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We consider the problem of communicating over a channel that randomly “tears” the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\mathrm{ Geometric}}(p_{n})$ , we characterize the capacity as $C = e^{-\alpha }$ , where $\alpha = \lim _{n\to \infty } p_{n} \log n$. Our results show that the case of ${\mathrm{ Geometric}}(p_{n})$ -length fragments and the case of deterministic length- $(1/p_{n})$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity. [ABSTRACT FROM AUTHOR]

The “carbon copy onto dirty paper” (CCDP) channel is the compound “writing on dirty paper” channel in which the channel output is obtained as the sum of the channel input, white Gaussian noise and a Gaussian state sequence randomly selected among a set possible realizations. The transmitter has non-causal knowledge of the set of possible state sequences but does not know which sequence is selected to produce the channel output. We study the capacity of the CCDP channel for two scenarios: 1) the state sequences are independent and identically distributed; and 2) the state sequences are scaled versions of the same sequence. In the first scenario, we show that a combination of superposition coding, time-sharing, and Gel’fand-Pinsker binning is sufficient to approach the capacity to within 3 bits per channel use for any number of possible state realizations. In the second scenario, we derive capacity to within 4 bits per channel use for the case of two possible state sequences. This result is extended to the CCDP channel with any number of possible state sequences under certain conditions on the scaling parameters, which we denote as “strong fading” regime. We conclude by providing some remarks on the capacity of the CCDP channel in which the state sequences have any jointly Gaussian distribution. [ABSTRACT FROM AUTHOR]

This paper studies the second-order coding rates for memoryless channels with a state sequence known non-causally at the encoder. In the case of finite alphabets, an achievability result is obtained using constant-composition random coding, and by using a small fraction of the block to transmit the empirical distribution of the state sequence. For error probabilities less than 0.5, it is shown that the second-order rate improves on an existing one based on independent and identically distributed random coding. In the Gaussian case (dirty paper coding) with an almost-sure power constraint, an achievability result is obtained using random coding over the surface of a sphere, and using a small fraction of the block to transmit a quantized description of the state power. It is shown that the second-order asymptotics are identical to the single-user Gaussian channel of the same input power without a state. [ABSTRACT FROM AUTHOR]

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper, we elaborate on the implication of these parameters and devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes, our bound is often tight. Furthermore, for these codes, the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. [ABSTRACT FROM AUTHOR]

An $(r, \delta)$ -locally repairable code ($(r, \delta)$ -LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al.. An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. , and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of , , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in , , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method. [ABSTRACT FROM AUTHOR]

We propose a successive cancellation list (SCL) encoding and decoding scheme for the Gelfand Pinsker (GP) problem based on the known nested polar coding scheme. It applies SCL encoding for the source coding part, and SCL decoding with a properly defined CRC for the channel coding part. The scheme shows improved performance compared to the existing method. A known issue with nested polar codes for binary dirty paper is the existence of frozen channel code bits that are not frozen in the source code. These bits need to be retransmitted in a second phase of the scheme, thus reducing the rate and increasing the required blocklength. We provide an improved bound on the size of this set, and on its scaling with respect to the blocklength, when the Bhattacharyya parameter of the test channel used for source coding is sufficiently large, or the Bhattacharyya parameter of the channel seen at the decoder is sufficiently small. The result is formulated for an arbitrary binary-input memoryless GP problem, since unlike the previous results, it does not require degradedness of the two channels mentioned above. Finally, we present simulation results for binary dirty paper and noisy write once memory codes. [ABSTRACT FROM AUTHOR]

The recipients of the 2019 IEEE Information Theory Society Paper Award are Emmanuel J. Candès, Xiaodong Li, and Mahdi Soltanolkotabi for the article “Phase Retrieval via Wirtinger Flow: Theory and Algorithms,” which appeared in the IEEE Transactions on Information Theory, vol. 61, no. 4, pp. 1985–2007, April 2015. [ABSTRACT FROM AUTHOR]

The recipients of the 2019 IEEE Communications Society and Information Theory Society Joint Paper Award are Yuyi Mao, Jun Zhang, and Khaled B. Letaief for the article “Dynamic computation offloading for mobile-edge computing with energy harvesting devices” which appeared in the IEEE Journal on Selected Areas in Communications, vol. 34, no. 12, pp. 3590–3605, December 2016. [ABSTRACT FROM AUTHOR]

The recipients of the 2018 IEEE Information Theory Society Paper Award are Mansoor I. Yousefi and Frank R. Kschischang for the three-part paper “Information Transmission Using the Nonlinear Fourier Transform, I, II, III” which appeared in the IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4312–4328 (Part I), pp. 4329–4345 (Part II), and pp. 4346–4369 (Part III), July 2014. [ABSTRACT FROM AUTHOR]

In wireless networks, the link connectivity may be intermittent due to shadowing, burstiness of data arrival, or uncoordinated resource allocation. In this paper, we model the intermittence of links as Bernoulli distributed random channel states, termed intermittence channel states, and study the impact of the corresponding channel state information at the transmitter (CSIT) in a two-user Gaussian broadcast channel (BC). Moreover, due to the heterogeneous timeliness of intermittence channel states, the CSIT considered in this paper is hybrid. More specifically, the CSIT of each link can be perfectly (causally or non-causally) available, delayed, or not available. When the links are connected, we adopt a general setting that the received signal-to-noise ratios can be different. Our contribution is the characterization of the capacity regions of intermittent Gaussian BC to within bounded gaps for all combinations of hybrid CSIT, except for scenario DN (delayed CSIT of receiver 1, no CSIT of receiver 2). For scenario DN, we propose an opportunistic physical layer network coding scheme that achieves a strictly larger degree-of-freedom (DoF) region than the no-CSIT DoF region. As a corollary, single-user CSIT is able to increase the sum DoF for intermittent Gaussian BC (also the capacity region for the erasure BC, as a by-product). This result is in sharp contrast to the recent negative result by Davoodi and Jafar, where it is shown that for fast-fading multiple-input single-output BC with continuous channel states, single-user CSIT does not help at all in terms of sum DoF. [ABSTRACT FROM AUTHOR]

The article announces the 2017 Information Theory Society Paper awards given to researchers including Charles Bennett, Aram Harrow, and Peter Shor.

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]

In its most elementary form, compressed sensing studies the design of decoding algorithms to recover a sufficiently sparse vector or code from a lower dimensional linear measurement vector. Typically it is assumed that the decoder has access to the encoder matrix, which in the combinatorial case is sparse and binary. In this paper we consider the problem of designing a decoder to recover a set of sparse codes from their linear measurements alone, that is without access to encoder matrix. To this end we study the matrix factorisation task of recovering both the encoder and sparse coding matrices from the associated linear measurement matrix. The contribution of this paper is a computationally efficient decoding algorithm, Decoder-Expander Based Factorisation, with strong performance guarantees. Under mild assumptions on the sparse coding matrix and by deploying a novel random encoder matrix, we prove that Decoder-Expander Based Factorisation recovers both the encoder and sparse coding matrix at the optimal measurement rate with high probability and from a near optimal number of measurement vectors. In addition, our experiments demonstrate the efficacy and computational efficiency of our algorithm in practice. Beyond compressed sensing, our results may be of interest for researchers working in areas as diverse as linear sketching, coding theory, matrix compression and dictionary learning. [ABSTRACT FROM AUTHOR]

The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [ABSTRACT FROM AUTHOR]

In this paper, we consider variable-length coding over the memoryless binary asymmetric channel (BAC) with full noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced a coding scheme, which we refer to as the small-enough-difference (SED) coding scheme. For symmetric binary-input channels, the deterministic variable-length feedback (VLF) code constructed with the SED coding scheme asymptotically achieves both capacity and Burnashev’s optimal error exponent. Building on the work of Naghshvar et al., this paper extends the SED coding scheme to the BAC and develops a non-asymptotic VLF achievability bound that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic VLF achievability bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. Numerical evaluations show that both new VLF achievability bounds outperform Polyanskiy’s achievability bound for variable-length stop-feedback codes. [ABSTRACT FROM AUTHOR]

The recipients of the 2018 IEEE Communications Society and Information Theory Society Joint Paper Award are Arash Gholami Davoodi and Syed A. Jafar for the paper “Aligned Image Sets Under Channel Uncertainty: Settling Conjectures on the Collapse of Degrees of Freedom Under Finite Precision CSIT” which appeared in the IEEE Transactions on Information Theory, vol. 62, no. 10, pp. 5603–5618, October 2016. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the generalization performance of the Iterative Hard Thresholding (IHT) algorithm widely used for sparse recovery problems. The parameter estimation and sparsity recovery consistency of IHT has long been known in compressed sensing. From the perspective of statistical learning, another fundamental question is how well the IHT estimation would predict on unseen data. This paper makes progress towards answering this open question by introducing a novel sparse generalization theory for IHT under the notion of algorithmic stability. Our theory reveals that: 1) under natural conditions on the empirical risk function over $n$ samples of dimension $p$ , IHT with sparsity level $k$ enjoys an $\tilde {\mathcal {O}}(n^{-1/2}\sqrt {k\log (n)\log (p)})$ rate of convergence in sparse excess risk; 2) a tighter $\tilde {\mathcal {O}}(n^{-1/2}\sqrt {\log (n)})$ bound can be established by imposing an additional iteration stability condition on a hypothetical IHT procedure invoked to the population risk; and 3) a fast rate of order $\tilde {\mathcal {O}}\left ({n^{-1}k(\log ^{3}(n)+\log (p))}\right)$ can be derived for strongly convex risk function under proper strong-signal conditions. The results have been substantialized to sparse linear regression and sparse logistic regression models to demonstrate the applicability of our theory. Preliminary numerical evidence is provided to support our theoretical predictions. [ABSTRACT FROM AUTHOR]

A functional $k$ -batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^{s}-1$ servers. This conjecture is verified for $s \leqslant 5$ but previous work could only show that codes with $n=2^{s}-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left \lfloor{ \frac { 2^{s/2}}{\sqrt {24}} }\right \rfloor $ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes. [ABSTRACT FROM AUTHOR]

This paper proposes using the uncertainty of information (UoI), measured by Shannon’s entropy, as a metric for information freshness. We consider a system in which a central monitor observes M binary Markov processes through m communication channels (m

In the coded caching problem, as originally formulated by Maddah-Ali and Niesen, a server communicates via a noiseless shared broadcast link to multiple users that have local storage capability. In order for a user to decode its demanded file from the coded multicast transmission, the demands of all the users must be globally known, which may violate the privacy of the users. To overcome this privacy problem, Wan and Caire recently proposed several schemes that attain coded multicasting gain while simultaneously guarantee information theoretic privacy of the users’ demands. In Device-to-Device (D2D) networks, the demand privacy problem is further exacerbated by the fact that each user is also a transmitter, which appears to be needing the knowledge of the files demanded by the remaining users in order to form its coded multicast transmission. This paper shows how to solve this seemingly infeasible problem. The main contribution of this paper is the development of new achievable and converse bounds for D2D coded caching that are to within a constant factor of one another when privacy of the users’ demands must be guaranteed even in the presence of colluding users (i.e., when some users share cached contents and demanded file indices). First, a D2D private caching scheme is proposed, whose key feature is the addition of virtual users in the system in order to “hide” the demands of the real users. By comparing the achievable D2D private load with an existing converse bound for the shared-link model without demand privacy constraint, the proposed scheme is shown to be order optimal, except for the very low memory size regime with more files than users. Second, in order to shed light into the open parameter regime, a new achievable scheme and a new converse bound under the constraint of uncoded cache placement (i.e., when each user stores directly a subset of the bits of the library) are developed for the case of two users, and shown to be to within a constant factor of one another for all system parameters. Finally, the two-user converse bound is extended to any number of users by a cut-set type argument. With this new converse bound, the virtual users scheme is shown to be order optimal in all parameter regimes under the constraint of uncoded cache placement and user collusion. [ABSTRACT FROM AUTHOR]

It is well known that for Gaussian channels, a nearest neighbor decoding rule, which seeks the minimum Euclidean distance between a codeword and the received channel output vector, is the maximum likelihood solution and hence capacity-achieving. Nearest neighbor decoding remains a convenient and yet mismatched solution for general channels, and the key message of this paper is that the performance of nearest neighbor decoding can be improved by generalizing its decoding metric to incorporate channel state dependent output processing and codeword scaling. Using generalized mutual information, which is a lower bound to the mismatched capacity under independent and identically distributed codebook ensemble, as the performance measure, this paper establishes the optimal generalized nearest neighbor decoding rule, under Gaussian channel input. Several restricted forms of the generalized nearest neighbor decoding rule are also derived and compared with existing solutions. The results are illustrated through several case studies for fading channels with imperfect receiver channel state information and for channels with quantization effects. [ABSTRACT FROM AUTHOR]

This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22. [ABSTRACT FROM AUTHOR]

It is known that there exist mathematical problems of practical relevance which cannot be computed on a Turing machine. An important example is the calculation of the first derivative of continuously differentiable functions. This paper precisely classifies the non-computability of the first derivative, and of the maximum-norm of the first derivative in the Zheng-Weihrauch hierarchy. Based on this classification, the paper investigates whether it is possible that a Turing machine detects this non-computability of the first derivative by observing the data of the problem, and whether it is possible to detect upper bounds for the peak value of the first derivative of continuously differentiable functions. So from a practical point of view, the question is whether it is possible to implement an exit-flag functionality for observing non-computability of the first derivative. This paper even studies two different types of exit-flag functionality. A strong one, where the Turing machine always has to stop, and a weak one, where the Turing machine stops if and only if the input lies within the corresponding set of interest. It will be shown that non-computability of the first derivative is not detectable by a Turing machine for two concrete examples, namely for the problem of computing the input–output behavior of simple analog circuits and for solutions of the three-dimensional wave equation. In addition, it is shown that it is even impossible to detect an upper bound for the maximum norm of the first derivative. In particular, it is shown that all three problems are not even semidecidable. Finally, we briefly discuss implications of these results for analog and quantum computing. [ABSTRACT FROM AUTHOR]

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of $\mathbb {Z}$ , from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for $w$ -demi-matroids defined over finite sets and $w$ -demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics. [ABSTRACT FROM AUTHOR]

In this paper, we consider a status updating system where updates are generated at a constant rate at $K$ sources and sent to the corresponding recipients through a noise-free broadcast channel. We assume that perfect channel state information (CSI) is available at the transmitter before each transmission, and the transmitter is able to utilize the CSI to precode the updates. Our object is to design optimal precoding schemes to minimize the summed average age of information (AoI) at the recipients. Under various assumptions on the size of each update $B$ , the number of transmit antennas $M$ , and the number of receive antennas $N$ at each user, this paper identifies the corresponding age-optimal precoding and transmission scheduling strategies. Specifically, for the case when $N=1$ , a round-robin based updating scheme is shown to be optimal. For the two-user systems with $N>B$ or $M\notin [N:2N]$ , framed updating schemes are proven to be optimal. For other cases in the two-user systems, a framed alternating updating scheme is proven to be 2-optimal. [ABSTRACT FROM AUTHOR]

For fixed integers $n$ and $b\geq k$ , let $A(b,k,n)$ the largest size of a subset of $\{1,2,\ldots,b\}^{n}$ such that, for any $k$ distinct elements in the set, there is a coordinate where they all differ. Bounding $A(b,k,n)$ is a problem of relevant interest in information theory and computer science, relating to the zero-error capacity with list decoding and to the study of $(b, k)$ -hash families of functions. It is known that, for fixed $b$ and $k$ , $A(b,k,n)$ grows exponentially in $n$. In this paper, we determine new exponential upper bounds for different values of $b$ and $k$. A first bound on $A(b,k,n)$ for general $b$ and $k$ was derived by Fredman and Komlós in the ’80s and improved for certain $b\neq k$ by Körner and Marton and by Arikan. Only very recently better bounds were derived for general $b$ and $k$ by Guruswami and Riazanov, while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan, and by Costa and Dalai. In this paper, we strengthen the bounds for some specific values of $b$ and $k$. Our contribution is a new computational method for obtaining upper bounds on the values of a quadratic form defined over discrete probability distributions in arbitrary dimensions, which emerged as a central ingredient in recent works. The proposed method reduces an infinite-dimensional problem to a finite one, which we manage to further simplify by means of a series of optimality conditions. [ABSTRACT FROM AUTHOR]

The article announces the recipients of the 2016 IEEE Information Theory Society Paper Award who include Yanlin Geng, assistant professor with the School of Information Science and Technology at Shanghai Tech University, Chandra Nair, an associate professor with the Information Engineering Department of the Chinese University of Hong Kong, and Mohammad Ali Maddah-Ali, communications network research scientist at Bell Laboratories, Alcatel-Lucent.

The article announces the recipients of the 2016 IEEE Communications Society and Information Theory Society Joint Paper Award who include Angel Lozano, professor of Information and Communication Technologies at Universitat Pompeu Fabra in Barcelona, Spain, Robert W. Heath Jr., chief innovation officer at Kuma Signals LLC, and Jeffrey G. Andrews, the Cullen Trust Endowed Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin.

This paper is concerned with the problem of recovering a structured signal from a relatively small number of corrupted random measurements. Sharp phase transitions have been numerically observed in practice when different convex programming procedures are used to solve this problem. This paper is devoted to presenting theoretical explanations for these phenomena by employing some basic tools from Gaussian process theory. Specifically, we identify the precise locations of the phase transitions for both constrained and penalized recovery procedures. Our theoretical results show that these phase transitions are determined by some geometric measures of structure, e.g., the spherical Gaussian width of a tangent cone and the Gaussian (squared) distance to a scaled subdifferential. By utilizing the established phase transition theory, we further investigate the relationship between these two kinds of recovery procedures, which also reveals an optimal strategy (in the sense of Lagrange theory) for choosing the tradeoff parameter in the penalized recovery procedure. Numerical experiments are provided to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Coded caching is a promising technique to smooth out network traffic by storing part of the library content at the users’ local caches. The seminal work on coded caching for single file retrieval by Maddah-Ali and Niesen (MAN) showed the existence of a global caching gain that scales with the total memory in the system, in addition to the known local caching gain in uncoded systems. This paper formulates a novel cache-aided matrix multiplication retrieval problem, relevant for data analytics and machine learning applications. In the considered problem, each cache-aided user requests the product of two matrices from the library. A structure-agnostic solution is to treat each possible matrix product as an independent file and use the MAN coded caching scheme for single file retrieval. This paper proposes two structure-aware schemes, which partition each matrix in the library by either rows or columns and let a subset of users cache some sub-matrices, that improve on the structure-agnostic scheme. For the case where the library matrices are “fat” matrices, the structure-aware row-partition scheme is shown to be order optimal under some constraint. [ABSTRACT FROM AUTHOR]

A common method for reading information stored in DNA molecules is shotgun sequencing. This method outputs a histogram of the frequencies of all the molecules’ substrings of a given length $\ell $. To protect against noisy readings, the rank-modulation scheme encodes the information in the relative ranking of the substring frequencies, instead of their absolute values. However, the best rank-modulation codes for shotgun sequencing have low rates which are asymptotically vanishing. In this paper we propose new constructions of rank-modulation codes for shotgun sequencing. The first code construction is systematic, allowing the user to arbitrarily set the frequencies of a large subset of the substrings, which the encoder then completes to a permutation that may be realized by a DNA molecule. The construction is then improved by allowing the user to set the frequencies of additional substrings, at the cost of imposing constraints on the frequencies. The resulting codes have higher, non-vanishing rates, compared with previously known codes. As an example, for histograms of substrings of length $\ell =2$ , and an alphabet of size 4 (as in DNA molecules), we are able to construct a code with rate $\approx 0.909$ , whereas previously, the best construction resulted in a code with rate $\approx 0.654$. Additionally, the encoded information in our construction may be written to shorter DNA molecules than possible before. We also prove that the systematic codes constructed in this paper are the largest possible among all systematic codes. [ABSTRACT FROM AUTHOR]

Recently, subfield codes of geometric codes over large finite fields ${\mathrm {GF}}(q)$ with dimension 3 and 4 were studied and distance-optimal subfield codes over ${\mathrm {GF}}(p)$ were obtained, where $q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over ${\mathrm {GF}}(q)$ , and then studies the subfield codes over ${\mathrm {GF}}(p)$ of some of the $[q+1, 2,q]$ MDS codes over ${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over ${\mathrm {GF}}(p)$ are obtained, and several families of nearly optimal codes over ${\mathrm {GF}}(p)$ are produced. Several open problems are also proposed in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the combinatorial setting where the maximum number of errors inflicted by an adversary is proportional to the number of transmissions, which goes to infinity. Without feedback, it is known that the rate of optimal asymmetric-error-correcting codes for the error fraction $\tau \ge 1/4$ vanishes as the blocklength grows. In this paper, we give an efficient feedback encoding scheme with $n$ transmissions that achieves a positive rate for any fraction of errors $\tau < 1$ and $n\to \infty $. Additionally, we state an upper bound on the rate of asymptotically long feedback asymmetric error-correcting codes. [ABSTRACT FROM AUTHOR]

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

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest $k$ -path problem ($k$ -VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph $G$ of order $n$ , BP algorithm converges to the unique optimal solution of $k$ -VDSP on $G$ within $O(n^{2}w_{max})$ iterations, provided that the weight $w_{e}$ is nonnegative integral for each arc $e\in E(G)$ , where $w_{max}=\max \{w_{e}: e\in E(G)\}$. To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of $k$ -VDSP to the case of multiple sources or sinks. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in consumer electronics, communications, data storage systems, secret sharing, authentication codes, and association schemes. As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure computation of data between two parties. The construction of minimal linear codes with new and desirable parameters is an interesting research topic in coding theory and cryptography. Recently, Mesnager et.al. stated that “constructing linear codes with good parameters from non-weakly regular bent functions is an interesting problem.” The goal of this paper is to construct linear codes with two or three weights from non-weakly regular bent functions over finite fields and analyze the minimality of the constructed linear codes. In doing so, we draw inspiration from a paper by Mesnager in which she constructed linear codes with small weights from weakly regular bent functions based on a generic construction method. First, we recall the definitions of the subsets $B_{+}(f)$ and $B_{-}(f)$ associated with a non-weakly regular bent function $f$. Next, we construct two- or three-weight linear $p$ -ary codes on these sets using duals of the non-weakly regular bent functions that are also bent. We note that the constructed linear codes are minimal in almost all cases. Moreover, when $f$ is a non-weakly regular bent function in a certain subclass of Generalized Maiorana-McFarland bent functions, we determine the weight distributions of the corresponding linear codes. As far as we know, the construction of linear codes from non-weakly regular bent functions over finite fields is first studied in the literature by the second author in his dissertation. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

We introduce a fundamental lemma called the Poisson matching lemma, and apply it to prove one-shot achievability results for various settings, namely channels with state information at the encoder, lossy source coding with side information at the decoder, joint source-channel coding, broadcast channels, distributed lossy source coding, multiple access channels and channel resolvability. Our one-shot bounds improve upon the best known one-shot bounds in most of the aforementioned settings (except multiple access channels and channel resolvability, where we recover bounds comparable to the best known bounds), with shorter proofs in some settings even when compared to the conventional asymptotic approach using typicality. The Poisson matching lemma replaces both the packing and covering lemmas, greatly simplifying the error analysis. This paper extends the work of Li and El Gamal on Poisson functional representation, which mainly considered variable-length source coding settings, whereas this paper studies fixed-length settings, and is not limited to source coding, showing that the Poisson functional representation is a viable alternative to typicality for most problems in network information theory. [ABSTRACT FROM AUTHOR]

In the scalar dirty multiple-access channel, in addition to Gaussian noise, two additive interference signals are present, each known non-causally to a single transmitter. It was shown by Philosof et al. that for strong interferences, an independent identically distributed ensemble of codes does not achieve the capacity region. Rather, a structured-codes approach was presented that was shown to be optimal in the limit of high signal-to-noise ratios, where the sum capacity is dictated by the minimal (“bottleneck”) channel gain. In this paper, we consider the multiple-input multiple-output (MIMO) variant of this setting. In order to incorporate structured codes in this case, one can utilize matrix decompositions that transform the channel into effective parallel scalar dirty multiple-access channels. This approach, however, suffers from a “bottleneck” effect for each effective scalar channel and, therefore, the achievable rates strongly depend on the chosen decomposition. It is shown that a recently proposed decomposition, where the diagonals of the effective channel matrices are equal up to a scaling factor, is optimal at high signal-to-noise ratios, under an equal rank assumption. This approach is then extended to any number of transmitters. Finally, an application to physical-layer network coding for the MIMO two-way relay channel is presented. [ABSTRACT FROM AUTHOR]

This paper considers a massive random access problem in which a large number of sporadically active devices wish to communicate with a base station (BS) equipped with massive multiple-input multiple-output (MIMO) antennas. Each device is preassigned a unique signature sequence, and the BS identifies the active devices by detecting which sequences are transmitted. This device activity detection problem can be formulated as a maximum likelihood estimation (MLE) problem for which the sample covariance matrix of the received signal is a sufficient statistic. The goal of this paper is to characterize the feasible set of problem parameters under which this covariance based approach is able to successfully recover the device activities in the massive MIMO regime. Through an analysis of the asymptotic behaviors of MLE via its associated Fisher information matrix, this paper derives a necessary and sufficient condition on the Fisher information matrix to ensure a vanishing probability of detection error as the number of antennas goes to infinity, based on which a numerical phase transition analysis is obtained. This condition is also examined from a perspective of covariance matching, which relates the phase transition analysis to a recently derived scaling law. Further, we provide a characterization of the distribution of the estimation error in MLE, based on which the error probabilities in device activity detection can be accurately predicted. Finally, this paper studies a random access scheme with joint device activity and data detection and analyzes its performance in a similar way. [ABSTRACT FROM AUTHOR]

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]

This paper derives the optimal test channel distribution and the complete characterization of the classical Gorbunov and Pinsker (1973), Gorbunov and Pinsker (1974) nonanticipatory epsilon entropy of multivariate Gaussian Markov sources with square-error fidelity, which remained an open problem since 1974. The paper also formulates a state dependent nonanticipatory epsilon entropy, in which past reproductions are available to the decoder and not to the encoder, the test channel is specified with respect to an auxiliary (state) random process, and the reproduction process is a causal function of past reproduction and the auxiliary random process. This variation is analogous to the Wyner and Ziv (1976) and Wyner (1978) rate distortion function (RDF), of memoryless sources. It is shown that the operational rate of zero-delay codes, with past reproductions available to the decoder but not to the encoder is bounded below by the state dependent nonanticipatory epsilon entropy rate. For the case of multivariate Gaussian Markov sources with square-error fidelity, the optimal test channel distribution and the complete characterization of the state dependent of nonanticipatory epsilon entropy are derived, and also shown that that the two nonanticipatory epsilon entropies coincide. The derivations are new; they are based on structural properties of the stochastic realizations of the reproduction process that induce the optimal test channel distributions. They are derived using, achievable lower bounds on information theoretic measures, properties of mean-square estimation theory, Hadamard’s inequality, and canonical correlation coefficients of a tuple of multivariate jointly Gaussian random processes. Applications of the nonanticipatory epsilon entropy and its state dependent variation are discussed to the areas of control of unstable Gaussian systems over limited memory channels, design of causal estimators for Gaussian Markov sources with a fidelity criterion, computation of the rate loss of causal and zero-delay codes of Gaussian Markov sources with respect to non-causal codes. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate linear approximations of random functions and permutations. Our motivation is twofold. First, before the distinguishability of a practical cipher from an ideal one can be analysed, the cryptanalyst must have an accurate understanding of the statistical behaviour of the ideal cipher. Secondly, this issue has been neglected both in old and in more recent studies, particularly when multiple linear approximations are being used simultaneously. Traditional models have been based on the average behaviour and simplified using other assumptions such as independence of the linear approximations. Multidimensional cryptanalysis was introduced to avoid making artificial assumptions about statistical independence of linear approximations. On the other hand, it has the drawback of including many trivial approximations that do not contribute to the attack but just cause a waste of time and memory. We show for the first time in this paper that the trivial approximations reduce the degree of freedom of the related $\chi ^{2}$ distribution. Previously, the affine linear cryptanalysis was proposed to allow removing trivial approximations and, at the same time, admitting a solid statistical model. In this paper, we identify another type of multidimensional linear approximation, called Davies-Meyer approximation, which has similar advantages, and present full statistical models for both the affine and the Davies-Meyer type of multidimensional linear approximations. The new models given in this paper are realistic, accurate and easy to use. They are backed up by standard statistical tools such as Pearson’s $\chi ^{2}$ test and finite population correction and demonstrated to work accurately using practical examples. [ABSTRACT FROM AUTHOR]

Caching is an efficient way to reduce network traffic congestion during peak hours by storing some content at the users’ local caches. For the shared-link network with end-user-caches, Maddah-Ali and Niesen proposed a two-phase coded caching strategy. In practice, users may communicate with the server through intermediate relays. This paper studies the tradeoff between the memory size ${\mathsf M}$ and the network load ${\mathsf R}$ for the networks where a server with ${\mathsf N}$ files is connected to ${\mathsf H}$ relays (without caches), which in turn are connected to ${\mathsf K}$ users equipped with caches of ${\mathsf M}$ files. When each user is connected to a different subset of ${\mathsf r}$ relays, i.e., ${\mathsf K}= \binom {\mathsf H}{\mathsf r}$ , the system is referred to as a combination network with end-user-caches. In this work, converse bounds are derived for the practically motivated case of uncoded cache contents, that is, bits of the various files are directly pushed into the user caches without any coding. In this case, once the cache contents and the users’ demands are known, the problem reduces to a general index coding problem. This paper shows that relying on a well-known “acyclic index coding converse bound” results in converse bounds that are not tight for combination networks with end-user-caches. A novel converse bound that leverages the network topology is proposed, which is the tightest converse bound known to date. As a result of independent interest, an inequality that generalizes the well-known sub-modularity of entropy is derived. Several novel caching schemes are proposed, based on the Maddah-Ali and Niesen cache placement. These schemes leverage the structure of the combination network or/and perform interference elimination at the end-users. The proposed schemes are proved: (i) to be (order) optimal for some $({\mathsf N}, {\mathsf M}, {\mathsf H}, {\mathsf r})$ parameters regimes under the constraint of uncoded cache placement, and (ii) to outperform the state-of-the-art schemes in numerical evaluations. [ABSTRACT FROM AUTHOR]

This paper formulates a distributed computation problem, where a master asks ${\mathsf N}$ distributed workers to compute a linearly separable function. The task function can be expressed as ${\mathsf K}_{\mathrm{ c}}$ linear combinations of ${\mathsf K}$ messages, where each message is a function of one dataset. Our objective is to find the optimal tradeoff between the computation cost (number of uncoded datasets assigned to each worker) and the communication cost (number of symbols the master must download), such that from the answers of any ${\mathsf N}_{\mathrm{ r}}$ out of ${\mathsf N}$ workers the master can recover the task function with high probability, where the coefficients of the ${\mathsf K}_{\mathrm{ c}}$ linear combinations are uniformly i.i.d. over some large enough finite field. The formulated problem can be seen as a generalized version of some existing problems, such as distributed gradient coding and distributed linear transform. In this paper, we consider the specific case where the computation cost is minimum, and propose novel achievability schemes and converse bounds for the optimal communication cost. Achievability and converse bounds coincide for some system parameters; when they do not match, we prove that the achievable distributed computing scheme is optimal under the constraint of a widely used ‘cyclic assignment’ scheme on the datasets. Our results also show that when ${\mathsf K}= {\mathsf N}$ , with the same communication cost as the optimal distributed gradient coding scheme proposed by Tandon et al. from which the master recovers one linear combination of ${\mathsf K}$ messages, our proposed scheme can let the master recover any additional ${\mathsf N}_{\mathrm{ r}}-1$ independent linear combinations of messages with high probability. [ABSTRACT FROM AUTHOR]

The article lists the recipients of the 2017 IEEE Communications Society and Information Theory Society Joint Paper Award including Ido Tal and Alexander Vardy.

Communication over a random-parameter quantum channel when the decoder is required to reconstruct the parameter sequence is considered. We study scenarios that include either strictly-causal, causal, or non-causal channel side information (CSI) available at the encoder, and also when CSI is not available. This model can be viewed as a form of quantum metrology, and as the quantum counterpart of the classical rate-and-state channel with state estimation at the decoder. Regularized formulas for the capacity-distortion regions are derived. In the special case of measurement channels, single-letter characterizations are derived for the strictly-causal and causal settings. Furthermore, in the more general case of entanglement-breaking channels, a single-letter characterization is derived when CSI is not available. As a consequence, we obtain regularized formulas for the capacity of random-parameter quantum channels with CSI, generalizing previous results by Boche et al., 2016, on classical-quantum channels. Bosonic dirty paper coding is introduced as a consequence, where we demonstrate that the optimal coefficient is not necessarily that of minimum mean-square error estimation as in the classical setting. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized pair weights of an $[n, k]$ -linear code over the finite field $\mathbb {F}_{q}$ and the notion of pair $r$ -equiweight codes, where $1\le r\le k-1$. Some basic properties of generalized pair weights of linear codes over finite fields are derived. We obtain a necessary and sufficient condition for an $[n,k]$ -linear code to be a pair equiweight code, and we characterize pair $r$ -equiweight codes for any $1\le r\le k-1$. A necessary and sufficient condition for a linear isomorphism to preserve pair weights between two linear codes is obtained. At the end of this paper, an application of generalized pair weights of linear codes to symbol-pair read wire-tap channels of type II is introduced. [ABSTRACT FROM AUTHOR]

We consider the fundamental problem of matching a template to a signal. We do so by M-estimation, which encompasses procedures that are robust to gross errors (i.e., outliers). Using standard results from empirical process theory, we derive the convergence rate and the asymptotic distribution of the M-estimator under relatively mild assumptions. We also discuss the optimality of the estimator, both in finite samples in the minimax sense and in the large-sample limit in terms of local minimaxity and relative efficiency. Although most of the paper is dedicated to the study of the basic shift model in the context of a random design, we consider many extensions towards the end of the paper, including more flexible templates, fixed designs, the agnostic setting, and more. [ABSTRACT FROM AUTHOR]

Distributed Arithmetic Coding (DAC) is a practical realization of Slepian-Wolf coding that partitions source space into cosets. Coset Cardinality Spectrum (CCS) is an important property of DAC that was defined in our previous work. In this paper, we give two applications of CCS. First, we find that DAC bitstream is not compact. The rate loss of DAC bitstream is caused by two factors: unequal coset partitioning and bit indivisibility. It is proved that as code length goes to infinity, the expected value of bit-indivisibility rate loss will tend to 0.5 for any irrational Rate Change Step (RCS), where the RCS refers to the rate change when one source bit is flipped. With the help of CCS, the bit-indivisibility rate loss can be compensated to some extend. Especially, for any irrational RCS, as code length goes to infinity, the expected value of the remaining bit-indivisibility rate loss after compensation will tend to about 0.47. The second application of CCS is DAC decoder design. We derive the formula of path metric and find that in the original paper on DAC, the intuitive formula of path metric is not correct. The backward-replacing algorithm is proposed to make full use of memory. Experimental results confirm the correctness of theoretical analyses. [ABSTRACT FROM AUTHOR]