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We consider a cooperative two-user multiaccess channel in which the transmission is controlled by a random state. Both encoders transmit a common message and, one of the encoders also transmits an individual message. We study the capacity region of this communication model for different degrees of availability of the states at the encoders, causally or strictly causally. In the case in which the states are revealed causally to both encoders but not to the decoder we find an explicit characterization of the capacity region in the discrete memoryless case. In the case in which the states are revealed only strictly causally to both encoders, we establish inner and outer bounds on the capacity region. The outer bound is nontrivial, and has a relatively simple form. It has the advantage of incorporating only one auxiliary random variable. In particular, it suggests that there is none, or at best only little, to gain from having the encoder that transmits both messages also sending an individual description of the state to the receiver, in addition to the compressed version that is sent cooperatively with the other encoder. We then introduce a class of cooperative multiaccess channels with states known strictly causally at both encoders for which the inner and outer bounds agree, and so we characterize the capacity region for this class. In this class of channels, the state can be obtained as a deterministic function of the channel inputs and output. We also study the model in which the states are revealed, strictly causally, in an asymmetric manner, to only one encoder. Throughout this paper, we discuss a number of examples, and compute the capacity region of some of these examples. The results shed more light on the utility of delayed channel state information for increasing the capacity region of state-dependent cooperative multiaccess channels, and tie with recent progress in this framework. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the minimum energy of transmitting correlated sources over the Gaussian multiple-access channel. Compared to other works on joint source-channel coding, we consider the fundamental problem of the minimum transmission energy, where the source and channel bandwidths are not naturally matched. Different models of correlated sources are studied. We first treat lossy transmission of Gaussian sources, including multiterminal sources and CEO sources. We then consider lossless transmission of correlated binary sources. In all cases, we lower bound the minimum energy using a cut-set argument that couples transmission energy and the distortions for the Gaussian cases (or source entropy for the discrete case). For the achievable schemes, separate source and channel coding and uncoded transmission are studied as benchmarks. In addition, we show that hybrid digital/analog transmission achieves the best known energy efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the structure of infeasibility results in network information theory, based on a notion of information state. We also discuss ideas for generalizing a known outer bound for lossless transmission of independent sources over a network to one of lossy transmission of dependent sources over the same network. To concretely demonstrate this, we apply our ideas and prove new results for lossy transmission of dependent sources by generalizing: 1) the cut-set bound; 2) the best known outer bound on the capacity region of a general broadcast channel; and 3) the outer bound part of the result of Maric, Yates, and Kramer on strong interference channels with a common message. [ABSTRACT FROM AUTHOR]

In this paper, we consider the finite-state multiple access channel (MAC) with partially cooperative encoders and delayed channel state information (CSI). Here, partial cooperation refers to the communication between the encoders via finite-capacity links. The channel states are assumed to be governed by a Markov process. Full CSI is assumed at the receiver, while at the transmitters, only delayed CSI is available. The capacity region of this channel model is derived by first solving the case of the finite-state MAC with a common message. Achievability for the latter case is established using the notion of strategies, however, we show that optimal codes can be constructed directly over the input alphabet. This results in a single codebook construction that is then leveraged to apply simultaneous joint decoding. Simultaneous decoding is crucial here because it circumvents the need to rely on the capacity region’s corner points, a task that becomes increasingly cumbersome with the growth in the number of messages to be sent. The common message result is then used to derive the capacity region for the case with partially cooperating encoders. Next, we apply this general result to the special case of the Gaussian vector MAC with diagonal channel transfer matrices, which is suitable for modeling, e.g., orthogonal frequency division multiplexing-based communication systems. The capacity region of the Gaussian channel is presented in terms of a convex optimization problem that can be solved efficiently using numerical tools. The region is derived by first presenting an outer bound on the general capacity region and then suggesting a specific input distribution that achieves this bound. Finally, numerical results are provided that give valuable insight into the practical implications of optimally using conferencing to maximize the transmission rates. [ABSTRACT FROM AUTHOR]

This paper develops a fundamental theory of realizations of linear and group codes on general graphs using elementary group theory, including basic group duality theory. Principal new and extended results include: normal realization duality; analysis of systems-theoretic properties of fragments of realizations and their connections; minimal \(\Leftrightarrow \) trim and proper theorem for cycle-free codes; results showing that all constraint codes except interface nodes may be assumed to be trim and proper, and that the interesting part of a cyclic realization is its 2-core; notions of observability and controllability for fragments, and related tests; and relations between state-trimness and controllability, and dual state-trimness and observability. [ABSTRACT FROM AUTHOR]

Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over certain finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-ring matrix channel \(Y = AX + BE\) , where \(X\) is the channel input, \(Y\) is the channel output, \(E\) is random error, and \(A\) and \(B\) are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under the assumption that \(A\) is uniform over all full-rank matrices and \(BE\) is uniform over all rank- \(t\) matrices, extending the work of Silva, Kschischang, and Kötter (2010), who handled the case of finite fields. This extension is based on several new results, which may be of independent interest, that generalize concepts and methods from matrices over finite fields to matrices over finite chain rings. [ABSTRACT FROM AUTHOR]

We propose a new coding scheme using only one lattice that achieves the \({1}/{2}~{\rm log(1+ \mathop {\mathrm {\mathsf {SNR}\kern 0pt}}\nolimits )}\) capacity of the additive white Gaussian noise (AWGN) channel with lattice decoding, which is provable for signal-to-noise ratio \( \mathop {\mathrm {\mathsf {SNR}\kern 0pt}}\nolimits >e\) at present. The scheme applies a discrete Gaussian distribution over an AWGN-good lattice, but otherwise does not require a shaping lattice or dither. Thus, it significantly simplifies the default lattice coding scheme of Erez and Zamir which involves a quantization-good lattice as well as an AWGN-good lattice. Using the flatness factor, we show that the error probability of the proposed scheme under minimum mean-square error lattice decoding is almost the same as that of Erez and Zamir, for any rate up to the AWGN channel capacity. We introduce the notion of good constellations, which carry almost the same mutual information as that of continuous Gaussian inputs. We also address the implementation of Gaussian shaping for the proposed lattice Gaussian coding scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study feedback optimization problems that maximize the users' signal to interference plus noise ratio (SINR) in a two-cell multiple-input multiple-output broadcast channel. Assuming the users learn their direct and interfering channels perfectly, they can feed back this information to the base stations (BSs) over the uplink channels. The BSs then use the channel information to design their transmission scheme. Two types of feedback are considered: 1) analog and 2) digital. In the analog feedback case, the users send their unquantized and uncoded channel state information (CSI) over the uplink channels. In this context, given a user's fixed transmit power, we investigate how he/she should optimally allocate it to feed back the direct and interfering (or cross) CSI for two types of BS cooperation schemes, namely, multicell processing (MCP) and coordinated beamforming. In the digital feedback case, the direct and cross link channel vectors of each user are quantized separately, each using the random vector quantization scheme, with different size codebooks. The users then send the index of the quantization vector in the corresponding codebook to the BSs. Similar to the feedback optimization problem for the analog feedback, we investigate the optimal bit partitioning for the direct and interfering link for both types of cooperation. We focus on regularized channel inversion precoding structures and perform our analysis in the large system limit in which the number of users per cell $(K)$ and the number of antennas per BS (N)$ tend to infinity with their ratio \beta=({K}/{N}) held fixed. We show that for both types of cooperation, for some values of interfering channel gain, usually at low values, no cooperation between the BSs is preferred. This is because, for these values of cross channel gain, the channel estimates for the cross link are not accurate enough for their knowledge to contribute to improving the SINR and there is no benefit in doing BS cooperation under that condition. We also show that for the MCP scheme, unlike in the perfect CSI case, the SINR improves only when the interfering channel gain is above a certain threshold. [ABSTRACT FROM PUBLISHER]

Binary codes over 2D arrays are very useful in data storage, where each array column represents a storage device or unit that may suffer failure. In this paper, we propose a new framework for probabilistic construction of codes on 2D arrays. Instead of a pure combinatorial erasure model used in traditional array codes, we propose a mixed combinatorial-probabilistic model of limiting the number of column failures, and assuming a binary erasure channel in each failing column. For this model, we give code constructions and detailed analysis that allow sustaining a large number of column failures with graceful degradation in the fraction of erasures correctable in failing columns. Another advantage of the new framework is that it uses low-complexity iterative decoding. The key component in the analysis of the new codes is to analyze the decoding graphs induced by the failed columns, and infer the decoding performance as a function of the code design parameters, as well as the array size and failure parameters. A particularly interesting class of codes, called probabilistically maximum distance separable (MDS) array codes, gives fault-tolerance that is equivalent to traditional MDS array codes. The results also include a proof that the 2D codes outperform standard 1D low-density parity-check codes. [ABSTRACT FROM PUBLISHER]

Caching is a technique to reduce peak traffic rates by prefetching popular content into memories at the end users. Conventionally, these memories are used to deliver requested content in part from a locally cached copy rather than through the network. The gain offered by this approach, which we term local caching gain, depends on the local cache size (i.e., the memory available at each individual user). In this paper, we introduce and exploit a second, global, caching gain not utilized by conventional caching schemes. This gain depends on the aggregate global cache size (i.e., the cumulative memory available at all users), even though there is no cooperation among the users. To evaluate and isolate these two gains, we introduce an information-theoretic formulation of the caching problem focusing on its basic structure. For this setting, we propose a novel coded caching scheme that exploits both local and global caching gains, leading to a multiplicative improvement in the peak rate compared with previously known schemes. In particular, the improvement can be on the order of the number of users in the network. In addition, we argue that the performance of the proposed scheme is within a constant factor of the information-theoretic optimum for all values of the problem parameters. [ABSTRACT FROM AUTHOR]

The interference channel with a cognitive relay (IFC-CR) consists of the classical IFC with two independent source-destination pairs whose communication are aided by an additional node, referred to as the CR, that has a priori knowledge of both sources' messages. This a priori message knowledge is termed cognition and idealizes the relay learning the messages of the two sources from their transmissions over a wireless channel. This paper presents improved outer and inner bounds on the capacity region of the general memoryless IFC-CR that are shown to be tight for certain classes of channels. The new outer bound follows from arguments originally devised for broadcast channels, among which Sato's observation that the capacity region of channels with noncooperative receivers only depends on conditional marginal distributions of the channel output, not on their conditional joint distribution. A simplified expression for the inner bound is derived, which contains all previously proposed coding schemes. The new inner and outer bounds coincide for a class of channels satisfying some strong interference condition, i.e., for these channels there is no loss in optimality if both destinations decode both messages. This result parallels analogous results for the classical interference channel and for the cognitive interference channel and is the first known capacity result for the general IFC-CR. Numerical evaluations of the proposed inner and outer bounds are presented for the additive white Gaussian noise case. [ABSTRACT FROM PUBLISHER]

We provide a refinement of the sphere-packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the subexponential factor in front of the exponent. The order of our subexponential factor is \Omega (N^-0.5(1+\epsilon+\rhoR^{\ast)}) for any \epsilon>0, where \rhoR^{\ast} is the left derivative of the sphere-packing exponent at rate R and N is the blocklength. [ABSTRACT FROM AUTHOR]

This paper studies multiple-input multiple-output linear precoding in the high-signal-to-noise-ratio regime under flat fading. The diversity at all fixed rates is analyzed for a number of linear precoders. The diversity-multiplexing tradeoffs (DMTs) are also obtained, discovering that for many linear precoders the DMT gives no direct insight into the intricate behavior of fixed-rate diversity. The zero-forcing (ZF), regularized ZF, matched filtering, and Wiener filtering precoders are analyzed. It is shown that regularized ZF (RZF) or matched filter (MF) suffers from error floors for all positive multiplexing gains. However, in the fixed rate regime, RZF and MF precoding achieve full diversity for spectral efficiencies up to a certain threshold and zero diversity at rates above it. When the regularization parameter in the RZF is optimized in the minimum mean square error sense, the structure is known as the Wiener precoder, which in the fixed-rate regime is shown to have diversity that depends not only on the number of antennas, but also on the spectral efficiency. The diversity in the presence of both precoding and equalization is also analyzed. [ABSTRACT FROM AUTHOR]

In this paper upper and lower bounds on the probability of decoding failure under maximum likelihood decoding are derived for different (nonbinary) Raptor code constructions. In particular four different constructions are considered; (i) the standard Raptor code construction, (ii) a multi-edge type construction, (iii) a construction where the Raptor code is nonbinary but the generator matrix of the LT code has only binary entries, (iv) a combination of (ii) and (iii). The latter construction resembles the one employed by RaptorQ codes, which at the time of writing this article represents the state of the art in fountain codes. The bounds are shown to be tight, and provide an important aid for the design of Raptor codes. [ABSTRACT FROM AUTHOR]

Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing solutions, both in the applicable range of parameters and in the complexity of the involved algorithms. Second, we introduce an approximate variant of the gradient coding problem, in which we settle for approximate gradient computation instead of the exact one. This approach enables graceful degradation, i.e., the $\ell _{2}$ error of the approximate gradient is a decreasing function of the number of stragglers. Our main result is that normalized adjacency matrices of expander graphs yield excellent approximate gradient codes, which enable significantly less computation compared to exact gradient coding, and guarantee faster convergence than trivial solutions under standard assumptions. We experimentally test our approach on Amazon EC2, and show that the generalization error of approximate gradient coding is very close to the full gradient while requiring significantly less computation from the workers. [ABSTRACT FROM AUTHOR]

The notion of Generalized Hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography. In this paper we extend this concept to the context of convolutional codes. In particular, we focus on column distances and introduce the novel notion of generalized column distances (GCD). We first show that the hierarchy of GCD is strictly increasing. We then provide characterizations of such distances in terms of the truncated parity-check matrix of the code, that will allow us to determine their values. Finally, the case in which the parity-check matrix is in systematic form is treated. [ABSTRACT FROM AUTHOR]

Let $P=([n],\leq _{P})$ be a poset on $[n] = \{1,2,\ldots,n\}$ , $\mathbb {F}_{q}^{n}$ be the linear space of $n$ -tuples over a finite field $\mathbb {F}_{q}$ and $w$ be a weight on $\mathbb {F}_{q}$. In this paper we consider metrics on $\mathbb {F}_{q}^{n}$ which are induced by posets over $[n]$ and weights over $\mathbb {F}_{q}$. Such family of metrics extend both the additive metric induced by the weight $w$ (when the poset is an anti-chain) and the poset metrics (when the weight is the Hamming weight). Furthermore, the pomset metrics is also a particular case of our construction, consequently, we provide a simpler approach to these metrics without using the multiset structure originally proposed. For the general case, we provide a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we (re)obtain the groups of linear isometries of the poset, pomset and additive metric spaces. When considering a chain order, we develop, for codes on these spaces, several of the invariants and properties found in the classical coding theory. Our construction of metrics based on partial orders and any weight over the base field, highlights the dependence of the poset metric over $\mathbb {F}_{q}^{n}$ with the Hamming metric on $\mathbb {F}_{q}$ and the additive property of its extension on $\mathbb {F}_{q}^{n}$. [ABSTRACT FROM AUTHOR]

We define a variable-length code having the property that no (non-empty) prefix of each its codeword is a suffix of any other one, and vice versa. This kind of code can be seen as an extension of two well-known codes in the literature, called, respectively, fix-free code and non-overlapping code. In this paper we present constructive algorithms for such codes and some numerical results about their cardinality. [ABSTRACT FROM PUBLISHER]

In this paper, the algebraic constructions of quasi-cyclic low-density parity-check (QC-LDPC) codes with the unequal error protection (UEP) property are considered. A criterion for constructing such codes via the masking technique is proposed, based on which explicit conditions on the base parity-check matrices and masking matrices to achieve UEP are provided. We also give three specific constructions of UEP QC-LDPC codes. Furthermore, a sufficient condition to ensure strict UEP is presented. Simulation results demonstrate the superiority of our constructed codes over time-sharing schemes. The constructed codes also have competitive error performance against randomly constructed equal-error-protection LDPC codes and irregular UEP LDPC codes designed based on the degree distribution. [ABSTRACT FROM PUBLISHER]

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over \mathop \mathrm GR\nolimits (2^a,m) of length 2^s have been characterized: the ring \mathcal R2(a,m,-1)= {\mathrm{ GR}}(2^{a},m)[x]/\langle x^{2^{s}}+1\rangle is a chain ring. Furthermore, these results have been generalized to \lambda -constacyclic codes for any unit \lambda of the form 4z-1 , . In this paper, we study more general cases and investigate all cases, where {\mathcal{ R}}_{p}(a,m,\gamma) = {\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle is a chain ring. In particular, the necessary and sufficient conditions for the ring {\mathcal{ R}}_{p}(a,m,\gamma) to be a chain ring are obtained. In addition, by using this structure we investigate all \gamma -constacyclic codes over \mathop {\mathrm {GR}}\nolimits (p^{a},m) when {\mathcal{ R}}_{p}(a,m,\gamma)$ is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual \gamma$ -constacyclic codes are also provided. Among others, for any prime p$ , the structure of is used to establish the Hamming and homogeneous distances of $\gamma$ -constacyclic codes. [ABSTRACT FROM PUBLISHER]

Modern compression codes exploit signals’ complex structures to encode them very efficiently. On the other hand, compressed sensing algorithms recover “structured” signals from their under-determined set of linear measurements. Currently, there is a noticeable gap between the types of structures used in the area of compressed sensing and those employed by state-of-the-art compression codes. Recent results in the literature on deterministic signals aim at bridging this gap through devising compressed sensing decoders that employ compression codes. This paper focuses on structured stochastic processes and studies application of lossy compression codes to compressed sensing of such signals. The performance of the formerly proposed compressible signal pursuit (CSP) optimization is studied in this stochastic setting. It is proved that in the low-distortion regime, as the blocklength grows to infinity, the CSP optimization reliably and robustly recovers $n$ instances of a stationary process from its random linear measurements as long as $n$ is slightly more than $n$ times the rate-distortion dimension (RDD) of the source. It is also shown that under some regularity conditions, the RDD of a stationary process is equal to its information dimension. This connection establishes the optimality of CSP at least for memoryless stationary sources, which have known fundamental limits. Finally, it is shown that CSP combined by a family of universal variable-length fixed-distortion compression codes yields a family of universal compressed sensing recovery algorithms. [ABSTRACT FROM PUBLISHER]

As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu et al. as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over F2^{2m} of any algebraic degree between 2 and $m$ . This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager. [ABSTRACT FROM PUBLISHER]

Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an [n,k,d] - (\alpha ) MSR code \mathbb C over \smash \mathbb F\!q stores a file \mathcal F consisting of \alpha k symbols over \smash {\mathbb {F}_{\!q}} among n$ nodes, each storing \alpha $ symbols, in such a way that: 1) the file k$ of the n$ nodes and 2) the content of any failed node can be reconstructed by accessing any d$ of the remaining symbols from each of these nodes. In practice, the file \mathcal F is typically available in uncoded form on some $k$ of the $n$ nodes, known as systematic nodes, and the defining node-repair condition above can be relaxed to requiring the optimal repair bandwidth for systematic nodes only. Such codes are called systematic–repair MSR codes. Unfortunately, finite– $\alpha $ constructions of $[n,k,d]$ MSR codes are known only for certain special cases: either low rate, namely $k/n \leqslant 0.5$ , or high repair connectivity, namely $d = n-1$ . Our main result in this paper is a finite– $\alpha $ construction of systematic-repair $[n,k,d]$ MSR codes for all possible values of parameters $n,k,d$ . We also introduce a generalized construction for $[n,k]$ MSR codes to achieve the optimal repair bandwidth for all values of $d$ simultaneously. [ABSTRACT FROM PUBLISHER]

The ability to store data in the DNA of a living organism has applications in a variety of areas including synthetic biology and watermarking of patented genetically modified organisms. Data stored in this medium are subject to errors arising from various mutations, such as point mutations, indels, and tandem duplication, which need to be corrected to maintain data integrity. In this paper, we provide error-correcting codes for errors caused by tandem duplications, which create a copy of a block of the sequence and insert it in a tandem manner, i.e., next to the original. In particular, we present two families of codes for correcting errors due to tandem duplications of a fixed length: the first family can correct any number of errors, while the second corrects a bounded number of errors. We also study codes for correcting tandem duplications of length up to a given constant $k$ , where we are primarily focused on the cases of $k=2,3$ . Finally, we provide a full classification of the sets of lengths allowed in tandem duplication that result in a unique root for all sequences. [ABSTRACT FROM AUTHOR]

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most $r$ ) of other symbols of the codeword. In this paper, we introduce a construction of LRC codes on algebraic curves, extending a recent construction of the Reed–Solomon like codes with locality. We treat the following situations: local recovery of a single erasure, local recovery of multiple erasures, and codes with several disjoint recovery sets for every coordinate (the availability problem). For each of these three problems we describe a general construction of codes on curves and construct several families of LRC codes. We also describe a construction of codes with availability that relies on automorphism groups of curves. We also consider the asymptotic problem for the parameters of the LRC codes on curves. We show that the codes obtained from asymptotically maximal curves (for instance, Garcia-Stichtenoth towers) improve upon the asymptotic versions of the Gilbert-Varshamov bound for LRC codes. [ABSTRACT FROM AUTHOR]

Write-once memory (WOM) is a storage device consisting of $q$ -ary cells that can only increase their value. A WOM code is a coding scheme that allows writing multiple times to the memory without decreasing the levels of the cells. In the conventional model, it is assumed that the encoder can read the memory state before encoding, while the decoder reads only the memory state after encoding. However, there are three more models in this setup, which depend on whether the encoder and the decoder are informed or uninformed with the previous state of the memory. These four models were first introduced by Wolf et al., where they extensively studied the WOM capacity in these models for the binary case. In the non-binary setup, only the model, in which the encoder is informed and the decoder is not, was studied by Fu and Vinck. In this paper, we first present constructions of WOM codes in the models where the encoder is uninformed with the memory state (that is, the encoder cannot read the memory prior to encoding). We then study the capacity regions and maximum sum-rates of non-binary WOM codes for all four models. We extend the results by Wolf et al. and show that the capacity regions for the models in which the encoder is informed and the decoder is informed or uninformed in both the $\epsilon $ -error and the zero-error cases are all identical. We also find the $\epsilon $ -error capacity region; in this case, the encoder is uninformed and the decoder is informed and show that, in contrary to the binary case, it is a proper subset of the capacity region in the first two models. Several more results on the maximum sum-rate are presented as well. [ABSTRACT FROM AUTHOR]

Caching is often used in content delivery networks as a mechanism for reducing network traffic. Recently, the technique of coded caching was introduced whereby coding in the caches and coded transmission signals from the central server were considered. Prior results in this area demonstrate that carefully designing the placement of content in the caches and designing appropriate coded delivery signals from the server allow for a system where the delivery rates can be significantly smaller than conventional schemes. However, matching upper and lower bounds on the transmission rate have not yet been obtained. In this paper, we derive tighter lower bounds on the coded caching rate than were known previously. We demonstrate that this problem can equivalently be posed as a combinatorial problem of optimally labeling the leaves of a directed tree. Our proposed labeling algorithm allows for significantly improved lower bounds on the coded caching rate. Furthermore, we study certain structural properties of our algorithm that allow us to analytically quantify improvements on the rate lower bound for general values of the problem parameters. This allows us to obtain a multiplicative gap of at most four between the achievable rate and our lower bound. [ABSTRACT FROM PUBLISHER]

The capacity region of the multicast Cognitive Inter-Ference Channel (CIFC) is investigated. This channel consists of two independent transmitters that wish to multicast two different messages, each to a different set of users. In addition, one of the transmitters—commonly referred to as the cognitive transmitter—has prior non-causal knowledge of both messages to be transmitted. This scenario subsumes some long-standing open problems, such as the interference channel, the broadcast channel, and multicast communications. The aim of this paper is the derivation of optimal interference mitigation techniques under different interference regimes. To this end, two settings, namely the multi-primary CIFC, i.e., $M=1$ and $N \geq 1$ , and its complementary, the multi-secondary CIFC, i.e., $N=1$ and $M \geq 1$ , are investigated as an attempt to build a thorough understanding for the more general multicast CIFC setting. It is shown that, for some interference regimes, well-known coding techniques for the standard CIFC remain still optimal under the constraint of multicasting to multiple users. However, in other interference regimes, capacity achieving coding and decoding schemes prove to be more involved. A careful combination of these coding schemes and new outer bounding techniques allows to characterize the capacity region for several classes of discrete memoryless and Gaussian multicast CIFC under different interference regimes. [ABSTRACT FROM PUBLISHER]

In this paper, we revisit the discrete lossy Gray-Wyner problem. In particular, we derive its optimal second-order coding rate region, its error exponent (reliability function), and its moderate deviations constant under mild conditions on the source. To obtain the second-order asymptotics, we extend some ideas from Watanabe’s work. In particular, we leverage the properties of an appropriate generalization of the conditional distortion-tilted information density, which was first introduced by Kostina and Verdú. The converse part uses a perturbation argument by Gu and Effros in their strong converse proof of the discrete Gray–Wyner problem. The achievability part uses two novel elements: 1) a generalization of various type covering lemmas and 2) the uniform continuity of the conditional rate-distortion function in both the source (joint) distribution and the distortion level. To obtain the error exponent, for the achievability part, we use the same generalized type covering lemma, and for the converse, we use the strong converse together with a change-of-measure technique. Finally, to obtain the moderate deviations constant, we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities. [ABSTRACT FROM PUBLISHER]

An information theoretical analysis of nonadaptive group testing schemes based on sparse pooling graphs is presented. A pooling graph is a bipartite graph for which the adjacency matrix is a pooling matrix. The binary status of the objects to be tested are modeled by independent identically distributed. Bernoulli random variables with probability $p$ . An $(l, r, n)$ -regular pooling graph is a bipartite graph in which the left nodes have degree $l$ , the right nodes have degree $r$ , and $n$ is the number of left nodes. The main contribution of this paper is a direct coding theorem that gives the conditions for the existence of an estimator that can achieve an arbitrarily small probability of error. An estimator is a function that uses observations to infer the state of an object. The direct coding theorem is proved by averaging the upper bound on the probability of the estimation error of the typical set estimator over an $(l, r, n)$ -regular pooling graph ensemble. Numerical results indicate sharp threshold behaviors in the asymptotic regime. These results can provide a concrete benchmark for nonadaptive group testing of existing and emerging detection algorithms over a noiseless system. [ABSTRACT FROM PUBLISHER]

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over \mathbb Z4 with type 4^k12^{k2} . [ABSTRACT FROM PUBLISHER]

In a system that performs both encryption and lossy compression, the conventional way is to compress first and then encrypt the compressed data. This separation approach has been proved to be optimal. In certain applications where sensitive information should be protected as early as possible, it is preferable to perform the encryption first and then compress the encrypted data, which leads to the concept of the reversed system. Johnson et al. proposed an achievable scheme for the reversed system, where a modulo-sum encryption is followed by a compression using the Wyner–Ziv distributed source coding with side information. However, in general, this reversed system performs worse than the conventional system in the sense that it requires more compression rate and secrecy key rate. In this paper, we propose a new achievable scheme for the reversed system, where the encryption is conducted by a permutation cipher, and then, the encrypted data is compressed using the optimal rate-distortion code. The proposed scheme can achieve the optimal compression rate and secret key rate. As a result, we show that reversing the order of the encryption and compression does not necessarily compromise the performance of an encryption-compression system. We show that the proposed system attains strong secrecy, and the information leakage vanishes exponentially. [ABSTRACT FROM PUBLISHER]

For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class \mathcal N of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil infinitely many new multicast networks linearly solvable over GF( q ) but not over GF( q' ) with q < q' , based on a subgroup order criterion. In particular: 1) for every k\geq 2 ) but not over GF( 2^{2k+1} ) and 2) for arbitrary distinct primes p and p' , there are infinitely many k and k' ) but not over GF( p'^k' ) with p^k < p'^k' . [ABSTRACT FROM PUBLISHER]

Projective Reed–Muller codes were introduced by Lachaud in 1988 and their dimension and minimum distance were determined by Serre and Sørensen in 1991. In coding theory, one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as the next-to-minimal weight) is completely determined. In this paper, we determine all the values of the next-to-minimal weight for the binary projective Reed–Muller codes, which we show to be equal to the next-to-minimal weight of Reed–Muller codes in most, but not all, cases. [ABSTRACT FROM PUBLISHER]

Here, we revisit the problem of exploring the secrecy capacity of minimum storage cooperative regenerating (MSCR) codes under the \l1,l2\ -eavesdropper model, where the eavesdropper can observe the data stored on l1 nodes and the repair downloads of an additional l2 nodes. Compared to minimum storage regenerating (MSR) codes which support only single node repairs, MSCR codes allow efficient simultaneous repairs of multiple failed nodes, referred to as a repair group. However, the repair data sent from a helper node to another failed node may vary with different repair groups or the sets of helper nodes, which would inevitably leak more data information to the eavesdropper and even render the storage system unable to maintain any data secrecy. In this paper, we introduce and study a special category of MSCR codes, termed “stable” MSCR codes, where the repair data from any one helper node to any one failed node is required to be independent of the repair group or the set of helper nodes. Our main contributions include: 1) Demonstrating that two existing MSCR codes inherently are not stable and thus have poor secrecy capacity; 2) Converting one existing MSCR code to a stable one, which offers better secrecy capacity when compared to the original one; and 3) Employing information theoretic analysis to characterize the secrecy capacity of stable MSCR codes in certain situations. [ABSTRACT FROM PUBLISHER]

This paper shows new general nonasymptotic achievability and converse bounds and performs their dispersion analysis for the lossy compression problem in which the compressor observes the source through a noisy channel. While this problem is asymptotically equivalent to a noiseless lossy source coding problem with a modified distortion function, nonasymptotically there is a noticeable gap in how fast their minimum achievable coding rates approach the common rate-distortion function, as evidenced both by the refined asymptotic analysis (dispersion) and the numerical results. The size of the gap between the dispersions of the noisy problem and the asymptotically equivalent noiseless problem depends on the stochastic variability of the channel through which the compressor observes the source. [ABSTRACT FROM PUBLISHER]

This paper has two goals. The first one is to discuss two related packing problems in the Lee and Manhattan metrics. One is to find good codes for error-correction (i.e., packings of Lee spheres) and the other is to transform the space in a way that volumes are preserved and each Lee sphere (or scaled cross-polytope) will be transformed into a shape inscribed in a small cube. The second goal is to consider weighing matrices for some of these coding problems. Weighing matrices have been used as building blocks for codes in the Hamming metric in various constructions. In this paper, we will consider mainly two types of weighing matrices, namely conference matrices and Hadamard matrices, to construct codes in the Lee (and Manhattan) metric. We will show that these matrices have some desirable properties when considered as generator matrices for codes in these metrics. [ABSTRACT FROM AUTHOR]

A two-terminal interactive function computation problem with alternating messages is studied within the framework of distributed block source coding theory. For any finite number of messages, a single-letter characterization of the sum-rate-distortion function was established in previous works using standard information-theoretic techniques. This, however, does not provide a satisfactory characterization of the infinite-message limit, which is a new, unexplored dimension for asymptotic analysis in distributed block source coding involving potentially an infinite number of infinitesimal-rate messages. In this paper, the infinite-message sum-rate-distortion function, viewed as a functional of the joint source distribution and the distortion levels, is characterized as the least element of a partially ordered family of functionals having certain convex-geometric properties. The new characterization does not involve evaluating the infinite-message limit of a finite-message sum-rate-distortion expression. This characterization leads to a family of lower bounds for the infinite-message sum-rate-distortion expression and a simple criterion to test the optimality of any achievable infinite-message sum-rate-distortion expression. The new convex-geometric characterization is used to develop an iterative algorithm for evaluating any finite-message sum-rate-distortion function. It is also used to construct the first examples which demonstrate that for lossy source reproduction, two messages can strictly improve the one-message Wyner–Ziv rate-distortion function settling an unresolved question from a 1985 paper. It is shown that a single backward message of arbitrarily small rate can lead to an arbitrarily large gain in the sum-rate. [ABSTRACT FROM AUTHOR]

This paper considers the problem of communicating over a relay channel with state when noncausal state information is partially available at the nodes. We first establish a lower bound on the achievable rates based on noisy network coding and Gelfand–Pinsker coding, and show that it provides an alternative characterization of a previously known bound. We then introduce the class of state-decoupled relay channels and show that our lower bound is tight for a subclass of semideterministic channels. We also compute the capacity for two specific examples of this subclass—a channel with multiplicative binary fading and a channel with additive Gaussian interference. These examples are not special cases of previous classes of semideterministic relay channels with known capacity. [ABSTRACT FROM PUBLISHER]

Consider a memoryless degraded broadcast channel (DBC) in which the channel output is a single-letter function of the channel input and the channel noise. As examples, for the Gaussian broadcast channel (BC), this single-letter function is real scalar addition and for the binary-symmetric BC, this single-letter function is modulo-two addition. This paper identifies several classes of discrete memoryless DBCs for which a relatively simple encoding scheme, which we call natural encoding, achieves capacity. Natural encoding (NE) combines symbols from independent codebooks (one for each receiver) using the same single-letter function that adds distortion to the channel. The alphabet size of each NE codebook is bounded by that of the channel input. This paper also defines the input-symmetric DBC, introduces permutation encoding for the input-symmetric DBC, and proves its optimality. Because it is a special case of permutation encoding, NE is capacity achieving for the two-receiver group-operation DBC. Combining the broadcast Z channel and group-operation DBC results yields a proof that NE is also optimal for the discrete multiplication DBC. Along the way, the paper also provides explicit parametric expressions for the two-receiver binary-symmetric DBC and broadcast Z channel. [ABSTRACT FROM AUTHOR]

Constant-weight codes play an important role in coding theory. The problem of determining the sizes for optimal quaternary constant-weight codes with length n, weight 4, and minimum Hamming distance 5 ((n,5,4)4 codes) has been investigated in several papers. Although some constructions and several infinite families for such codes with length n\equiv 0,1\pmod4 have been given, the problem is still far from complete. In this paper, we determine the size of an optimal (n,5,4)4 code for each integer n\geq 4 leaving 55 lengths unsolved. Especially, for length n\equiv 0,1\pmod 4, the existence problem of the equivalent combinatorial object, namely the generalized Steiner system, is solved leaving only seven values undetermined. [ABSTRACT FROM AUTHOR]

Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of this paper is to fill this gap by constructing near-explicit “polar” codes that are capacity-achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by “channel combining” and “channel splitting,” in which a fraction of the synthesized channels are perfect for data transmission, while the other channels are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and “frozen” bits through the useless ones. The main technical contributions of this paper are threefold. First, we leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and the encoding complexity is O(N\log N), where N is the blocklength of the code. We also demonstrate that a quantum successive cancellation decoder works well, in the sense that the word error rate decays exponentially with the blocklength of the code. For this last result, we exploit Sen's recent “noncommutative union bound” that holds for a sequence of projectors applied to a quantum state. [ABSTRACT FROM AUTHOR]

Recently, network error correction coding (NEC) has been studied extensively. Several bounds in classical coding theory have been extended to NEC, especially the Singleton bound. In this paper, following the research line using the extended global encoding kernels proposed by Zhang in 2008, the refined Singleton bound of NEC can be proved more explicitly. Moreover, we give a constructive proof of the attainability of this bound and indicate that the required field size for the existence of network maximum distance separable (MDS) codes can become smaller further. By this proof, an algorithm is proposed to construct general linear network error correction codes including the linear network error correction MDS codes. Finally, we study the error correction capability of random linear NEC. Motivated partly by the performance analysis of random linear network coding, we evaluate the different failure probabilities defined in this paper in order to analyze the performance of random linear NEC. Several upper bounds on these probabilities are obtained and they show that these probabilities will approach to zero as the size of the base field goes to infinity. Using these upper bounds, we slightly improve on the probability mass function of the minimum distance of random linear network error correction codes in a paper by Balli and colleagues, as well as the upper bound on the field size required for the existence of linear network error correction codes with degradation at most d. [ABSTRACT FROM AUTHOR]

This paper considers the relationship between code-rate selection and queueing performance for communication systems subject to time-varying channel conditions. While error-correcting codes offer protection against channel uncertainties, there exists a natural tradeoff between the enhanced protection of low-rate codes and the rate penalty imposed by additional redundancy. In the limiting regime where codewords are asymptotically long, this tradeoff is well understood and characterized by the Shannon capacity. However, for delay-sensitive communication systems and finite block lengths, a complete characterization of this tradeoff is not fully developed. This paper offers a new perspective on the queueing performance of communication systems with finite block lengths operating over correlated erasure channels. A rigorous framework that links code rate to overall system performance for random codes is presented. Guidelines for code-rate selection in delay-sensitive systems are identified. These findings are supported by a numerical study. [ABSTRACT FROM AUTHOR]

Determining the minimum distance of a linear code is one of the most important problems in algorithmic coding theory. The exact version of the problem was shown to be NP-complete by Vardy. The gap version of the problem was shown to be NP-hard for any constant factor under a randomized reduction in an earlier work. It was shown in the same paper that the minimum distance problem is not approximable in randomized polynomial time to the factor 2^\log^1-\epsilonn unless NP\subseteq RTIME(2^polylog(n)). In this paper, we derandomize the reduction and thus prove that there is no deterministic polynomial time algorithm to approximate the minimum distance to any constant factor unless P=NP. We also prove that the minimum distance is not approximable in deterministic polynomial time to the factor 2^\log^1-\epsilonn unless NP\subseteq DTIME(2^polylog(n)). As the main technical contribution, for any constant 2/3 < \rho < 1, we present a deterministic algorithm that given a positive integer s, runs in time poly(s) and constructs a code \cal C of length poly(s) with an explicit Hamming ball of radius \rho d(\cal C), such that the projection at the first s coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension s, where d(\cal C) denotes the minimum distance of \cal C. The codes are obtained by concatenating Reed–Solomon codes with Hadamard codes. [ABSTRACT FROM PUBLISHER]

The achievable information rate region problem of multisource multisink network coding for general acyclic networks with arbitrary transmission requirements has previously been studied, where inner and outer bounds on the region were derived in terms of \Gamma^\ast, the fundamental region of entropy functions. In this paper, we derive the exact characterization of the achievable rate region in terms of entropic functions, thus closing the gap between the existing inner and outer bounds. [ABSTRACT FROM AUTHOR]

Inductive coupling is a technique wherein one device (the reader) induces an electrical current in another device (the tag), thereby providing not only power for the tag, but also a communication channel. In this paper, we focus exclusively on the reader-to-tag channel. The first part of this paper presents modulation codes that possess a high minimum and a high average power. This is important, since the tag gets its entire power from the received signal, and the information should be modulated in a way that maximizes the power transferred to the tag. The presented modulation codes compare favorably to codes used in radio frequency identification applications today. The second part of the paper describes modulation codes with some error-correcting capabilities. In fact, most errors in the reader-to-tag channel are due to incorrect timing. Here, we propose to model the timing errors in the reader-to-tag communication channel by a simple bit-shift channel, and we will present optimal (in the sense of maximizing the code rate for a given block length) single bit-shift error-correcting codes for this simple bit-shift channel that also have large average power. [ABSTRACT FROM PUBLISHER]

In 1995, Best published a formula for the exact bit error probability for Viterbi decoding of the rate R=1/2, memory m=1 (two-state) convolutional encoder with generator matrix G(D)=(1 \quad 1+D) when used to communicate over the binary symmetric channel. Their formula was later extended to the rate R=1/2, memory m=2 (four-state) convolutional encoder with generator matrix G(D)=(1+D^2 \quad 1+D+D^2) by Lentmaier In this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed-form expression for the exact bit error probability. As special cases, the expressions obtained by Best for the two-state encoder and by Lentmaier for a four-state encoder are used. The closed-form expression derived in this paper is evaluated for various realizations of encoders, including rate R=1/2 and R=2/3 encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to study the achievable rates for a K-user Gaussian interference channel (G-IFC) for any signal-to-noise ratio using a combination of lattice and algebraic codes. Lattice codes are first used to transform the G-IFC into a discrete input--output noiseless channel, and subsequently algebraic codes are developed to achieve good rates over this new alphabet. In this context, a quantity called efficiency is introduced which reflects the effectiveness of the algebraic coding strategy. This paper first addresses the problem of finding high-efficiency algebraic codes. A combination of these codes with Construction-A lattices is then used to achieve nontrivial rates for the original G-IFC. [ABSTRACT FROM PUBLISHER]

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, the two-prime sequence is employed to construct several classes of cyclic codes over GF(q). Lower bounds on the minimum weight of these cyclic codes are developed. Some of the codes obtained are optimal or almost optimal. The p-ranks of the twin-prime difference sets and a class of almost difference sets are computed. [ABSTRACT FROM PUBLISHER]