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

In this paper, we introduce the concept of spatially coupled turbo-like codes (SC-TCs) as the spatial coupling of a number of turbo-like code ensembles. In particular, we consider the spatial coupling of parallel concatenated codes, introduced by Berrou et al., and that of serially concatenated codes (SCCs), introduced by Benedetto et al. Furthermore, we propose two extensions of braided convolutional codes (BCCs), and a class of turbo-like codes which have an inherent spatially coupled structure, to higher coupling memories, and show that these yield improved belief propagation (BP) thresholds as compared with the original BCC ensemble. We derive the exact density evolution (DE) equations for SC-TCs and analyze their asymptotic behavior on the binary erasure channel. We also consider the construction of families of rate-compatible SC-TC ensembles. Our numerical results show that the threshold saturation of the BP decoding threshold to the maximum a posteriori threshold of the underlying uncoupled ensembles occurs for large enough coupling memory. The improvement of the BP threshold is especially significant for SCCs and BCCs, whose uncoupled ensembles suffer from a poor BP threshold. For a wide range of code rates, SC-TCs show close-to-capacity performance as the coupling memory increases. We further give a proof of threshold saturation for SC-TC ensembles with identical component encoders. In particular, we show that the DE of SC-TC ensembles with identical component encoders can be properly rewritten as a scalar recursion. This allows us to define potential functions and prove threshold saturation using the proof technique recently introduced by Yedla et al. [ABSTRACT FROM PUBLISHER]

This paper studies the performance of sparse regression codes for lossy compression with the squared-error distortion criterion. In a sparse regression code, code words are linear combinations of subsets of columns of a design matrix. It is shown that with minimum-distance encoding, sparse regression codes achieve the Shannon rate-distortion function for i.i.d. Gaussian sources R^ (D) as well as the optimal excess-distortion exponent. This completes a previous result which showed that R^ (D) and the optimal exponent were achievable for distortions below a certain threshold. The proof of the rate-distortion result is based on the second moment method, a popular technique to show that a non-negative random variable $X$ is strictly positive with high probability. In our context, $X$ is the number of code words within target distortion $D$ of the source sequence. We first identify the reason behind the failure of the standard second moment method for certain distortions, and illustrate the different failure modes via a stylized example. We then use a refinement of the second moment method to show that R^ (D) is achievable for all distortion values. Finally, the refinement technique is applied to Suen’s correlation inequality to prove the achievability of the optimal Gaussian excess-distortion exponent. [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]

This paper studies the problem of secure communication over the broadcast channel with receiver-side information under the lens of individual secrecy constraints, that is, the transmitter wants to send two independent messages to two receivers, which have, respectively, the desired message of the other receiver as side information, while keeping the eavesdropper ignorant of each message (i.e., the information leakage rate from each message to the eavesdropper is made vanishing). Building upon one-time pad, secrecy coding, and broadcasting schemes, achievable rate regions are investigated, and the capacity region for special cases of either a weak or strong eavesdropper (compared to both legitimate receivers) is characterized. Interestingly, the capacity region for the former corresponds to a line and the latter corresponds to a rectangle with missing corners; a phenomenon occurring due to the coupling between user’s rates. Moreover, the individual secrecy capacity region is also fully characterized for the case where the eavesdropper’s channel is deterministic. In addition to discrete memoryless setup, Gaussian scenarios are studied. For the Gaussian model, in addition to the strong and weak eavesdropper cases, the capacity region is characterized for the low and high SNR regimes when the eavesdropper’s channel is stronger than one receiver but weaker than the other. Remarkably, positive secure transmission rates are always guaranteed under the individual secrecy constraint, unlike the case of the joint secrecy constraint (i.e., the information leakage rate from both messages to the eavesdropper is made vanishing). Thus, this notion of secrecy serves as an appropriate candidate for trading off secrecy level and transmission rate, making secrecy more affordable but still acceptable to the end user. [ABSTRACT FROM PUBLISHER]

It was previously shown by Hashimoto that Forney’s optimal erasure decoder can be significantly simplified, in the sense that a simplified decoder achieves the same random coding bounds, for the ensemble of independent and identically distributed codewords. In this paper, the analysis of simplified decoders is refined and generalized in several aspects. First, tighter random coding bounds for simplified decoders are derived, which equal the exact exponential behavior of the fixed composition ensemble average. Second, the exponential bounds are valid both in the erasure mode and in the list mode. Third, the analysis pertains to a rather general class of simplified decoders, including the case of mismatch in the threshold function of the decoder. Fourth, expurgated exponents, which are larger than the random coding exponents at low rates, are shown to be achievable using a significantly simpler decoder than Forney’s optimal decoder. It is shown numerically that, from the aspect of exact random coding exponents, a decoder in the spirit of Hashimoto’s is as good as Forney’s in the erasure mode, as well as in the list mode. [ABSTRACT FROM PUBLISHER]

Guessing games for directed graphs were introduced by Riis for studying multiple unicast network coding problems. In a guessing game, the players toss generalised dice and can see some of the other outcomes depending on the structure of an underlying digraph. They later guess simultaneously the outcome of their own die. Their objective is to find a strategy, which maximizes the probability that they all guess correctly. The performance of the optimal strategy for a graph is measured by the guessing number of the digraph. Christofides and Markström studied guessing numbers of undirected graphs and defined a strategy which they conjectured to be optimal. One of the main results of this paper is a disproof of this conjecture. The main tool so far for computing guessing numbers of graphs is information theoretic inequalities. The other main result of this paper is that Shannon’s information inequalities, which work particularly well for a wide range of graph classes, are not sufficient for computing the guessing number. Finally, we pose a few more interesting questions some of which we can answer and some which we leave as open problems. [ABSTRACT FROM PUBLISHER]

The objective of low-delay codes is to protect communication streams from erasure bursts by minimizing the time between the packet erasure and its reconstruction. Previous work has concentrated on the constant-delay scenario, where all erased packets need to exhibit the same decoding delay. We consider the case of heterogeneous delay, where the objective is to minimize the average delay across the erased packets in a burst. We derive delay lower bounds for the average case, and show that they match the constant-delay bounds only at a single rate point $R=0.5$ . We then construct codes with optimal average delays for the entire range of code rates. The construction for rates $R\leq 0.5$ achieves optimality for every erasure instance, while the construction for rates $R>0.5$ is optimal for a $(1-R)/R$ fraction of all burst instances and close to optimal for the remaining fraction. The paper also studies the benefits of delay heterogeneity within the application of sensor communications. It is shown that a carefully designed code can significantly improve the temporal precision at the receiving node following erasure-burst events. [ABSTRACT FROM PUBLISHER]

The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper, we study E\gamma -resolvability, in which total variation is replaced by the more general E\gamma distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let Q\sf X|U be a random transformation, n be an integer, and E\in (0,+\infty ) . We show that in the asymptotic setting where \gamma =\exp (nE) , a (nonnegative) randomness rate above is sufficient to approximate the output distribution {\pi }_{\sf X}^{\otimes n} using the channel Q_{\sf X|U}^{\otimes n} , where Q\sf U\to Q\sf X|U\to Q\sf X , and is also necessary in the case of finite \mathcal {U} and \mathcal X . In particular, a randomness rate of \inf Q\sf UI(Q\sf U,Q\sf X|U)-E is always sufficient. We also study the convergence of the approximation error under the high-probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating E\gamma and other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth Rényi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive: 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d. setting; 2) a one-shot version of the mutual covering lemma; and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight. [ABSTRACT FROM AUTHOR]

Lattice codes used under the compute-and-forward paradigm suggest an alternative strategy for the standard Gaussian multiple-access channel (MAC): the receiver successively decodes the integer linear combinations of the messages until it can invert and recover all messages. In this paper, a multiple-access technique called compute-forward multiple access (CFMA) is proposed and analyzed. For the two-user MAC, it is shown that without time-sharing, the entire capacity region can be attained using CFMA with a single-user decoder as soon as the signal-to-noise ratios are above 1+\sqrt 2 . A partial analysis is given for more than two users. Finally, the strategy is extended to the so-called dirty MAC, where two interfering signals are known non-causally to the two transmitters in a distributed fashion. Our scheme extends the previously known results and gives new achievable rate regions. [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]

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following tradeoffs (for any \varepsilon > 0 ): 1) codes that can correct a fraction 1- \varepsilon of deletions with rate \mathop \mathrm poly\nolimits ( \varepsilon ) over an alphabet of size \mathop \mathrm poly\nolimits (1/ \varepsilon ) ; 2) binary codes of rate 1-\tilde O(\sqrt \varepsilon ) that can correct a fraction $ \varepsilon $ of deletions; and 3) Binary codes that can be list-decoded from a fraction (1/2- \varepsilon )$ of deletions with rate \mathop {\mathrm {poly}}\nolimits ( \varepsilon ) . This paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet. The above-mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors. [ABSTRACT FROM PUBLISHER]

Current deterministic algorithms for the construction of polar codes can only be argued to be practical for channels with small input alphabet sizes. In this paper, we show that any construction algorithm for channels with moderate input alphabet size, which follows the paradigm of “degrading after each polarization step,” will inherently be impractical with respect to a certain “hard” underlying channel. This result also sheds light on why the construction of low-density parity-check codes using density evolution is impractical for channels with moderate-sized input alphabets. [ABSTRACT FROM PUBLISHER]

Transmissions across asynchronous communication channels are subject to delay injection attacks, which can cause an arbitrary number of skews. That is, such attacks can cause an arbitrary number of transmitted signals to arrive after the first signal of the next transmission has arrived. The (common) assumption that despite the delays, all signals from the $i$ th transmission arrive at the decoder before any signal from the (i+2){\text{nd}} transmission arrives is called a no switch assumption. This paper presents a self-synchronizing, zero-latency, zero-error coding scheme that requires no acknowledge and can decode transmissions distorted by an arbitrary number of skews that obey this no switch assumption. The rate associated with the coding scheme provides a lower bound of 0.6942 for the (zero-error) capacity of such a channel. It is further shown that zero-error channel capacity of the channel is upper bounded by 0.7248. Finally, this paper presents bounds on the (zero-error) capacity of a channel for which the number of transmissions that can mix with one another is large. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) array codes are widely used in storage systems due to their computationally efficient encoding and decoding procedures. An MDS code with $r$ redundancy nodes can correct any $r$ node erasures by accessing (reading) all the remaining information in the surviving nodes. However, in practice, $e$ erasures are a more likely failure event, for some $1 \le e < r$ . Hence, a natural question is how much information do we need to access in order to rebuild $e$ storage nodes. We define the rebuilding ratio as the fraction of remaining information accessed during the rebuilding of $e$ erasures. In our previous work, we constructed MDS codes, called zigzag codes, that achieve the optimal rebuilding ratio of $1/r$ for the rebuilding of any systematic node when $e=1$ ; however, all the information needs to be accessed for the rebuilding of the parity node erasure. The (normalized) repair bandwidth is defined as the fraction of information transmitted from the remaining nodes during the rebuilding process. For codes that are not necessarily MDS, Dimakis et al. proposed the regenerating codes framework where any $r$ erasures can be corrected by accessing some of the remaining information, and any $e=1$ erasure can be rebuilt from some subsets of surviving nodes with optimal repair bandwidth. In this paper, we present three results on rebuilding of codes: 1) we show a fundamental outer bound on the storage size of the node and the repair bandwidth similar to the regenerating codes framework, and show that zigzag codes achieve the optimal rebuilding ratio of $e/r$ for systematic nodes of MDS codes, for any $1 \le e \le r$ ; 2) we construct systematic codes that achieve optimal rebuilding ratio of $1/r$ , for any systematic or parity node erasure; and 3) we present error correction algorithms for zigzag codes, and in particular demonstrate how these codes can be corrected beyond their minimum Hamming distances. [ABSTRACT FROM AUTHOR]

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arıkan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (second) part provides a foundation of polarization theory based on the ergodic theory of binary operations which we developed in the first part. We show that a binary operation is polarizing if and only if it is uniformity preserving and its right-inverse is strongly ergodic. The rate of polarization of single user channels is studied. It is shown that the exponent of any polarizing operation cannot exceed \frac 12 , which is the exponent of quasi-group operations. We also study the polarization of multiple access channels (MAC). In particular, we show that a sequence of binary operations is MAC-polarizing if and only if each binary operation in the sequence is polarizing. It is shown that the exponent of any MAC-polarizing sequence cannot exceed \frac 12 , which is the exponent of sequences of quasi-group operations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Gaussian diamond-wiretap channel that consists of an orthogonal broadcast channel from a source to two relays and a Gaussian fast-fading multiple access-wiretap channel from the two relays to a legitimate destination and an eavesdropper. For the multiple access part, we consider both the case with full channel state information (CSI) and the case with no eavesdropper’s CSI, at the relays and the legitimate destination. For both the cases, we establish the exact secure degrees of freedom and generalize the results for multiple relays. Our results show: 1) how to strike a balance between sending message symbols and common noise symbols from the source to the relays in the broadcast component and 2) how to combine artificial noise-beamforming and noise-alignment techniques at the relays in the multiple access component. In the case with full CSI, we propose a scheme where the relays simultaneously beamform common noise signals in the null space of the legitimate destination’s channel, and align them with the message signals at the eavesdropper. In the case with no eavesdropper’s CSI, we present a scheme that efficiently utilizes the broadcast links by incorporating computation between the message and common noise symbols at the source. Finally, most of our achievability and converse techniques can also be adapted to the Gaussian (non-fading) channel model. [ABSTRACT FROM PUBLISHER]

In this paper, we develop achievability schemes for symmetric $K$ -user interference channels with a rate-limited feedback from each receiver to the corresponding transmitter. We study this problem under two different channel models: the linear deterministic model, and the Gaussian model. For the deterministic model, the proposed scheme achieves a symmetric rate that is the minimum of the symmetric capacity with infinite feedback, and the sum of the symmetric capacity without feedback and the symmetric amount of feedback. For the Gaussian interference channel, we use lattice codes to propose a transmission strategy that incorporates the techniques of Han–Kobayashi message splitting, interference decoding, and decode and forward. This strategy achieves a symmetric rate, which is within a constant number of bits to the minimum of the symmetric capacity with infinite feedback, and the sum of the symmetric capacity without feedback and the amount of symmetric feedback. This constant is obtained as a function of the number of users, $K$ . We note that for the special case of Gaussian IC with $K = 2$ , our proposed achievability scheme results in a symmetric rate that is within at most 21.085 bits/s/Hz of the outer bound, which is the first constant gap bound despite the constant gap claim in [1]. The symmetric achievable rate is used to characterize the achievable generalized degrees of freedom, which exhibits a gradual increase from no feedback to perfect feedback in the presence of feedback links with limited capacity. [ABSTRACT FROM PUBLISHER]

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion— $q$ -ary sequences with fixed Manhattan weight—is briefly discussed, and an application in coding for molecular timing channels is illustrated. [ABSTRACT FROM AUTHOR]

Coded computation techniques provide robustness against straggling workers in distributed computing. However, most of the existing schemes require exact provisioning of the straggling behavior and ignore the computations carried out by straggling workers. Moreover, these schemes are typically designed to recover the desired computation results accurately, while in many machine learning and iterative optimization algorithms, faster approximate solutions are known to result in an improvement in the overall convergence time. In this paper, we first introduce a novel coded matrix-vector multiplication scheme, called coded computation with partial recovery (CCPR), which benefits from the advantages of both coded and uncoded computation schemes, and reduces both the computation time and the decoding complexity by allowing a trade-off between the accuracy and the speed of computation. We then extend this approach to distributed implementation of more general computation tasks by proposing a coded communication scheme with partial recovery, where the results of subtasks computed by the workers are coded before being communicated. Numerical simulations on a large linear regression task confirm the benefits of the proposed scheme in terms of the trade-off between the computation accuracy and latency. [ABSTRACT FROM AUTHOR]

Just as rank-metric or Gabidulin codes may be used to construct rate–diversity tradeoff optimal space–time codes, a recently introduced generalization for the sum-rank metric—linearized Reed–Solomon codes—accomplishes the same in the case of multiple fading blocks. In this paper, we provide the first explicit construction of minimal delay rate–diversity optimal multiblock space–time codes as an application of linearized Reed–Solomon codes. We also provide sequential decoders for these codes and, more generally, space–time codes constructed from finite field codes. Simulation results show that the proposed codes can outperform full diversity codes based on cyclic division algebras at low SNRs as well as utilize significantly smaller constellations. [ABSTRACT FROM AUTHOR]

A new family of codes, called clustering-correcting codes, is presented in this paper. This family of codes is motivated by the special structure of the data that is stored in DNA-based storage systems. The data stored in these systems has the form of unordered sequences, also called strands, and every strand is synthesized thousands to millions of times, where some of these copies are read back during sequencing. Due to the unordered structure of the strands, an important task in the decoding process is to place them in their correct order. This is usually accomplished by allocating part of the strand for an index. However, in the presence of errors in the index field, important information on the order of the strands may be lost. Clustering-correcting codes ensure that if the distance between the index fields of two strands is small, their data fields have large distance. It is shown how this property enables to place the strands together in their correct clusters even in the presence of errors. We present lower and upper bounds on the size of clustering-correcting codes and an explicit construction of these codes which uses only a single symbol of redundancy. The results are first presented for the Hamming metric and are then extended for the edit distance. [ABSTRACT FROM AUTHOR]

A functional PIR array code is a coding scheme which encodes some $s$ information bits into a $t\times m$ array such that every linear combination of the $s$ information bits has $k$ mutually disjoint recovering sets. Every recovering set consists of some of the array’s columns while it is allowed to read at most $\ell $ encoded bits from every column in order to receive the requested linear combination of the information bits. Functional batch array codes impose a stronger property where every multiset request of $k$ linear combinations has $k$ mutually disjoint recovering sets. Locality functional array codes demand that the size of every recovering set is restrained to be at most $r$. Given the values of $s, k, t, \ell,r$ , the goal of this paper is to study the optimal value of the number of columns $m$ such that these codes exist. Several lower bounds are presented as well as explicit constructions for several of these parameters. [ABSTRACT FROM AUTHOR]

The majority of works in distributed storage networks assume a simple network model with a collection of identical storage nodes with the same communication cost between the nodes. In this paper, we consider a realistic multi-rack distributed data storage network and present a code design framework for this model. Considering the cheaper data transmission within the racks, our code construction method is able to locally repair the nodes failure within the same rack by using only the survived nodes in the same rack. However, in the case of severe failure patterns when the information content of the survived nodes is not sufficient to repair the failures, other racks will participate in the repair process. By employing the criteria of our multi-rack storage code, we establish a linear programming bound on the size of the code in order to maximize the code rate. [ABSTRACT FROM AUTHOR]

The Gauss–Markov source produces $U_{i} = aU_{i-1} + Z_{i}$ for $i\geq 1$ , where $U_{0} = 0$ , $|a|< 1$ and $Z_{i}\sim \mathcal {N}(0, \sigma ^{2})$ are i.i.d. Gaussian random variables. We consider lossy compression of a block of $n$ samples of the Gauss–Markov source under squared error distortion. We obtain the Gaussian approximation for the Gauss–Markov source with excess-distortion criterion for any distortion $d>0$ , and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function $R(n,d,\epsilon)$ approaches the rate-distortion function $\mathbb {R}(d)$ as $R(n,d,\epsilon) = \mathbb {R}(d) + \sqrt {\frac {V(d)}{n}}Q^{-1}(\epsilon) + o\left ({\frac {1}{\sqrt {n}}}\right)$ , where $V(d)$ is the dispersion, $\epsilon \in (0,1)$ is the excess-distortion probability, and $Q^{-1}$ is the inverse $Q$ -function. We give a reverse waterfilling integral representation for the dispersion $V(d)$ , which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all $0 < d\leq \frac {\sigma ^{2}}{(1+|a|)^{2}}$ , $R(n,d,\epsilon)$ of the Gauss–Markov source coincides with that of $Z_{i}$ , the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of $n$ samples of the Gauss–Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter $a$ based on $n$ observations. [ABSTRACT FROM AUTHOR]

In this paper, new two-user multiple access wiretap channel models are studied. First, the multiple access wiretap channel II with a discrete memoryless main channel under different wiretapping scenarios is introduced. The wiretapper, as in the classical wiretap channel II model, chooses a fixed-size subset of the channel uses, in which it obtains noise-free observations of one of the codewords: a deterministic function, e.g., superposition, of the two codewords or each of the two codewords. A fourth wiretapping scenario is considered, in which the wiretapper, in each position it chooses, decides to observe either one of the codewords or both codewords, with an overall budget on the number of its noiselessly observed symbols. These, thus, extend the recently examined wiretap channel II with a noisy main channel to a multiple access setting with a variety of attack models for the wiretapper. Next, the proposed multiple access wiretap channel II models are further generalized to the case when the wiretapper observes the outputs of a discrete memoryless channel, instead of erasures, outside the subset of noiseless observations. Achievable strong secrecy rate regions for all the proposed models are derived. Achievability is established by solving dual multi-terminal secret key agreement problems in the source model and converting the solution to the original channel models using probability distribution approximation arguments. The derived achievable rate regions quantify the secrecy cost due to the additional capabilities of the wiretapper with respect to the previous multiple access wiretap models. [ABSTRACT FROM AUTHOR]

Biometric authentication systems, based on secret key generation, work as follows. In the enrollment stage, an individual provides a biometric signal that is mapped into a secret key and a helper message, the former being prepared to become available to the system at a later time (for authentication), and the latter is stored in a public database. When an authorized user requests authentication, claiming his/her identity to be one of those of the subscribers, he/she has to provide a biometric signal again, and then the system, which retrieves also the helper message of the claimed subscriber, produces an estimate of the secret key that is finally compared to the secret key of the claimed user. In the case of a match, the authentication request is approved; otherwise, it is rejected. Evidently, there is an inherent tension between two desired, but conflicting, properties of the helper message encoder; on the one hand, the encoding should be informative enough concerning the identity of the real subscriber, in order to approve him/her in the authentication stage, but on the other hand, it should not be too informative, as otherwise, unauthorized imposters could easily fool the system and gain access. A good encoder should then trade off the two kinds of errors: the false reject (FR) error and the false accept (FA) error. In this paper, we investigate trade-offs between the random coding FR error exponent and the best achievable FA error exponent. We compare two types of ensembles of codes: fixed-rate codes and variable-rate codes, and we show that the latter class of codes offers considerable improvement compared to the former. In doing this, we characterize ensemble-optimal rate functions for both types of codes. We also examine the effect of privacy leakage constraints for both fixed-rate codes and variable-rate codes [ABSTRACT FROM AUTHOR]

Erasure coding is widely used for massive storage in data centers to achieve high fault tolerance and low storage redundancy. Since the cross-rack communication cost is often high, it is critical to design erasure codes that minimize the cross-rack repair bandwidth during failure repair. In this paper, we analyze the optimal trade-off between storage redundancy and cross-rack repair bandwidth specifically for data centers, subject to the condition that the original data can be reconstructed from a sufficient number of any non-failed nodes. We characterize the optimal trade-off curve under functional repair, and propose a general family of erasure codes called rack-aware regenerating codes (RRC), which achieve the optimal trade-off. We further propose exact repair constructions of RRC that have minimum storage redundancy and minimum cross-rack repair bandwidth, respectively. We show that (i) the minimum storage redundancy constructions support a wide range of parameters and have cross-rack repair bandwidth that is strictly less than that of the classical minimum storage regenerating codes in most cases, and (ii) the minimum cross-rack repair bandwidth constructions support all the parameters and have less cross-rack repair bandwidth than that of the minimum bandwidth regenerating codes for almost all of the parameters. [ABSTRACT FROM AUTHOR]

Permutation codes have recently garnered substantial research interest due to their potential in various applications, including cloud storage systems, genome resequencing, and flash memories. In this paper, we study the theoretical bounds and constructions of permutation codes in the generalized Cayley metric. The generalized Cayley metric captures the number of generalized transposition errors in a permutation and subsumes previously studied error types, including transpositions and translocations, without imposing restrictions on the lengths and positions of the translocated segments. Based on the so-called breakpoint analysis method proposed by Chee and Vu, we first present a coding framework that leads to order-optimal constructions, thus improving upon the existing constructions that are not order-optimal. We then use this framework to also develop an order-optimal coding scheme that is additionally explicit and systematic. [ABSTRACT FROM AUTHOR]

In large-scale distributed computing clusters, such as Amazon EC2, there are several types of “system noise” that can result in major degradation of performance: system failures, bottlenecks due to limited communication bandwidth, latency due to straggler nodes, and so on. There have been recent results that demonstrate the impact of coding for efficient utilization of computation and storage redundancy to alleviate the effect of stragglers and communication bottlenecks in homogeneous clusters. In this paper, we focus on general heterogeneous distributed computing clusters consist of a variety of computing machines with different capabilities. We propose a coding framework for speeding up distributed computing in heterogeneous clusters by trading redundancy for reducing the latency of computation. In particular, we propose heterogeneous coded matrix multiplication (HCMM) algorithm for performing distributed matrix multiplication over heterogeneous clusters that are provably asymptotically optimal for a broad class of processing time distributions. Moreover, we show that HCMM is unboundedly faster than any uncoded scheme that partitions the total workload among the workers. To demonstrate how the proposed HCMM scheme can be applied in practice, we provide results from numerical studies and Amazon EC2 experiments comparing HCMM with three benchmark load allocation schemes—uniform uncoded, load-balanced uncoded, and uniform coded. In particular, in our numerical studies, HCMM achieves speedups of up to 73%, 56%, and 42%, respectively, over the three benchmark schemes mentioned earlier. Furthermore, we carry out experiments over Amazon EC2 clusters and demonstrate how HCMM can be combined with rateless codes with nearly linear decoding complexity. In particular, we show that HCMM combined with the Luby transform codes can significantly reduce the overall execution time. HCMM is found to be up to 61%, 46%, and 36% faster than the aforementioned three benchmark schemes, respectively. Additionally, we provide a generalization to the problem of optimal load allocation in heterogeneous settings, where we take into account the monetary costs associated with distributed computing clusters. We argue that HCMM is asymptotically optimal for budget-constrained scenarios as well. In particular, we characterize the minimum possible expected cost associated with a computation task over a given cluster of machines. Furthermore, we develop a heuristic algorithm for (HCMM) load allocation for the distributed implementation of budget-limited computation tasks. [ABSTRACT FROM AUTHOR]

The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the $(0,k)$ -runlength limited (RLL) constraint, i.e., no more than $k$ consecutive ‘0’s are allowed. The feedback capacity of this channel is derived for all $k\geq 1$ , and is given by $C^{\mathrm {fb}}_{(0,k)}(\varepsilon) = \max \frac {\overline {\varepsilon }H_{2}(\delta _{0})+\sum _{i=1}^{k-1}\left ({\overline {\varepsilon }^{i+1}H_{2}(\delta _{i})\prod _{m=0}^{i-1}\delta _{m}}\right)}{1+\sum _{i=0}^{k-1}\left ({\overline {\varepsilon }^{i+1} \prod _{m=0}^{i}\delta _{m}}\right)}$ , where $\varepsilon $ is the erasure probability, $\overline {\varepsilon }=1-\varepsilon $ and $H_{2}(\cdot)$ is the binary entropy function. The maximization is only over $\delta _{k-1}$ , while the parameters $\delta _{i}$ for $i\leq k-2$ are straightforward functions of $\delta _{k-1}$. The lower bound is obtained by constructing a simple coding for all $k\geq 1$. It is shown that the feedback capacity can be achieved using zero-error, variable length coding. For the converse, an upper bound on the non-causal setting, where the erasure is available to the encoder just prior to the transmission, is derived. This upper bound coincides with the lower bound and concludes the search for both the feedback capacity and the non-causal capacity. As a result, non-causal knowledge of the erasures at the encoder does not increase the feedback capacity for the $(0,k)$ -RLL input-constrained BEC. This property does not hold in general: the $(2,\infty)$ -RLL input-constrained BEC, where every ‘1’ is followed by at least two ‘0’s, is used to show that the feedback capacity can be strictly smaller than the non-causal capacity. [ABSTRACT FROM AUTHOR]

Grassmannian codes are known to be useful in error correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this paper, we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for the network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts, it is proved that the codes in the Hamming scheme form a subfamily of the Grassmannian codes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a binary erasure channel (BEC) with an unknown erasure probability of ${\delta }$ at a transmitter. In addition, we consider that ${\delta }$ has a constant value in each transmission block and the transmitter knows the probability distribution of ${\delta }$. For this problem with the infinite block length, based on the distribution of the random variable ${\delta }$ , a multi-layer encoding strategy at the transmitter and a successive decoding strategy at the receiver are proposed. In our proposed scheme, based on the value of ${\delta }$ , the receiver can decode a part of transmitted data from different transmission layers. In order to have a comparison with other erasure correction codes, we generalize the result of the infinite block length to the finite block-length transmission. We show that considering the average transmission rate, our scheme has a better performance than the other erasure correction codes such as the Raptor and the MDS codes. We also analyze both the decoding complexity and bit error rate versus the erasure probability of the channel. Besides, by using a numerical comparison, we try to show that our proposed strategy outperforms other transmission strategies. [ABSTRACT FROM AUTHOR]

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criterion for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring the subcodes of Gabidulin codes, if each of the rightmost $\delta -1$ columns in the Ferrers diagram $\cal F$ has at least $n-r$ dots, where $r$ is taken in a range, then the conditions that an FDRM code in $\cal F$ is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams. [ABSTRACT FROM AUTHOR]

Mutually uncorrelated (MU) codes are a class of codes in which no proper prefix of one codeword is a suffix of another codeword. These codes were originally studied for synchronization purposes and recently, Yazdi et al. showed their applicability to enable random access in DNA storage. In this paper,we follow the research of Yazdi et al. and study MU codes along with their extensions to correct errors and balanced codes. We first review a well-known construction of MU codes and study the asymptotic behavior of its cardinality. This task is accomplished by studying a special class of run-length limited codes that impose the longest run of zeros to be at most some function of the codewords length. We also present an efficient algorithm for this class of constrained codes and show how to use this analysis for MU codes. Next, we extend the results on the run-length limited codes in order to study $(d_{h},d_{m})$ -MU codes that impose a minimum Hamming distance of $d_{h}$ between different codewords and $d_{m}$ between prefixes and suffixes. In particular, we show an efficient construction of these codes with nearly optimal redundancy. We also provide similar results for the edit distance and balanced MU codes. Last, we draw connections to the problems of comma-free and prefix synchronized codes. [ABSTRACT FROM AUTHOR]

Regenerating codes are efficient methods for distributed storage in storage networks, where node failures are common. They guarantee low cost data reconstruction and repair through accessing only a predefined number of arbitrarily chosen storage nodes in the network. In this paper, we consider two simultaneous extensions to the original regenerating codes framework introduced by Dimakis et al.; 1) both data reconstruction and repair are resilient to the presence of a certain number of erroneous nodes in the network and 2) the number of helper nodes in every repair is not fixed, but is a flexible parameter that can be selected during the run-time. We study the fundamental limits of required total repair bandwidth and provide an upper bound for the storage capacity of these codes under these assumptions. We then focus on the minimum repair bandwidth (MBR) case and derive the exact storage capacity by presenting explicit coding schemes with exact repair, which achieve the upper bound of the storage capacity in the considered setup. To this end, we first provide a more natural extension of the well-known product matrix (PM) MBR codes, modified to provide flexibility in choosing the number of helpers in each repair, and simultaneously be robust to erroneous nodes in the network. This is achieved by proving the non-singularity for a family of matrices in large enough finite fields. We next provide another extension of the PM codes, based on a novel repair scheme which enables flexibility in the number of helpers and robustness against erroneous nodes without any extra cost in field size compared with the original PM codes. [ABSTRACT FROM AUTHOR]

In distributed storage, erasure codes (like Reed–Solomon Codes) are often employed to provide reliability. In this setting, it is desirable to be able to repair one or more failed nodes while minimizing the repair bandwidth. In this paper, motivated by Reed-Solomon codes, we study the problem of repairing multiple failed nodes in a scalar MDS code. We extend the framework of (Guruswami and Wootters, 2017) to give a framework for constructing repair schemes for multiple failures in general scalar MDS codes in the centralized repair model. We then specialize our framework to Reed–Solomon codes, and also extend and improve upon recent results of (Dau et al., 2017). [ABSTRACT FROM AUTHOR]

The two-user multiple-input multiple-output Gaussian broadcast channel with common, private, and confidential messages is considered. The transmitter sends a common message to both users, a confidential message to the User 1 and a private (non-confidential) message to the User 2. The secrecy-capacity region is characterized by showing that certain inner and outer bounds coincide and that the boundary points are achieved by Gaussian inputs, which enables the development of a tight converse. The proof relies on the factorization of upper concave envelopes and a variant of dirty-paper coding (DPC). It is shown that the entire region is exhausted by using DPC to cancel out the signal of the non-confidential message at Receiver 1, thus making DPC against the signal of the confidential message unnecessary. A numerical example illustrates the secrecy-capacity results. [ABSTRACT FROM AUTHOR]

It is a well-known fact that feedback does not increase the capacity of point-to-point memoryless channels, however, its effect in secure communications is not fully understood yet. In this paper, an achievable scheme for the wiretap channel with generalized feedback is presented. This scheme, which uses the feedback signal to generate a shared secret key between the legitimate users, encrypts the message to be sent at the bit level. New capacity results for a class of channels are provided, as well as some new insights into the secret key agreement problem. Moreover, this scheme recovers previously reported rate regions from the literature, and thus it can be seen as a generalization that unifies several results in the field. [ABSTRACT FROM AUTHOR]

Gaussian thermal loss channels are of particular importance to quantum communication theory since they model realistic optical communication channels. Except for special cases, the quantum capacity of Gaussian thermal loss channels is not yet quantified completely. In this paper, we provide improved upper bounds of the Gaussian thermal loss channel capacity, both in energy-constrained and unconstrained scenarios. We briefly review Gottesman-Kitaev-Preskill (GKP) codes and discuss their experimental implementation. We then prove, in the energy-unconstrained case, that a family of GKP codes achieves the quantum capacity of Gaussian thermal loss channels up to at most a constant gap from the improved upper bound. In the energy-constrained case, we formulate a biconvex encoding and decoding optimization problem to maximize entanglement fidelity. Then, we solve the biconvex optimization heuristically by an alternating semi-definite programming method and report that, starting from Haar random initial codes, our numerical optimization yields a hexagonal GKP code as an optimal encoding in a practically relevant regime. [ABSTRACT FROM AUTHOR]

Every channel can be expressed as a convex combination of deterministic channels with each deterministic channel corresponding to one particular intrinsic state. Such convex combinations are, in general, not unique, each giving rise to a specific intrinsic-state distribution. In this paper, we study the maximum and minimum capacities of a channel when the realization of its intrinsic state is causally available at the encoder and/or the decoder. Several conclusive results are obtained for binary-input channels and binary-output channels. By-products of our investigation include a generalization of the Birkhoff–von Neumann theorem and a condition on the uselessness of causal state information at the encoder. [ABSTRACT FROM AUTHOR]

Communication in integrated circuits faces two major impediments: inter-wire capacitive coupling and noise. Coding can be used to address both these problems. So-called crosstalk-avoidance codes mitigate capacitive coupling, and traditional error-correction codes introduce resilience against channel errors. Unfortunately, crosstalk-avoidance and error-correction codes cannot be combined in a straightforward manner. On the one hand, crosstalk-avoidance encoding followed by error-correction encoding destroys the crosstalk-avoidance property. On the other hand, error-correction encoding followed by crosstalk-avoidance encoding causes the crosstalk-avoidance decoder to fail in the presence of errors. Existing approaches circumvent this difficulty by using additional bus wires to protect the parities generated from the output of the error-correction encoder, and are therefore inefficient. In this paper, we propose a novel joint crosstalk-avoidance and error-correction coding and decoding scheme that provides higher bus transmission rates compared with existing approaches. Our joint approach carefully embeds the parities such that the crosstalk-avoidance property is preserved. We analyze the rate and minimum distance of the proposed scheme. We also provide a density evolution analysis and predict iterative decoding thresholds for reliable communication under random bus erasures. This density evolution analysis is nonstandard, since the crosstalk-avoidance constraints are inherently nonlinear. [ABSTRACT FROM AUTHOR]

We propose two classes of universal codes that are suited to two asymptotic regimes when the output alphabet is possibly continuous. The first class has the property that the error probability decays exponentially fast, and we identify an explicit lower bound on the error exponent. The other class attains the epsilon-capacity of the channel, and we also identify the second-order term in the asymptotic expansion. The proposed encoder is essentially based on the packing lemma of the method of types. For the decoder, we first derive a Rényi-relative-entropy version of Clarke and Barron’s formula the distance between the true distribution and the Bayesian mixture, which is of independent interest. The universal decoder is stated in terms of this formula and quantities used in the information spectrum method. The methods contained herein allow us to analyze universal codes for channels with continuous and discrete output alphabets in a unified manner and to analyze their performances in terms of the exponential decay of the error probability and the second-order coding rate. [ABSTRACT FROM AUTHOR]

An important problem in secret sharing schemes is minimizing the share size. For ($k$ , $n$)-threshold schemes and ($k$ , $L$ , $n$)-ramp schemes, constructions that minimize the share size are known. This paper presents optimal constructions for a more general class of access structures in which subsets with the same cardinality have the same amount of information about the secret. We refer to schemes with such uniform access structures as uniform secret sharing. We first derive a tight lower bound for share entropy and then present an optimal construction. Our lower bound exceeds that previously reported. The optimal construction encodes the secret value using one or more ramp schemes. [ABSTRACT FROM AUTHOR]

Synthesis of DNA molecules offers unprecedented advances in storage technology. Yet, the microscopic world in which these molecules reside induces error patterns that are fundamentally different from their digital counterparts. Hence, to maintain reliability in reading and writing, new coding schemes must be developed. In a reading technique called shotgun sequencing, a long DNA string is read in a sliding window fashion, and a profile vector is produced. It was recently suggested by Kiah et al. that such a vector can represent the permutation which is induced by its entries, and hence a rank-modulation scheme arises. Although this interpretation suggests high error tolerance, it is unclear which permutations are feasible and how to produce a DNA string whose profile vector induces a given permutation. In this paper, by observing some necessary conditions, an upper bound for the number of feasible permutations is given. Furthermore, a technique for deciding the feasibility of a permutation is devised. By using insights from this technique, an algorithm for producing a considerable number of feasible permutations is given, which applies to any alphabet size and any window length. [ABSTRACT FROM AUTHOR]

We investigate the maximum coding rate for a given average blocklength and error probability over a $K$ -user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiyet al.(2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e.,zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message $K$ -user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaardet al.(2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case. [ABSTRACT FROM AUTHOR]

We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a “solitonic behavior.” Namely, a uniquely shaped wavelike solution develops, which propagates with a constant velocity. Under this assumption, we derive an analytical formula for the velocity in the framework of a continuum limit of the spatially coupled system. The general formalism is developed for spatially coupled low-density parity-check codes on general binary memoryless symmetric channels, which form the main systems of interest in this paper. We apply the formula for special channels and illustrate that it matches the direct numerical evaluation of the velocity for a wide range of noise values. A possible application of the velocity formula to the evaluation of finite size scaling law parameters is also discussed. We conduct a similar analysis for general scalar systems and illustrate the findings with applications to compressive sensing and generalized low-density parity-check codes on the binary erasure or binary symmetric channels. [ABSTRACT FROM AUTHOR]

In this paper, we consider two extensions of the Gács–Körner common information to three variables, the conditional common information (cCI) and the coarse-grained conditional common information (ccCI). Both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations. In particular, the ccCI is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical source $p_{XYZ}$ when Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness with the eavesdropper Eve ($Z$). Moving to the quantum setting, we consider two different ways of generating a tripartite quantum state from classical correlations $p_{XYZ}$ : 1) coherent encodings $\sum _{xyz}\sqrt {p_{xyz}} |xyz\rangle $ and 2) incoherent encodings $\sum _{xyz}p_{xyz} |xyz\rangle \langle xyz|$. We study how well can Alice and Bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sources $p_{XYZ}$ using classical operations. While the power of quantum mechanics increases Alice and Bob’s ability to generate shared randomness, it also equips Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one. We first demonstrate that the classical key rate of $p_{XYZ}$ is equivalent to the quantum key rate for an incoherent quantum encoding of the distribution. For coherent encodings, we next show that the classical and quantum rates are generally incomparable, and in fact, their difference can be arbitrarily large in either direction. Finally, we introduce a “zoo” of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions. Remarkably, for these states almost all entanglement measures, such as Alice and Bob’s entanglement cost, squashed entanglement, and relative entropy of entanglement, can be sharply bounded or even exactly expressed in terms of the conditional common information. In the latter case, we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated. [ABSTRACT FROM AUTHOR]

In frequency hopping (FH) communication systems, frequency hopping sequences (FHSs) are crucial in determining the system’s anti-jamming performance. If FHSs can ensure a wide-gap between two adjacent frequency points to avoid the frequency points with high interference probability, it will significantly improve the FH communication system’s anti-interference ability. Moreover, if each frequency point appears at the same number of times in a sequence period, the system’s anti-electromagnetic interference will be enhanced. Therefore, it is desirable to employ FHSs with low Hamming autocorrelation, wide frequency-hopping gap, and good uniformity in practical applications. However, to the best of our knowledge, no such infinite classes of FHSs have been reported in the literature to date. This paper aims to present two constructions of uniform wide-gap frequency-hopping sequences (WGFHSs) by concatenating two or three adequately designed sequences. For the first time, we obtain two infinite classes of WGFHSs, which are optimal with respect to the well-known Lempel-Greenberger bound. [ABSTRACT FROM AUTHOR]

In this paper, we study an information-theoretic secret sharing problem, where a dealer distributes shares of a secret among a set of participants under the following constraints: (i) authorized sets of users can recover the secret by pooling their shares, and (ii) non-authorized sets of colluding users cannot learn any information about the secret. We assume that the dealer and participants observe the realizations of correlated Gaussian random variables and that the dealer can communicate with participants through a one-way, authenticated, rate-limited, and public channel. Unlike traditional secret sharing protocols, in our setting, no perfectly secure channel is needed between the dealer and the participants. Our main result is a closed-form characterization of the fundamental trade-off between secret rate and public communication rate. [ABSTRACT FROM AUTHOR]