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In this paper, we consider communication on a two-hop channel in which a source wants to send information reliably and securely to the destination via a relay. We consider both the untrusted relay case and the external eavesdropper case. In the untrusted relay case, the relay behaves as an eavesdropper, and there is a cooperative node, which sends a jamming signal to confuse the relay when it is receiving from the source. In the external eavesdropper case, the relay is trusted, and there is an external node eavesdropping the communication. We propose two secure transmission schemes using the scaled compute-and-forward technique. One of the schemes is based on a random binning code, and the other one is based on a lattice chain code. It is proved that in the high signal-to-noise-ratio (SNR) scenario and/or the limited relay power scenario, if the destination is used as the jammer, both schemes outperform all existing schemes and achieve the upper bound. In particular, if the SNR is large and the source, the relay, and the cooperative jammer have identical power and channels, both schemes achieve the upper bound for secrecy rate, which is merely 1/2 bit per channel use lower than the channel capacity without secrecy constraints. We also prove that one of our schemes achieves a positive secrecy rate in the external eavesdropper case in which the relay is trusted and there exists an external eavesdropper. [ABSTRACT FROM AUTHOR]

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that a complementary channel can be obtained from the channel by applying a degrading channel. In this paper, we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite $\varepsilon \geq 0$ . That is, there exists a degrading channel which upon composition with the channel is $\varepsilon $ -close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such $\varepsilon $ can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication. [ABSTRACT FROM AUTHOR]

Frameproof codes are used to fingerprint digital data. They can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for $w$ -frameproof codes of length N$ over an alphabet of size q$ . The upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction. Both bounds can improve one of the previous results when q$ is small compared with w$ , say cq\leq w$ for some constant c\leq q$ . Furthermore, we pay special attention to binary frameproof codes. We show a binary w . [ABSTRACT FROM PUBLISHER]

A regular Hilberg process is a stationary process that satisfies both a hyperlogarithmic growth of maximal repetition and a power-law growth of topological entropy, which are a kind of dual conditions. The hyperlogarithmic growth of maximal repetition has been experimentally observed for texts in natural language, whereas the power-law growth of topological entropy implies a vanishing Shannon entropy rate and thus probably does not hold for natural language. In this paper, we provide a constructive example of regular Hilberg processes, which we call random hierarchical association (RHA) processes. Our construction does not apply the standard cutting and stacking method. For the constructed RHA processes, we demonstrate that the expected length of any uniquely decodable code is the orders of magnitude larger than the Shannon block entropy of the ergodic component of the RHA process. Our proposition supplements the classical result by Shields concerning nonexistence of universal redundancy rates. [ABSTRACT FROM PUBLISHER]

The capacity region of the $K$ -user Gaussian interference channel (GIC) is a long-standing open problem and even capacity outer bounds are little known in general. A significant progress on degrees-of-freedom (DoF) analysis, a first-order capacity approximation, for the $K$ -user GIC has provided new important insights into the problem of interest in the high signal-to-noise ratio (SNR) limit. However, such capacity approximation has been observed to have some limitations in predicting the capacity at finite SNR. In this paper, we develop a new upper-bounding technique that utilizes a new type of genie signal and applies time sharing to genie signals at $K$ receivers. Based on this technique, we derive new upper bounds on the sum capacity of the three-user GIC with constant, complex channel coefficients and then generalize to the $K$ -user case to better understand sum-rate behavior at finite SNR. We also provide closed-form expressions of our upper bounds on the capacity of the $K$ -user symmetric GIC easily computable for any $K$ . From the perspectives of our results, some sum-rate behavior at finite SNR is in line with the insights given by the known DoF results, while some others are not. In particular, the well-known $K/2$ DoF achievable for almost all constant real channel coefficients turns out to be not embodied as a substantial performance gain over a certain range of the cross-channel coefficient in the $K$ -user symmetric real case especially for large $K$ . We further investigate the impact of phase offset between the direct-channel coefficient and the cross-channel coefficients on the sum-rate upper bound for the three-user complex GIC. As a consequence, we aim to provide new findings that could not be predicted by the prior works on DoF of GICs. [ABSTRACT FROM PUBLISHER]

Caching is a promising solution to satisfy the ever-increasing demands for the multi-media traffics. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains over the uncoded caching schemes. However, to implement the coded caching schemes, each file has to be split into F packets, which usually increases exponentially with the number of users K . Thus, designing caching schemes that decrease the order of F is meaningful for practical implementations. In this paper, by reviewing the Ali-Niesen caching scheme, the placement delivery array (PDA) design problem is first formulated to characterize the placement issue and the delivery issue with a single array. Moreover, we show that, through designing appropriate PDA, new centralized coded caching schemes can be discovered. Second, it is shown that the Ali-Niesen scheme corresponds to a special class of PDA, which realizes the best coding gain with the least F . Third, we present a new construction of PDA for the centralized coded caching system, wherein the cache size M at each user (identical cache size is assumed at all users) and the number of files N ( q is an integer, such that q\geq 2 ). The new construction can decrease the required F of Ali-Niesen scheme to O(e^{K\cdot ({M}/{N})\ln \!\,({N}/\!{M})}) or O(e^{K\cdot (1-({M}/\!{N}))\ln \!\,(N/(N-M))})$ , respectively, while the coding gain loss is only 1. [ABSTRACT FROM AUTHOR]

We study the channel capacity of a general discrete energy harvesting channel with a finite battery. Contrary to traditional communication systems, the transmitter of such a channel is powered by a device that harvests energy from a random exogenous energy source and has a finite-sized battery. As a consequence, at each transmission opportunity, the system can only transmit a symbol whose energy is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is simultaneously random, instantaneous, and influenced by the full history of the inputs and the energy harvesting process. Furthermore, naturally, in such a channel, the energy information is observed causally at the transmitter. Both of these characteristics pose great challenges for the analysis of the channel capacity. In this paper, we use techniques developed for channels with side information and finite-state channels, to obtain lower and upper bounds on the capacity of energy harvesting channels. In particular, in a general case with Markov energy harvesting processes, we use stationarity and ergodicity theory to compute and optimize the achievable rates for the channels, and derive a series of computable capacity upper and lower bounds. [ABSTRACT FROM AUTHOR]

A phase detection sequence is a length- $n$ cyclic sequence, such that the location of any length- $k$ contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible $k$ in the limit of $n\to \infty $ , and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length- $k$ contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed. [ABSTRACT FROM AUTHOR]

Binary almost instantaneous fixed-to-variable length (AIFV) codes are lossless codes that generalize the class of instantaneous fixed-to-variable length codes. The code uses two code trees and assigns source symbols to incomplete internal nodes as well as to leaves. AIFV codes are empirically shown to attain better compression ratio than Huffman codes. Nevertheless, an upper bound on the redundancy of optimal binary AIFV codes is only known to be 1, which is the same as the bound of Huffman codes. In this paper, the upper bound is improved to 1/2, which is shown to coincide with the worst-case redundancy of the codes. Along with this, the worst-case redundancy is derived for sources with p\max \geq 1 /2, where p\max is the probability of the most likely source symbol. In addition, we propose an extension of binary AIFV codes, which use $m$ code trees and allow at most $m$ -bit decoding delay. We show that the worst-case redundancy of the extended binary AIFV codes is $1/m$ for $m \leq 4$ . [ABSTRACT FROM AUTHOR]

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]

The best outer bound on the capacity region of the two-user Gaussian interference channel (GIC) is known to be the intersection of regions of various bounds, including genie-aided outer bounds, in which a genie provides noisy input signals to the intended receiver. The Han and Kobayashi (HK) scheme provides the best known inner bound. The rate difference between the best known lower and upper bounds on the sum capacity remains as large as 1 b/channel use, especially around g^2=P^-1/3 , where P is the symmetric power constraint and g is the symmetric real cross-channel coefficient. In this paper, we pay attention to the moderate interference regime where g^2\in ~(\max (0.086, P^-1/3),1) . We propose a new upper-bounding technique that utilizes noisy observation of interfering signals as genie signals and applies time sharing to the genie signals at the receivers. A conditional version of the worst additive noise lemma is also introduced to derive new capacity bounds. The resulting upper (outer) bounds on the sum capacity (capacity region) are shown to be tighter than the existing bounds in a certain range of the moderate interference regime. Using the new upper bounds and the HK lower bound, we show that R\text {sym}^{ }=\frac {1}2\log \big (|g|P+|g|^{-1}(P+1)\big ) characterizes the capacity of the symmetric real GIC to within 0.104 b/channel use in the moderate interference regime at any signal-to-noise ratio (SNR). We further establish a high-SNR characterization of the symmetric real GIC, where the proposed upper bound is at most 0.1 b far from a certain HK achievable scheme with Gaussian signalling and time sharing for g^2\in ~(0,1] . In particular, R\text {sym}^{ } is achievable at high SNR by the proposed HK scheme and turns out to be the high-SNR capacity at least at g^2=0.25, 0.5 . We finally point out that there are two subregimes at high SNR in the weak interference regime, where g^2\in (0,1] . [ABSTRACT FROM AUTHOR]

This paper revisits the Gaussian degraded relay channel, where the link that carries information from the source to the destination is a physically degraded version of the link that carries information from the source to the relay. The source and the relay are subject to expected power constraints. The $\varepsilon $ -capacity of the channel is characterized and it is strictly larger than the capacity for any $\varepsilon >0$ , which implies that the channel does not possess the strong converse property. The proof of the achievability part is based on several key ideas: block Markov coding, which is used in the classical decode-forward strategy, power control for Gaussian channels under expected power constraints, and a careful scaling between the block size and the total number of block uses. The converse part is proved by first establishing two non-asymptotic lower bounds on the error probability, which are derived from the type-II errors of some binary hypothesis tests. Subsequently, each lower bound is simplified by conditioning on an event related to the power of some linear combination of the codewords transmitted by the source and the relay. Lower and upper bounds on the second-order term of the optimal coding rate are also obtained. [ABSTRACT FROM PUBLISHER]

Suppose Y^n is obtained by observing a uniform Bernoulli random vector X^n through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Y^n and a Boolean function \mathsf b(X^n) could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of Y^n under logarithmic loss, given \mathsf b(X^n) . In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function—the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than a dictator, and that for a strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels. [ABSTRACT FROM PUBLISHER]

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for the MCWCs, which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson-type upper bounds of the MCWCs are asymptotically tight for fixed Hamming weights and distances. Finally, we provide bounds and constructions of the 2-D MCWCs. [ABSTRACT FROM AUTHOR]

Handling interference is one of the main challenges in the design of wireless networks. In this paper, we study the application of cooperation for interference management in the weak interference (WI) regime, focusing on the Z-interference channel with a causal relay (Z-ICR), in which the channel coefficients are subject to ergodic phase fading, all transmission powers are finite, and the relay is full-duplex. The phase fading model represents many practical communications systems in which the transmission path impairments mainly affect the phase of the signal, such as non-coherent wireless communications and fiber optic channels. In order to provide a comprehensive understanding of the benefits of cooperation in the WI regime, we characterize, for the first time, two major performance measures for the ergodic phase fading Z-ICR in the WI regime: the sum-rate capacity and the maximal generalized degrees-of-freedom (GDoF). In the capacity analysis, we obtain conditions on the channel coefficients, subject to which the sum-rate capacity of the ergodic phase fading Z-ICR is achieved by treating interference as noise at each receiver, and explicitly state the corresponding sum-rate capacity. In the GDoF analysis, we derive conditions on the exponents of the magnitudes of the channel coefficients, under which treating interference as noise achieves the maximal GDoF, which is explicitly characterized as well. It is shown that under certain conditions on the channel coefficients, relaying strictly increases both the sum-rate capacity and the maximal GDoF of the ergodic phase fading Z-interference channel in the WI regime. Our results demonstrate for the first time the gains from relaying in the presence of interference, when interference is weak and the relay power is finite, both in increasing the sum-rate capacity and in increasing the maximal GDoF, compared with the channel without a relay. [ABSTRACT FROM AUTHOR]

Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. A simple but accurate limiting volume formula is then obtained by invoking Szegő’s strong limit theorem for large Toeplitz matrices. The derived asymptotic volume formula enables analytical evaluation of some coding-theoretic bounds of unitary codes. In particular, the Gilbert–Varshamov lower bound and the Hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived. Moreover, bounds on the scaling law of code rate are found. Finally, a closed-form bound on the diversity sum relevant to unitary space-time codes is obtained, which was only computed numerically in the literature. [ABSTRACT FROM AUTHOR]

This paper addresses the detection of a suddenly arriving dynamic profile of a finite duration often called a transient change. In contrast to the traditional abrupt change detection, where the post-change period is assumed to be infinitely long, the detection of a suddenly arriving transient change should be done before it disappears. The detection of transient changes after their disappearance is considered as missed. Hence, the traditional quickest change detection criterion, minimizing the average detection delays provided a prescribed false alarm rate, is compromised. The proposed optimality criterion minimizes the worst case probability of missed detection provided that the worst case probability of false alarm during a certain period is upper bounded. A suboptimal CUSUM-type transient change detection algorithm, based on a subclass of truncated Sequential Probability Ratio Tests, is proposed. The optimization of the proposed algorithm in this subclass leads to a specially designed Finite Moving Average Test. The proposed method is analyzed theoretically and by simulation. A special attention is paid to the case of Gaussian observations with a dynamic profile. [ABSTRACT FROM AUTHOR]

In order to scale economically, data centers are increasingly evolving their data storage methods from simple data replication to more powerful erasure codes, which provide the same level of reliability as replication, but at a significantly lower storage cost. In particular, it is well known that maximum-distance-separable (MDS) codes, such as Reed–Solomon codes, can achieve a target reliability with the maximum storage efficiency. While the use of codes for providing improved reliability in archival storage systems, where data is less frequently accessed (or so-called “cold data”), is well understood, the role of codes in storing more frequently accessed and active “hot data”, where latency is the key metric, is less clear. In this paper, we study data storage systems based on MDS codes through the lens of queueing theory, and term the queueing system arising under codes as an “MDS queue.” We provide lower and upper bounds on the average job latency for both centralized and decentralized versions of MDS queues. We also provide extensive simulations to corroborate our analysis as well as obtain additional insights. [ABSTRACT FROM PUBLISHER]

This paper focuses on the problem of broadcasting information in the most efficient manner in a large 2-D ad hoc wireless network at low SNR and under line-of-sight propagation. A new communication scheme is proposed, where source nodes first broadcast their data to the entire network, despite the lack of sufficient available power. The signal’s power is then reinforced via successive back-and-forth beamforming transmissions between different groups of nodes in the network, so that all nodes are able to decode the transmitted information at the end. This scheme is shown to achieve asymptotically the broadcast capacity of the network, which is expressed in terms of the largest singular value of the matrix of fading coefficients between the nodes in the network. A detailed mathematical analysis is then presented to evaluate the asymptotic behavior of this largest singular value. [ABSTRACT FROM PUBLISHER]

In this paper, we address the problem of the universal coding of natural numbers. A new numeration system is introduced, which is based on variable- $r$ meta-Fibonacci sequences and it is a generalization of the Zeckendorf numeration system. This new numeration system is used to construct binary, prefix-free, uniquely decodable universal codes called meta-Fibonacci codes. The main advantage of these codes is that they are parametrized by a sequence of numbers, the sequence o. By controlling the growth of the values of this sequence, we can control the length of the code word. This means that we can provide a general framework for building efficient universal coders for natural numbers. Such framework is applied to the upper bounds of the code word length defined by Leung-Yan-Cheong and Cover (1978), Levenshtein (1968), and Ahlswede (1997). There is no other code meeting these bounds. In each case, we build meta-Fibonacci codes and demonstrate that the upper bound of their code word length is satisfied up to an additive constant, thereby solving these open problems. The framework may be applied to other upper bounds that satisfy Kraft inequality. [ABSTRACT FROM PUBLISHER]

This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion/deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion/insertion-correcting Varshamov–Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion/deletion channels and probabilistic channels. [ABSTRACT FROM PUBLISHER]

Shannon’s analysis of the fundamental capacity limits for memoryless communication channels has been refined over time. In this paper, the maximum volume M_ \mathrm avg^*(n,\epsilon ) of length- n codes subject to an average decoding error probability \epsilon is shown to satisfy the following tight asymptotic lower and upper bounds as n \to \infty : , where C is the Shannon capacity, V_\epsilon is the \epsilon -channel dispersion, or second-order coding rate, Q and \overline A_\epsilon are explicitly identified. This expression holds under mild regularity assumptions on the channel, including nonsingularity. The gap \overline A_\epsilon - \underline A_\epsilon is one nat for weakly symmetric channels in the Cover–Thomas sense, and typically a few nats for other symmetric channels, for the binary symmetric channel, and for the $Z$ channel. The derivation is based on strong large-deviations analysis and refined central limit asymptotics. A random coding scheme that achieves the lower bound is presented. The codewords are drawn from a capacity-achieving input distribution modified by an O(1/\sqrt {n}) correction term. [ABSTRACT FROM PUBLISHER]

This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here, we use this approach along with a “privacy test” to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel’s relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and the receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels. [ABSTRACT FROM PUBLISHER]

In this paper, we revisit the problem of characterizing the secrecy capacity of minimum storage regenerating (MSR) codes under the passive (l1,l2) -eavesdropper model, where the eavesdropper has access to data stored on l1 nodes and the repair data for an additional l2 nodes. We study it from the information-theoretic perspective. First, some general properties of MSR codes as well as a simple and generally applicable upper bound on secrecy capacity are given. Second, a new concept of stable MSR codes is introduced, where the stable property is shown to be closely linked with secrecy capacity. Finally, a comprehensive and explicit result on secrecy capacity in the linear MSR scenario is present, which generalizes all related works in the literature and also predicts certain results for some unexplored linear MSR codes. [ABSTRACT FROM PUBLISHER]

Efficient list decoding of rank-metric codes seems more difficult compared with classical block codes although list decodability of random rank-metric codes is completely determined by Ding. For example, it was shown by Raviv and Wachter-Zeh that the list decoding radius of Gabidulin codes is the same as the unique decoding radius, i.e., half the minimum distance for some instances of parameters. On the other hand, Guruswami and Xing give an explicit construction of subcodes of Gabidulin codes, which can be list decoded up to the Singleton bound. This implies that subcodes of Gabidulin codes are good candidates for list decoding. In this paper, we confirm that, with overwhelming probability, a random subcode of a Gabidulin code can be list decoded with decoding radius far beyond half of the minimum distance. [ABSTRACT FROM PUBLISHER]

In many applications including communications, one may encounter a linear model where the parameter vector \hat x is an integer vector in a box. To estimate \hat x , a typical method is to solve a box-constrained integer least squares problem. However, due to its high complexity, the box-constrained Babai integer point x^ \scriptscriptstyle \text BB is commonly used as a suboptimal solution. In this paper, we first derive formulas for the success probability P^ \scriptscriptstyle \text BB of x^ \scriptscriptstyle \text BB and the success probability P^ \scriptscriptstyle \text OB of the ordinary Babai integer point x^ \scriptscriptstyle \text OB when \hat x is uniformly distributed over the constraint box. Some properties of P^ \scriptscriptstyle \text BB and P^ \scriptscriptstyle \text OB and the relationship between them are studied. Then, we investigate the effects of some column permutation strategies on P^ \scriptscriptstyle \text BB . In addition to V-BLAST and SQRD, we also consider the permutation strategy involved in the LLL lattice reduction, to be referred to as LLL-P. On the one hand, we show that when the noise is relatively small, LLL-P always increases P^ \scriptscriptstyle \text BB and argue why both V-BLAST and SQRD often increase P^ \scriptscriptstyle \text BB ; and on the other hand, we show that when the noise is relatively large, LLL-P always decreases P^ \scriptscriptstyle \text BB and argue why both V-BLAST and SQRD often decrease P^ \scriptscriptstyle \text BB . We also derive a column permutation invariant bound on P^ \scriptscriptstyle \text BB , which is an upper bound and a lower bound under these two opposite conditions, respectively. Numerical results demonstrate our findings. Finally, we consider a conjecture concerning x^ \scriptscriptstyle \text OB proposed by Ma et al. We first construct an example to show that the conjecture does not hold in general, and then show that it does hold under some conditions. [ABSTRACT FROM PUBLISHER]

Treating interference as noise (TIN) as the most appropriate approach in dealing with interference and the conditions on its optimality has attracted the interest of researchers recently. However, our knowledge on necessary and sufficient conditions of TIN is restricted to a few setups with limited number of users. In this paper, we study the optimality of TIN in terms of the generalized degrees of freedom (GDoF) for a fundamental network, namely, the $M\times 2$ X-channel. To this end, the achievable GDoF of TIN with power allocations at the transmitters is studied. It turns out that the transmit power allocation maximizing the achievable GDOF is given by on–off signaling as long as the receivers use TIN. This leads to two variants of TIN, namely, P2P-TIN and 2-IC-TIN. While in the first variant the $M\times 2$ X-channel is reduced to a point-to-point (P2P) channel, in the second variant, the setup is reduced to a two-user interference channel in which the receivers use TIN. The optimality of these two variants is studied separately. To this end, novel genie-aided upper bounds on the capacity of the X-channel are established. The conditions on the optimality of P2P-TIN can be summarized as follows. P2P-TIN is GDoF-optimal if there exists a dominant multiple access channel or a dominant broadcast channel embedded in the X channel. Furthermore, the necessary and sufficient conditions on the GDoF-optimality of 2-IC-TIN are presented. Interestingly, it turns out that operating the $M\times 2$ X-channel in the 2-IC-TIN mode might be still GDOF optimal, although the conditions given by Geng et al. are violated. However, 2-IC-TIN is sub-optimal if there exists a single interferer which causes sufficiently strong interference at both receivers. The comparison of the results with the state of the art shows that the GDOF optimality of TIN is expanded significantldy. [ABSTRACT FROM PUBLISHER]

In this paper, list decoding of crisscross errors in arrays over finite fields is considered. For this purpose, the so-called cover metric is used, where the cover of a matrix is a set of rows and columns which contains all non-zero elements of the matrix. A Johnson-like upper bound on the maximum list size in the cover metric is derived, showing that the list of codewords has polynomial size up to a certain radius. Furthermore, a simple list decoding algorithm for a known optimal code construction is presented, which decodes errors in the cover metric up to our upper bound. These results reveal significant differences between the cover metric and the rank metric and show that the cover metric is more suitable for correcting crisscross errors. [ABSTRACT FROM PUBLISHER]

The impact of cooperation on interference management is investigated by studying an elemental wireless network, the so-called symmetric interference relay channel (IRC), from a generalized degrees of freedom (GDoF) perspective. This is motivated by the fact that the deployment of relays is considered as a remedy to overcome the bottleneck of current systems in terms of achievable rates. The focus of this paper is on the regime in which the interference link is weaker than the source-relay link in the IRC. Our approach toward studying the GDoF goes through the capacity analysis of the linear deterministic IRC (LD-IRC). New upper bounds on the sum capacity of the LD-IRC based on genie-aided approaches are established. These upper bounds together with some existing upper bounds are achieved by using four novel transmission schemes. Extending the upper bounds and the transmission schemes to the Gaussian case, the GDoF of the Gaussian IRC is characterized for the aforementioned regime. This completes the GDoF results available in the literature for the symmetric GDoF. It turns out that even if the incoming and outgoing links of the relay are both weaker than the desired channel, involving a relay can increase the GDoF. Interestingly, utilizing the relay in this case can increase the slope of the GDoF from −2 [in the interference channel (IC)] to −1 or 0. This shrinks the regime where ignoring the interference by treating it as noise is optimal. Furthermore, the analysis shows that if the relay ingoing and outgoing links are sufficiently strong, the relay is able to neutralize the interference completely. In this case, the bottleneck of the transmission will be the interference links, and hence, the GDoF increases if the interference link gets stronger. It is shown that in the strong interference regime, in contrast to the IC, the GDoF can be a monotonically decreasing function of the interference level. [ABSTRACT FROM PUBLISHER]

Consider the transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W and let be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters R , N , , and the quality of the channel W quantified by its capacity I(W) and its Bhattacharyya parameter Z(W) . In previous work, two main regimes were studied. In the error exponent regime, the channel W and the rate scales roughly as 2^{-\sqrt {N}} . In the scaling exponent approach, the channel W and the error probability P\mathrm{ e} are fixed and it was proved that the gap to capacity I(W)-R scales as N^-1/\mu . Here, $\mu $ is called scaling exponent and this scaling exponent depends on the channel W$ . A heuristic computation for the binary erasure channel (BEC) gives \mu =3.627$ and it was shown that, for any channel W$ , 3.579 \le \mu \le 5.702$ . Our contributions are as follows. First, we provide the tighter upper bound \mu \le 4.714$ valid for any WN$ . In other words, we neither fix the gap to capacity (error exponent regime) nor the error probability (scaling exponent regime), but we do consider a moderate deviations regime in which we study how fast both quantities, as the functions of the block length N$ , simultaneously go to 0. Third, we prove that polar codes are not affected by error floors. To do so, we fix a polar code of block length N$ and rate R$ . Then, we vary the channel W scales as the Bhattacharyya parameter Z(W)$ raised to a power that scales roughly like {\sqrt {N}} . This agrees with the scaling in the error exponent regime. [ABSTRACT FROM PUBLISHER]

In this paper, we consider optimal q -ary cyclic constant-weight codes of length n , minimum distance d , and weight w codes. We introduce the pure and mixed difference method to present a combinatorial description for a cyclic (n,d,w)_{q} code and then obtain some tight upper bounds on the sizes of optimal cyclic (n,d,w)_{q} codes. Finally, by using Skolem-type sequences, we completely determine the sizes of optimal cyclic (n,d,3)3 codes with minimum distance 1\leq d\leq 6 . [ABSTRACT FROM AUTHOR]

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. First, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee et al. for some parameters. The asymptotic lower bound of MCWCs is also examined. Then, we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances 2\sum i=1^{m}wi-2 or $2mw-2w$ are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely. [ABSTRACT FROM AUTHOR]

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

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

This paper starts with new aliasing-truncation error upper bounds in the sampling theorem for non-bandlimited stochastic signals. Then, it investigates Lp([0,T]) approximations of the sub-Gaussian random signals. Explicit truncation error upper bounds are established. The obtained rate of convergence provides a constructive algorithm for determining the sampling rate and the sample size in the truncated Whittaker-Kotel’nikov-Shannon expansions to ensure the approximation of the sub-Gaussian signals with given accuracy and reliability. Some numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an instance of the Heegard–Berger problem with two sources and arbitrarily correlated side information sequences at two decoders, in which the reconstruction sets at the decoders are degraded. Specifically, two sources are to be encoded in a manner that one of the two is reproduced losslessly by both the decoders, and the other is reproduced within some prescribed distortion level at one of the two decoders. We establish a single-letter characterization of the rate-distortion function for this model. In particular, we show that the optimal coding scheme for this setting is one in which the common description to be recovered by both the decoders should allow to involve all or part of the source that is to be reproduced at only one decoder. Furthermore, we also generalize our result to the setting in which the source component that is to be recovered by both users is reconstructed in a lossy fashion, under the requirement that all terminals (i.e., the encoder and both the decoders) can share an exact copy of the compressed version of this source component, i.e., a common encoder–decoder reconstruction constraint. For this model as well, we establish a single-letter characterization of the associated rate-distortion function. [ABSTRACT FROM PUBLISHER]

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

The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a regularizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This paper provides a simple characterization of all operators saturating the upper and lower bounds. [ABSTRACT FROM AUTHOR]

Compressed sensing deals with the recovery of sparse signals from linear measurements. Without any additional information, it is possible to recover an s -sparse signal using m \gtrsim s \log (d/s) measurements in a robust and stable way. Some applications provide additional information, such as on the location the support of the signal. Using this information, it is conceivable that the threshold amount of measurements can be lowered. A proposed algorithm for this task is weighted \ell 1 -minimization. Put shortly, one modifies standard \ell 1 -minimization by assigning different weights to different parts of the index set $[1, \ldots d]$ . The task of choosing the weights is, however, non-trivial. This paper provides a complete answer to the question of an optimal choice of the weights. In fact, it is shown that it is possible to directly calculate unique weights that are optimal in the sense that the threshold amount of measurements needed for exact recovery is minimized. The proof uses recent results about the connection between convex geometry and compressed sensing-type algorithms. [ABSTRACT FROM PUBLISHER]

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

We revisit the problem of secret key agreement using interactive public communication for two parties and propose a new secret key agreement protocol. The protocol attains the secret key capacity for general observations and attains the second-order asymptotic term in the maximum length of a secret key for independent and identically distributed observations. In contrast to the previously suggested secret key agreement protocols, the proposed protocol uses interactive communication. In fact, the standard one-way communication protocol used prior to this paper fails to attain the asymptotic results above. Our converse proofs rely on a recently established upper bound for secret key lengths. Both our lower and upper bounds are derived in a single-shot setup and the asymptotic results are obtained as corollaries. [ABSTRACT FROM PUBLISHER]

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

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert–Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets. [ABSTRACT FROM AUTHOR]

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

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

In this paper, we derive non-asymptotic achievability and converse bounds on the random number generation with/without side-information. Our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length. We also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds, which implies that our bounds are asymptotically tight in those regimes. We also show the second-order rates of those problems, and derive single letter forms of the variances characterizing the second-order rates. Furthermore, we address the relative entropy rate and the modified mutual information rate for these problems. [ABSTRACT FROM AUTHOR]

The classical problem in a network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers that decode the same set of messages. In this paper, computation over multicast networks is considered: each receiver decodes an identical function of the original messages. For a countably infinite class of two-transmitter two-receiver single-hop linear deterministic networks, the computation capacity is characterized for a linear function (modulo-2 sum) of Bernoulli sources. A new upper bound is derived that is tighter than cut-set-based and genie-aided bounds. A matching inner bound is established via the development of a network decomposition theorem, which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality. The decomposition theorem provides a conceptually simple proof of achievability that generalizes to $L$ -transmitter $L$ -receiver networks. [ABSTRACT FROM AUTHOR]

Gabidulin codes can be seen as the rank-metric equivalent of Reed–Solomon codes. It was recently proved, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially sized lists exist for codes with specific parameters. The second bound presents exponentially sized lists explicitly for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes. [ABSTRACT FROM AUTHOR]

Random forests are ensemble methods which grow trees as base learners and combine their predictions by averaging. Random forests are known for their good practical performance, particularly in high-dimensional settings. On the theoretical side, several studies highlight the potentially fruitful connection between the random forests and the kernel methods. In this paper, we work out this connection in detail. In particular, we show that by slightly modifying their definition, random forests can be rewritten as kernel methods (called KeRF for kernel based on random forests) which are more interpretable and easier to analyze. Explicit expressions of KeRF estimates for some specific random forest models are given, together with upper bounds on their rate of consistency. We also show empirically that the KeRF estimates compare favourably to the random forest estimates. [ABSTRACT FROM AUTHOR]

Frequency-hopping sequences (FHSs) with favorable partial Hamming correlation properties are desirable in many synchronization and multiple-access systems. An FHS set is said to be strictly optimal if it has optimal partial Hamming correlation for any correlation window. In this paper, we derive upper bounds on the family sizes of FHS sets with respect to partial Hamming correlation from some classical bounds on error-correcting codes. We then present strictly optimal FHS sets having optimal family sizes with respect to one of the new bounds. In particular, our construction gives new parameters not covered in the literature. [ABSTRACT FROM AUTHOR]