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The main problem of constant-dimension subspace coding is to determine the maximal possible size ${\mathbf{A}}_{q}(n,d,k)$ of a set of $k$ -dimensional subspaces in ${\mathbf{F}}_{q}^{n}$ such that the subspace distance satisfies $d(U,V) \geq d$ for any two different subspaces $U$ and $V$ in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of $L_{1}(L_{2}(x))-x$ where $L_{1}$ and $L_{2}$ are $q$ -polynomials over the extension field ${\mathbf{F}}_{q^{n}}$. New lower bounds for ${\mathbf{A}}_{q}(4k,2k,2k)$ , ${\mathbf{A}}_{q}(4k+2,2k,2k+1)$ , and ${\mathbf{A}}_{q}(4k+2,2(k-1),2k+1)$ are presented. Many new constant-dimension subspace codes better than previously best known codes with small parameters are constructed. [ABSTRACT FROM AUTHOR]

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate $R$ and minimum distance $\delta $ of a binary code. The best known tradeoff is the Gilbert–Varshamov bound, and says that for every $\delta \in (0, 1/2)$ , there are codes with minimum distance $\delta $ and rate $R = {R_{\mathsf {GV}}}(\delta) > 0$ (for a certain simple function ${R_{\mathsf {GV}}}(\cdot)$ ). In this paper, we show that the Gilbert–Varshamov bound can be achieved by codes, which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1) Local testing: for all $\delta \in (0,1/2)$ and all $R < {R_{\mathsf {GV}}}(\delta)$ , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally testable with $\mathsf {quasipolylog}(n)$ query complexity. 2) Local correction: for all $\epsilon > 0$ , for all $\delta < 1/2$ sufficiently large, and all $R < (1-\epsilon) {R_{\mathsf {GV}}}(\delta)$ , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally correctable from $({\delta }/{2}) - o(1)$ fraction errors with $O(n^{\epsilon })$ query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen’s random concatenation technique and the best known locally testable codes by Kopparty et al. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami–Sudan–Indyk list decoding strategy for concatenated codes, Alon–Edmonds–Luby distance amplification, and the local list-decodability, local list-recoverability, and local testability of Reed–Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code. [ABSTRACT FROM AUTHOR]

This paper contains two main contributions concerning the asymmetric broadcast channel. The first is an analysis of the exact random coding error exponents for both users, and the second is the derivation of universal decoders for both users. These universal decoders are certain variants of the maximum mutual information universal decoder, and they achieve the corresponding random coding exponents of optimal decoding. In addition, we introduce some lower bounds, which involve optimizations over very few parameters, unlike the original, exact exponents, which involve minimizations over auxiliary probability distributions. Numerical results for the binary symmetric broadcast channel show improvements over previously derived error exponents for the same model. [ABSTRACT FROM AUTHOR]

For non-binary codes the Elias bound is a good upper bound for the asymptotic information rate at low-relative minimum distance, whereas the Plotkin bound is better at high-relative minimum distance. In this paper, we obtain a hybrid of these bounds, which improves both. This in turn is based on the anticode bound, which is a hybrid of the Hamming and Singleton bounds and improves both bounds. The question of convexity of the asymptotic rate function is an important open question. We conjecture a much weaker form of the convexity, and we show that our bounds follow immediately if we assume the conjecture. [ABSTRACT FROM AUTHOR]

Compared with classical block codes, list decoding rank-metric codes efficiently seems more difficult. The evidences to support this view include: 1) so far the only known efficient list decoding of rank-metric codes \mathcal C gives decoding radius beyond (1-R)/2 with positive rate R when the ratio of the number of rows over the number of columns is extremely small, 2) the Johnson bound for rank-metric codes does not exist as opposed to classical codes, and 3) the Gabidulin codes of square matrices cannot be list decoded beyond half of the minimum distance. Although the list decodability of random rank-metric codes and the limits to the list decodability have been determined completely, little work on efficient list decoding of rank-metric codes has been done. The only known efficient list decoding of rank-metric codes 1-R- \varepsilon with positive rate R is extremely small, i.e., O(\varepsilon ^{2}) , where \rho (\mathcal {C}) denotes the ratio of the number of rows over the number of columns of \rho (\mathcal C) close to 1. The main purpose of this paper is to explicitly construct a class of rank-metric codes \mathcal C with the ratio \rho (\mathcal C) up to 1/2 and efficiently list decode these codes beyond unique decoding radius (1-R)/2 . Furthermore, encoding and list decoding algorithms run in polynomial time {\mathrm{ poly}}(n,\exp (1/{ \varepsilon })) . The list size can be reduced to O(1/{ \varepsilon })$ by randomizing the algorithm. Our key idea is to employ bivariate polynomials $f(x,y)$ , where $f$ is linearized in variable $y$ and the variable $x$ is used to “fold” the code. In other words, the rows are used to correct rank errors and the columns are used to “fold” the code to enlarge the decoding radius. Apart from the above algebraic technique, we have to prune down the list. The algebraic idea enables us to pin down the messages into a structured subspace whose dimension is linear in the number $n$ of columns. This “periodic” structure allows us to pre-encode the messages to prune down the list. More precisely, we use subspace design introduced by Guruswami and Xing to obtain a deterministic algorithm with a larger constant list size and employ hierarchical subspace-evasive sets introduced by Guruswami et al. to obtain a randomized algorithm with a smaller constant list size. [ABSTRACT FROM PUBLISHER]

This paper studies channel coding for the discrete memoryless multiple-access channel with a given (possibly suboptimal) decoding rule. A multi-letter successive decoding rule depending on an arbitrary non-negative decoding metric is considered, and achievable rate regions and error exponents are derived both for the standard MAC (independent codebooks), and for the cognitive MAC (one user knows both messages) with superposition coding. In the cognitive case, the rate region and error exponent are shown to be tight with respect to the ensemble average. The rate regions are compared with those of the commonly considered decoder that chooses the message pair maximizing the decoding metric, and numerical examples are given for which successive decoding yields a strictly higher sum rate for a given pair of input distributions. [ABSTRACT FROM AUTHOR]

In this paper, a new wiretap channel model is proposed, where the legitimate transmitter and receiver communicate over a discrete memoryless channel. The wiretapper has perfect access to a fixed-length subset of the transmitted code word symbols of her choosing. Additionally, she observes the remainder of the transmitted symbols through a discrete memoryless channel. This new model subsumes the classical wiretap channel and wiretap channel II with noisy main channel as its special cases, and is termed as the generalized wiretap channel for that reason. The strong secrecy capacity of the proposed channel model is identified. Achievability is established by solving a dual secret key agreement problem in the source model, and converting the solution to the original channel model using probability distribution approximation arguments. In the dual problem, a source encoder and decoder, who observe random sequences independent and identically distributed according to the input and output distributions of the legitimate channel in the original problem, communicate a confidential key over a public error-free channel using a single forward transmission, in the presence of a compound wiretapping source who has perfect access to the public discussion. The security of the key is guaranteed for the exponentially many possibilities of the subset chosen at the wiretapper by deriving a lemma which provides a doubly-exponential convergence rate for the probability that, for a fixed choice of the subset, the key is uniform and independent from the public discussion and the wiretapping source’s observation. The converse is derived by using Sanov’s theorem to upper bound the secrecy capacity of the generalized wiretap channel by the secrecy capacity when the tapped subset is randomly chosen by nature. [ABSTRACT FROM PUBLISHER]

Let W=\w1, \ldots , wr\ be a set of positive integers, \lambda c a positive integer, \Lambda a=(\lambda a^{(1)}, \ldots \lambda a^{(r)}) an r -tuple of positive integers, and Q=(q_{1}, \ldots q_{r}) an r -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code, (n, W, \Lambda a, \lambda c, Q) -OOC, for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal (n, W, \Lambda a, 1, Q) -OOCs with unequal auto-correlation constraints for W=\3, 4\ and {3, 5}, while little is known on optimal (n, W, \Lambda a, 1, Q) -OOCs for |W|\geq 3 . In this paper, we focus our main attentions on (n, \{3, 4, 5\}, \Lambda _{a}, 1, Q) -OOCs with \Lambda _{a}\in \{(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)\} . Tight upper bounds on the maximum code size of (n, \3, 4, 5\, \Lambda a, 1, Q) -OOCs are obtained, and infinite classes of optimal (n, \3, 4, 5\, \Lambda a, 1, Q) -OOCs are constructed. [ABSTRACT FROM PUBLISHER]

The Johnson-type upper bound on the maximum size of a code of length n , distance d=2w-1 , and constant composition \overline w is \lfloor \dfrac \vphantom R^.nw1\rfloor , where w is the total weight and w_{1} is the largest component of {\overline {w}} . Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let Nccc({\overline {w}}) be the smallest such length. The determination of Nccc({\overline {w}}) is trivial for binary codes. This paper provides a lower bound on Nccc({\overline {w}}) , which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of Nccc({\overline {w}}) , for all $q$ -ary constant-composition codes, provided that 3w_{1}\geq w with finite possible exceptions. [ABSTRACT FROM AUTHOR]

BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding. Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a given maximum number, and provide new insights on the decoding performance. Specifically, for a fixed number of input symbols and a range of the number of batches used for decoding, we obtain recursive formulae to calculate the stopping time distribution of BP decoding and the inactivation probability in inactivation decoding. We also find that both the failure probability of BP decoding and the expected number of inactivations in inactivation decoding can be expressed in a power-sum form where the number of batches appears only as the exponent. This power-sum expression reveals clearly how the decoding failure probability and the expected number of inactivation decrease with the number of batches. When the number of batches used for decoding follows a Poisson distribution, we further derive recursive formulas with potentially lower computational complexity for both decoding algorithms. For the BP decoder that consumes batches one by one, three formulae are provided to characterize the expected number of consumed batches until all the input symbols are decoded. [ABSTRACT FROM PUBLISHER]

Caching plays an important role in reducing the backbone traffic when serving high-volume multimedia content. Recently, a new class of coded caching schemes have received significant interest, because they can exploit coded multi-cast opportunities to further reduce backbone traffic. Without considering file popularity, prior works have characterized the fundamental performance limits of coded caching through a deterministic worst-case analysis. However, when heterogeneous file popularity is considered, there remain open questions regarding the fundamental limits of coded caching performance. In this paper, for an arbitrary popularity distribution, we first derive a new information-theoretic lower bound on the expected transmission rate of any coded caching schemes. We then show that a simple coded-caching scheme attains an expected transmission rate that is at most a constant factor away from the lower bound. Unlike other existing studies, the constant factor that we derived is independent of the popularity distribution. [ABSTRACT FROM PUBLISHER]

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum–minimum-distance codes. Computing a code’s density follows from computing: 1) the normalized volume of a metric ball and 2) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes. [ABSTRACT FROM PUBLISHER]

Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes, and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error correction in network coding. In this paper, we study their security behavior against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst case information leakage to the wire tapper; 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal) and use the previous bounds to estimate their higher GRWs; 3) we show that all linear (over the extension field) codes, whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence; and 4) we show that the information leaked to a wire tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: conditions to be rank equivalent to Cartesian products of linear codes and conditions to be rank degenerate, duality properties, and MRD ranks. [ABSTRACT FROM PUBLISHER]

In this paper we study stopping sets of AG codes over a family of curves, denoted by Xƒ,g, that includes several important curves with applications in coding theory. We present results concerning stopping sets of one-point and m-point codes over Xƒ,g , generalizing some results for Hermitian codes presented by Anderson and Matthews. [ABSTRACT FROM AUTHOR]

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 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]

Algebraic network information theory is an emerging facet of network information theory, studying the achievable rates of random code ensembles that have algebraic structure, such as random linear codes. A distinguishing feature is that linear combinations of codewords can sometimes be decoded more efficiently than codewords themselves. The present work further develops this framework by studying the simultaneous decoding of multiple messages. Specifically, consider a receiver in a multi-user network that wishes to decode several messages. Simultaneous joint typicality decoding is one of the most powerful techniques for determining the fundamental limits at which reliable decoding is possible. This technique has historically been used in conjunction with random i.i.d. codebooks to establish achievable rate regions for networks. Recently, it has been shown that, in certain scenarios, nested linear codebooks in conjunction with “single-user” or sequential decoding can yield better achievable rates. For instance, the compute–forward problem examines the scenario of recovering $L \le K$ linear combinations of transmitted codewords over a $K$ -user multiple-access channel (MAC), and it is well established that linear codebooks can yield higher rates. This paper develops bounds for simultaneous joint typicality decoding used in conjunction with nested linear codebooks, and applies them to obtain a larger achievable region for compute–forward over a $K$ -user discrete memoryless MAC. The key technical challenge is that competing codeword tuples that are linearly dependent on the true codeword tuple introduce statistical dependencies, which requires careful partitioning of the associated error events. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, mathematics and information theory. In this paper we deduce a new method for constructing APN functions by studying the isotopic equivalence, concept defined for quadratic planar functions in fields of odd characteristic. In particular, we construct a family of quadratic APN functions which provides a new example of an APN mapping over ${\mathbb F}_{2^{9}}$ and includes an example of another APN function $x^{9}+ \mathop {\mathrm {Tr}}\nolimits (x^{3})$ over ${\mathbb F}_{2^{8}}$ , known since 2006 and not classified up to now. We conjecture that the conditions for this family are satisfied by infinitely many APN functions. [ABSTRACT FROM AUTHOR]

The Slepian-Wolf (SW) coding system is a source coding system with two encoders and a decoder, where these encoders independently encode source sequences from two correlated sources into codewords, and the decoder reconstructs both source sequences from the codewords. In this paper, we consider the situation in which the SW coding system is asynchronous, i.e., each encoder samples a source sequence with some unknown delay. We assume that delays are unknown but maximum and minimum values of possible delays are known to encoders and the decoder. We also assume that sources are discrete stationary memoryless and the probability mass function (PMF) of the sources is unknown but the system knows that it belongs to a certain set of PMFs. For this asynchronous SW coding system, we clarify the achievable rate region which is the set of rate pairs of encoders such that the decoding error probability vanishes as the blocklength tends to infinity. We show that this region does not always coincide with that of the synchronous SW coding system in which each encoder samples a source sequence without any delay. [ABSTRACT FROM AUTHOR]

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field ${\mathbb F}_{q}$ is competely solved for $q$ is even. In the literature, there are many known constructions of MDS self-dual codes for $q$ is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when $q$ is not a square. [ABSTRACT FROM AUTHOR]

Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree ≤ 6 over finite fields $\mathbb {F}_{q}$ for all $q$ and the inverses of all PPs of degree 7 over $\mathbb {F}_{2^{n}}$. The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas’ theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields. [ABSTRACT FROM AUTHOR]

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for $x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$ being involutions over the finite field ${\mathbb F}_{q}$ , where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ from given involutions over some subgroups of ${\mathbb F}_{q}^{*}$ by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ are obtained. [ABSTRACT FROM AUTHOR]