splitting and placement, producing a particular pattern of content overlap at the caches. This overlap allows to implement interference cancellation. Furthermore, it allows us to create several virtual transmitters, each transmitting a part of the requested content, which increases interference-alignment possibilities. [ABSTRACT FROM AUTHOR]

In 2010, Silva et al. studied certain classes of finite-field matrix channels in order to model random linear network coding where exactly $t$ random errors are introduced. In this paper, we consider a generalization of these matrix channels where the number of errors is not required to be constant, indeed the number of errors may follow any distribution. We show that a capacity-achieving input distribution can always be taken to have a very restricted form (the distribution should be uniform given the rank of the input matrix). This result complements, and is inspired by a paper of Nobrega et al., which establishes a similar result for a class of matrix channels that model network coding with link erasures. Our result shows that the capacity of our channels can be expressed as maximization over probability distributions on the set of possible ranks of input matrices: a set of linear rather than exponential size. [ABSTRACT FROM AUTHOR]

In this paper, we propose a lattice-based robust distributed source coding system for two correlated sources and provide a detailed performance analysis under the high resolution assumption. It is shown, among other things, that, in the asymptotic regime where: 1) the side distortion approaches 0 and 2) the ratio between the central and side distortions approaches 0, our scheme is capable of achieving the information-theoretic limit of quadratic multiple description coding when the two sources are identical, whereas a variant of the random coding scheme by Chen and Berger with Gaussian codes has a performance loss of 0.5 bits relative to this limit. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained. [ABSTRACT FROM AUTHOR]

Consider a channel with a given input alphabet size and input distribution. Our aim is to degrade or upgrade it to a channel with at most L output letters. Channel Q is degraded with respect to a channel W if Q can be obtained from W by processing the output of W. Upgrading is the inverse relation. This paper contains four main results. The first result, from which the paper title is derived, deals with the so called “greedy–merge” algorithm. We derive an upper bound on the reduction in mutual information between input and output, as a function of L. This upper bound is within a constant factor of an algorithm-independent lower bound. Thus, we establish that greedy–merge is optimal in the power-law sense (i.e. the power of L). The other main results deal with upgrading. The second result shows that a certain sequence of channels, which was previously shown to be “hard” for degrading, displays the same hardness in the context of upgrading. That is, suppose we are given such a channel and a corresponding input distribution. If we upgrade (degrade) to a new channel with L output letters, we incur an increase (decrease) in mutual information between input and output. We show that a previously derived bound on the decrease in mutual information for the degrading case is also a lower bound on the increase for the upgrading case. The third result is an efficient algorithm for optimal upgrading, in the binary-input case. That is, we are given a channel and an input distribution. We must find an upgraded channel with L output letters, for which the increase in mutual information is minimal. We give a simple characterization of such a channel, which implies an efficient algorithm. The fourth result is an analog of the first result for the upgrading case when the input is binary. That is, we first present a sub-optimal algorithm for the setting considered in the third result. The main advantage of the sub-optimal algorithm is that it is amenable to analysis. We carry out the analysis and show that the increase incurred in mutual information is within a constant factor of the lower bound derived in the second result. [ABSTRACT FROM AUTHOR]

Maximum distance profile codes over finite non-binary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of $p$ -encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over ${\mathbb Z}_{p^{r}}$ is presented. [ABSTRACT FROM AUTHOR]

Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some t errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are N channels, which cause at most t1 or t2 errors, where t1 < t2, and the number of channels with at most t1 errors is at least [pN], for some fixed 0 < p < 1. If the information codeword belongs to a code with minimum distance d, the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance d that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions. [ABSTRACT FROM AUTHOR]

In this paper, we study outer bounds on the capacity region of the downlink multicell processing model with finite backhaul capacity for the simple case of two base stations and two mobile users. It is modeled as a two-user multiple access diamond channel. It consists of a first hop from the central processor to the base stations via orthogonal links of finite capacity and the second hop from the base stations to the mobile users via a Gaussian interference channel. The outer bound is derived using the converse tools of the multiple access diamond channel and that of the Gaussian MIMO broadcast channel. Through numerical results, it is shown that our outer bound improves upon the existing outer bounds greatly in the medium backhaul capacity range, and as a result, the gap between the outer bounds and the rate of the time-sharing of the known achievable schemes is significantly reduced. [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]

Tietäväinen’s upper and lower bounds assert that for block-length- $n$ linear codes with dual distance $d$ , the covering radius $R$ is at most $({n}/{2})-(({1}/{2})-o(1))\sqrt {dn}$ and typically at least $({n}/{2})-\Theta (({dn\log {({n}/{d})}})^{1/2})$. The gap between those bounds on $R -({n}/{2})$ is a $\Theta (({\log {({n}/{d})}})^{1/2})$ factor related to the gap between the worst covering radius given $d$ and the sphere-covering bound. Our focus in this paper is on the case when $d = o(n)$ , i.e., when the code size is subexponential and the gap is $w(1)$. We show that up to a constant, the gap can be eliminated by relaxing the covering requirement to allow for missing $o(1)$ fraction of points. Namely, if the dual distance $d = o(n)$ , then for sufficiently large $d$ , almost all points can be covered with radius $R\leq ({n}/{2})-\Theta (({dn\log {({n}/{d})}})^{1/2})$. Compared with random linear codes, our bound on $R-({n}/{2})$ is asymptotically tight up to a factor less than 3. We give applications to dual-BCH codes. The proof builds on the author’s previous work on the weight distribution of cosets of linear codes, which we simplify in this paper and extend from codes to probability distributions on $\{0,1\}^{n}$ , thus enabling the extension of the earlier result to $(d-1)$ -wise independent distributions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the cluster estimation problem under the stochastic block model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the censored block model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing and improves upon existing polynomially decaying error bounds proved using the Grothendieck’s inequality. Our analysis builds on two key ingredients: 1) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges and 2) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation. [ABSTRACT FROM AUTHOR]

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

The linear model, in which a set of observations is assumed to be given by a linear combination of columns of a matrix (often termed a dictionary), has long been the mainstay of the statistics and signal processing literature. One particular challenge for inference under linear models is understanding the conditions on the dictionary under which reliable inference is possible. This challenge has attracted renewed attention in recent years, since many modern inference problems (e.g., high-dimensional statistics and compressed sensing) deal with the underdetermined setting, in which the number of observations is much smaller than the number of columns in the dictionary. This paper makes several contributions for this setting when the set of observations is given by a linear combination of a small number of groups of columns of the dictionary, termed the block-sparse case. First, it specifies conditions on the dictionary under which most block submatrices of the dictionary (often termed block subdictionaries) are well conditioned. This result is fundamentally different from prior work on block-sparse inference because: 1) it provides conditions that can be explicitly computed in polynomial time; 2) the given conditions translate into near-optimal scaling of the number of columns of the block subdictionaries as a function of the number of observations for a large class of dictionaries; and 3) it suggests that the spectral norm, rather than the column/block coherences of the dictionary, fundamentally limits the scaling of dimensions of the well-conditioned block subdictionaries. Second, in order to help understand the significance of this result in the context of block-sparse inference, this paper investigates the problems of block-sparse recovery and block-sparse regression in underdetermined settings. In both of these problems, this paper utilizes its result concerning conditioning of block subdictionaries and establishes that near-optimal block-sparse recovery and block-sparse regression is possible for a large class of dictionaries as long as the dictionary satisfies easily computable conditions and the coefficients describing the linear combination of groups of columns can be modeled through a mild statistical prior. Third, the paper reports extensive numerical experiments that highlight the effects of different measures of the dictionary in block-sparse inference problems. [ABSTRACT FROM PUBLISHER]

The rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage. Error correction in the rank modulation scheme is done by considering permutation codes. In this paper, we consider codes in the set of all permutations on n elements, S_{n} , using the Kendall \tau -metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose, we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall \tau -metric, structures which have their own considerable interest. We prove that there are no perfect single-error-correcting codes in S_{n} , where n>4 is a prime or 4\leq n\leq 10 . We present lower bounds on the size of optimal anticodes with odd diameter. As a consequence, we obtain a new upper bound on the size of codes in S_{n} with even minimum Kendall \tau -distance. We present larger single-error-correcting codes than the known ones in S5 and S7 . [ABSTRACT FROM AUTHOR]

This paper investigates the symmetric degrees of freedom (DoF) of multiple-input multiple-output (MIMO) cellular networks with $G$ cells and $K$ users per cell, having $N$ antennas at each base station and $M$ antennas at each user. In particular, we investigate techniques for achievability that are based on either decomposition with asymptotic interference alignment or linear beamforming schemes and show that there are distinct regimes of $(G,K,M,N)$ , where one outperforms the other. We first note that both one-sided and two-sided decomposition with asymptotic interference alignment achieve the same DoF. We then establish specific antenna configurations under which the DoF achieved using decomposition-based schemes is optimal by deriving a set of outer bounds on the symmetric DoF. Using these results, we completely characterize the optimal DoF of any $G$ -cell network with single-antenna users. For linear beamforming schemes, we first focus on small networks and propose a structured approach to linear beamforming based on a notion called packing ratios. Packing ratio describes the interference footprint or shadow cast by a set of transmit beamformers and enables us to identify the underlying structures for aligning interference. Such a structured beamforming design can be shown to achieve the optimal spatially normalized DoF (sDoF) of two-cell two-user/cell network and the two-cell three-user/cell network. For larger networks, we develop an unstructured approach to linear interference alignment, where transmit beamformers are designed to satisfy conditions for interference alignment without explicitly identifying the underlying structures for interference alignment. The main numerical insight of this paper is that such an approach appears to be capable of achieving the optimal sDoF for MIMO cellular networks in regimes where linear beamforming dominates asymptotic decomposition, and a significant portion of sDoF elsewhere. Remarkably, polynomial identity test appears to play a key role in identifying the boundary of the achievable sDoF region in the former case. [ABSTRACT FROM AUTHOR]

This paper deals with Hadamard \mathbb Z2 \mathbb Z4 {Q}8 -codes, which are binary codes after a Gray map from a subgroup of direct products of \mathbb Z2 , \mathbb Z4 , and Q8 , where Q8 is the quaternionic group. In a previous work, these codes were classified in five shapes. In this paper, we analyze the allowable range of values for the rank and dimension of the kernel, which depends on the particular shape of the code. We show that all these codes can be represented in a standard form, from a set of generators, which can help in understanding the characteristics of each shape. The main results we present are the characterization of Hadamard \mathbb Z2 \mathbb Z4 {Q}8 -codes as a quotient of a semidirect product of \mathbb Z2 \mathbb Z4 -linear codes and the construction of Hadamard \mathbb Z2 \mathbb Z4 {Q}8 -codes with each allowable pair of values for the rank and dimension of the kernel. [ABSTRACT FROM AUTHOR]

This paper considers a distributed storage system, where multiple storage nodes can be reconstructed simultaneously at a centralized location. This centralized multi-node repair (CMR) model is a generalization of regenerating codes that allow for bandwidth efficient repair of a single-failed node. This paper focuses on the tradeoff between the amount of data stored and repair bandwidth in the CMR model. In particular, repair bandwidth bounds are derived for the minimum storage multi-node repair (MSMR) and the minimum bandwidth multi-node repair (MBMR) operating points. The tightness of these bounds is analyzed via code constructions. The MSMR point is characterized by codes achieving this point under functional repair for the general set of CMR parameters, as well as with codes enabling exact repair for certain CMR parameters. The MBMR point, on the other hand, is characterized with exact repair codes for all CMR parameters for systems that satisfy a certain entropy accumulation property. Finally, the model proposed here is utilized for the secret sharing problem, where the codes for the multi-node repair problem are used to construct communication efficient secret sharing schemes with the property of bandwidth efficient share repair. [ABSTRACT FROM AUTHOR]

Fundamental relations between information and estimation have been established in the literature for the continuous time Gaussian and Poisson channels. In this paper, we demonstrate that such relations hold for a much larger family of continuous-time channels. We introduce the family of semi-martingale channels where the channel output is a semi-martingale stochastic process, and the channel input modulates the characteristics of the semi-martingale. For these channels, which includes as a special case the continuous time Gaussian and Poisson models, we establish new representations relating the mutual information between the channel input and output to an optimal causal filtering loss, thereby unifying and considerably extending results from the Gaussian and Poisson settings. Extensions to the setting of mismatched estimation are also presented where the relative entropy between the laws governing the output of the channel under two different input distributions is equal to the cumulative difference between the estimation loss incurred by using the mismatched and optimal causal filters, respectively. The main tool underlying these results is the Doob–Meyer decomposition of a class of sub-martingales. The results in this paper can be viewed as the continuous-time analogues of recent generalizations for relations between information and estimation for discrete-time Lévy channels. [ABSTRACT FROM AUTHOR]

We show an explicit (that is, efficient and deterministic) capacity approaching interactive coding scheme that simulates any interactive protocol under random errors with nearly optimal communication rate. Specifically, over the binary symmetric channel with crossover probability $\epsilon $ , our coding scheme achieves a communication rate of $1 - O(\sqrt {H({\epsilon })})$ , together with negligible $\exp (-\Omega (\epsilon ^{4}\,\,n/\log n))$ failure probability (over the randomness of the channel). A rate of $1 - \tilde \Theta (\sqrt {H({\epsilon })})$ is likely asymptotically optimal as a result of Kol and Raz (2013) suggests. Prior to this paper, such a communication rate was achievable only using randomized coding schemes [Kol and Raz (2013); Hauepler (2014)]. [ABSTRACT FROM AUTHOR]

Reed–Muller codes belong to the family of affine-invariant codes. As such codes, they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper, we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced by Bernal and Simón to construct information sets for them from their defining set. For first- and second-order Reed–Muller codes, we describe a direct method to construct information sets in terms of their basic parameters. [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]

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict avoiding codes. In this paper, we prove some nonexistence results for nonsingular perfect splitter sets. We also give some necessary conditions for the existence of purely singular perfect splitter sets. Finally, we apply these results to purely singular perfect $B[-1,k](m)$ and $B[-2,k](m)$ sets for small $k$. In particular, we solve completely the problems left by Schwartz (European J. Combin., vol. 36, pp.130–142, Feb. 2014). [ABSTRACT FROM AUTHOR]

A function computation problem over a directed acyclic network has been considered in the literature, where a sink node is required to compute a target function correctly with the inputs arbitrarily generated at multiple source nodes. The network links are error free but capacity limited, and the intermediate nodes perform network coding. The computing rate of a network code is the average number of times that the target function is computed for one use of the network, i.e., each link in the network is used at most once. In the existing papers, two cut-set bounds were proposed on the computing rate. However, we in this paper show that these bounds are not valid for general network function computation problems. We analyze the reason of the invalidity and propose a general cut-set bound by using a new equivalence relation associated with the inputs of the target function. Moreover, some results in the existing papers were proved by applying the invalid upper bound. We also justify the validity of these results. [ABSTRACT FROM AUTHOR]

High-performance steganography requires large embedding rate and small distortion, i.e., high embedding efficiency. Steganographic codes (stego-codes) derived from covering codes can improve embedding efficiency. In this paper, a new method is proposed to construct binary stego-codes for LSB embedding in gray-scale signals, which shows that not just one but a family of stego-codes can be generated from a covering code by combining Hamming codes and wet paper codes. This method can greatly expand the set of embedding schemes as applied to steganography. Performances of stego-code families (SCF) of structured codes and random codes are analyzed. SCFs of random codes can approach the rate-distortion bound on LSB embedding for any chosen embedding rate. Furthermore, SCFs are modified for applications in ±1 embedding, and a treble layered embedding method for ±2 embedding is obtained. By combining the modified SCFs and the treble layered method, a near-optimal scheme for ±2 embedding is presented. [ABSTRACT FROM AUTHOR]

The article discusses various papers published in this issue including one by Don H. Johnson on information theory and neural information processing and another by Aurel A. Lazar on population encoding with Hodgkin-Huxley neurons.

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the \mathrm L^1 -risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the \mathrm L^2 -risk of the nonparametric spectral estimators. In the case where the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions. [ABSTRACT FROM AUTHOR]

Blind deconvolution (BD) arises in many applications. Without assumptions on the signal and the filter, BD does not admit a unique solution. In practice, subspace or sparsity assumptions have shown the ability to reduce the search space and yield the unique solution. However, existing theoretical analysis on uniqueness in BD is rather limited. In an earlier paper, we provided the first algebraic sample complexities for BD that hold for Lebesgue almost all bases or frames. We showed that for BD of a pair of vectors in \mathbb C^n , with subspace constraints of dimensions m1 and m2 , respectively, a sample complexity of n\geq m1m2 is sufficient. This result is suboptimal, since the number of degrees of freedom is merely m1+m2-1 . We provided analogous results, with similar suboptimality, for BD with sparsity or mixed subspace and sparsity constraints. In this paper, taking advantage of the recent progress on the information-theoretic limits of unique low-rank matrix recovery, we finally bridge this gap, and derive an optimal sample complexity result for BD with generic bases or frames. We show that for BD of an arbitrary pair (respectively, all pairs) of vectors in \mathbb C ^n , with sparsity constraints of sparsity levels s1 and s2 , a sample complexity of n > s1+s2 [respectively, n > 2(s1+s2) ] is sufficient. We also present analogous results for BD with subspace constraints or mixed constraints, with the subspace dimension replacing the sparsity level. Last but not least, in all the above scenarios, if the bases or frames follow a probabilistic distribution specified in this paper, the recovery is not only unique, but also stable against small perturbations in the measurements, under the same sample complexities. [ABSTRACT FROM PUBLISHER]

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes—a class of linear codes—have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper. [ABSTRACT FROM PUBLISHER]

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Pathogen identification involves the identification of disease-causing biomolecules. Drug discovery involves finding chemical compounds, called lead compounds, that bind to pathogenic proteins and eventually inhibit the function of the protein. In this paper, the lead compounds are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of “ineffective” chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive if it does not contain its “associated” inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their “associations” with high probability, or in other words, learn the immune defectives graph (IDG) efficiently through group tests. We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design, and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives, and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. To be precise, lower and upper bounds of $\Omega ((r+d)\log n + rd)$ and O(rd \log n)$ tests, respectively, are identified for classifying r$ inhibitors and d$ defectives amongst n tests) exceeds the proposed lower bound (given by \max \{\Omega ((r+d)\log n + rd),\Omega (({r^{2}}/{\log r})\log n),\Omega (d^{2})\} tests) by at most a logarithmic multiplicative factor in the number of items. [ABSTRACT FROM AUTHOR]

This paper studies codes that correct a burst of deletions or insertions. Namely, a code will be called a $b$ -burst-deletion/insertion-correcting code if it can correct a burst of deletions/insertions of any $b$ consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically $\log (n)+b-1$ , the redundancy of the best code construction by Cheng et al. is $b(\log (n/b+1))$ . In this paper, we close on this gap and provide codes with redundancy at most $\log (n) + (b-1)\log (\log (n)) +b -\log (b)$ . We first show that the models of insertions and deletions are equivalent and thus it is enough to study codes correcting a burst of deletions. We then derive a non-asymptotic upper bound on the size of $b$ -burst-deletion-correcting codes and extend the burst deletion model to two more cases: 1) a deletion burst of at most $b$ consecutive bits and 2) a deletion burst of size at most $b$ (not necessarily consecutive). We extend our code construction for the first case and study the second case for $b=3,4$ . [ABSTRACT FROM PUBLISHER]