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In this paper, the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of signals in both forward and feedback channels. It is shown that the generalized Schalkwijk–Kailath scheme, a capacity-achieving coding scheme for the Gaussian feedback channel, achieves the same maximum rate for the same channel even with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the Gaussian feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. Our result implies that as the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel. [ABSTRACT FROM AUTHOR]

Because of their efficient encoding and decoding algorithms, cyclic codes—an interesting class of linear codes—are widely used in communication systems, storage devices, and consumer electronics. BCH codes form a special class of cyclic codes, and are usually among the best cyclic codes. A subclass of good BCH codes is the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The main objective of this paper is to study the dimension and minimum distances of a subclass of the narrow-sense primitive BCH codes with design distance \delta =(q-\ell 0)q^{m-\ell 1-1}-1 for certain pairs (\ell 0, \ell 1) , where 0 \leq \ell 0 \leq q-2 and 0 \leq \ell 1 \leq m-1 . The parameters of other related classes of BCH codes are also investigated, and some open problems are proposed in this paper. [ABSTRACT FROM PUBLISHER]

We investigate a pilot allocation problem in wireless networks over Markovian fading channels. In wireless systems, the channel state information (CSI) is collected at the base station, in particular, this paper considers a pilot-aided channel estimation method (TDD mode). Typically, there are less available pilots than users, hence at each slot the scheduler needs to decide an allocation of pilots to users with the goal of maximizing the long-term average throughput. There is an inherent tradeoff in how the limited pilots are used: assign a pilot to a user with up-to-date CSI and good channel condition for exploitation, or assign a pilot to a user with outdated CSI for exploration. As we show, the arising pilot allocation problem is a restless bandit problem and thus its optimal solution is out of reach. In this paper, we propose an approximation based on the Lagrangian relaxation method, which provides a low-complexity Whittle index policy. We prove this policy to be asymptotically optimal in the many users regime (when the number of users in the system and the available pilots for channel sensing grow large). We evaluate the performance of Whittle’s index policy in various scenarios and illustrate its remarkably good performance for small number of users, where it is not guaranteed to be optimal. [ABSTRACT FROM AUTHOR]

A nonlinear feedback shift register (NFSR) of $n$ -stage is called irreducible if, the family of output sequences of any NFSR of stage less than $n$ is not included in that of the NFSR. Tian and Qi in this paper [IEEE-IT, 2013(6),4006–4012] gave a lower bound on the density of irreducible NFSRs. In this paper, we improve their lower bound and also give an upper bound on the density of irreducible NFSRs. Moreover, the gap between our upper and lower bounds is less than 0.04. [ABSTRACT FROM PUBLISHER]

This paper analyzes the asymptotic performances of fully distributed sequential hypothesis testing procedures as the type-I and type-II error rates approach zero, in the context of a sensor network without a fusion center. In particular, the sensor network is defined by an undirected graph, where each sensor can observe samples over time, access the information from the adjacent sensors, and perform the sequential test based on its own decision statistic. Different from most literature, the sampling process and the information exchange process in our framework take place simultaneously (or, at least in comparable time-scales), thus cannot be decoupled from one another. Our goal is to achieve second-order asymptotically optimal performance at every sensor, i.e., the average detection delay is within a constant gap from the centralized optimal sequential test as the error rates approach zero for the fixed number of sensors. To that end, a type of test procedure that resembles the well-known sequential probability ratio test (SPRT), termed as distributed SPRT (DSPRT) in this paper, is studied based on two message-passing schemes, respectively. The first scheme features the dissemination of the raw samples. In specific, every sample propagates over the network by being relayed from one sensor to another until it reaches all the sensors in the network. Although the sample propagation-based DSPRT is shown to yield the asymptotically optimal performance at each sensor, it incurs excessive inter-sensor communication overhead due to the exchange of raw samples with index information. The second scheme adopts the consensus algorithm, where the local decision statistic is exchanged between sensors instead of the raw samples, thus significantly lowering the communication requirement compared with the first scheme. In particular, the decision statistic for DSPRT at each sensor is updated by the weighted average of the decision statistics in the neighborhood at every message-passing step. We show that, under certain regularity conditions, the consensus algorithm-based DSPRT also yields the second-order asymptotically optimal performance at all sensors given a fixed number of sensors. Our asymptotic analyses of the two message-passing-based DSPRTs are then corroborated by simulations using the Gaussian and Laplacian samples. [ABSTRACT FROM AUTHOR]

A code is said to be an $r$ -local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most r$ other coordinates. When some of the r$ coordinates are also erased, the r$ -local LRC cannot accomplish the local repair, which leads to the concept of (r,\delta)$ -locality. A q is said to have (r, \delta)$ -locality ( \delta \ge 2$ ) if for each coordinate i with support containing $i$ , whose length is at most $r + \delta - 1$ , and whose minimum distance is at least $\delta$ . The $(r, \delta)$ -LRC can tolerate $\delta -1$ erasures in every local code (i.e., punctured subcode), which degenerates to an $r$ -local LRC when $\delta =2$ . A $q$ -ary $(r,\delta)$ LRC is called optimal if it meets the singleton-like bound for $(r,\delta)$ -LRCs. A class of optimal $q$ -ary cyclic $r$ -local LRCs with lengths $n\mid q-1$ were constructed by Tamo, Barg, Goparaju, and Calderbank based on the $q$ -ary Reed-Solomon codes. In this paper, we construct a class of optimal $q$ -ary cyclic $(r,\delta)$ -LRCs ( $\delta \ge 2$ ) with length $n\mid q-1$ , which generalizes the results of Tamo et al. Moreover, we construct a new class of optimal $q$ -ary cyclic $r$ -local LRCs with lengths $n\mid q+1$ and a new class of optimal $q$ -ary cyclic $(r,\delta)$ -LRCs ( $\delta \ge 2$ ) with lengths $n\mid q+1$ . The constructed optimal LRCs with length $n=q+1$ have the best-known length for a given finite field with size $q$ when the minimum distance is larger than 4. [ABSTRACT FROM AUTHOR]

In this paper, we show that the maximum number of bent component functions of a vectorial function F:GF(2)^n\to GF(2)^n is 2^n-2^n/2 . We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F\in GF(2^n)[x] , where F has only a few terms. The only known power functions having such a large number of bent components are x^{d} , where d=2^{n/2}+1 . In this paper, we show that the binomials F^i(x)=x^2^i(x+x^2^n/2) also have such a large number of bent components, and these binomials are inequivalent to the monomials x^2^n/2+1 if 0

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]

The paper deals with linear information inequalities valid for entropy functions induced by discrete random variables. Specifically, the so-called conditional Ingleton inequalities are in the center of interest: these are valid under conditional independence assumptions on the inducing random variables. We discuss five inequalities of this particular type, four of which has appeared earlier in the literature. Besides the proof of the new fifth inequality, simpler proofs of (some of) former inequalities are presented. These five information inequalities are used to characterize all conditional independence structures induced by four discrete random variables. [ABSTRACT FROM AUTHOR]

This paper considers the problem of minimum cost communication of correlated sources over a network with multiple sinks, which consists of distributed source coding followed by routing. We introduce a new routing paradigm called dispersive information routing (DIR), wherein the intermediate nodes are allowed to split a packet and forward subsets of the received bits on each of the forward paths. This paradigm opens up a rich class of research problems, which focus on the interplay between encoding and routing in a network. Unlike conventional routing methods such as in [1], DIR ensures that each sink receives just the information needed to reconstruct the sources it is required to reproduce. We demonstrate using simple examples that the proposed approach offers better asymptotic performance than conventional routing techniques. We show that, under certain assumptions on the cost function, the problem of finding the minimum cost under DIR essentially reduces to characterizing an achievable rate region for a new multiterminal information theoretic setup. While it is possible to derive an achievable region for this setup using prior results from general multiterminal source coding [3], these techniques do not exploit the underlying problem structure and thereby lead to suboptimal regions. In this paper, we propose a new coding scheme, using principles from multiple descriptions encoding [2], and show that it strictly improves upon a corresponding variant of coding scheme in [3]. We further show that the new coding scheme achieves the complete rate region for certain special cases of the general setup and thereby achieves the minimum communication cost under this routing paradigm. [ABSTRACT FROM PUBLISHER]

This paper studies the problem of the distributed compression of correlated sources with an action-dependent joint distribution. This class of problems is, in fact, an extension of the Slepian–Wolf model, but where cost-constrained actions taken by the encoder or the decoder affect the generation of one of the sources. The purpose of this paper is to study the impact of actions on the achievable rates. In particular, two cases where transmission occurs over a rate-limited link are studied; case A for actions taken at the decoder and case B where actions are taken at the encoder. A complete single-letter characterization of the set of achievable rates is given in both cases. Furthermore, a network coding setup for the case where actions are taken at the encoder is investigated. The sources are generated at different nodes of the network and are required at a set of terminal nodes, yet transmission occurs over a general, acyclic, directed network. For this setup, generalized cut-set bounds are derived, and a full characterization of the set of achievable rates using single-letter expressions is provided. For this scenario, random linear network coding is proved to be optimal, even though this is not a classical multicast problem. In addition, two binary examples are investigated and demonstrate how actions taken at different nodes of the system have a significant effect on the achievable rate region, when compared with a naive time-sharing strategy. [ABSTRACT FROM AUTHOR]

In general, it is hard to find the cross correlation distribution between two sequences. In particular, there are only few cases that the cross correlation is three-valued. In this paper, new pairs of binary sequences with three cross correlation values are presented. These two sequences have different lengths: 2^2k-1 and 3(2^k-1) , respectively. We determine the cross correlation distribution between these two sequences. We also give the possible correlation values for the shift decimated sequences. The magnitude of cross correlation values is shown to be low. The techniques presented in this paper may be useful to handle other similar cases. [ABSTRACT FROM PUBLISHER]

Golay complementary sequences and complementary sets have been proposed to deal with the high peak-to-average power ratio (PAPR) problem in orthogonal frequency division multiplexing (OFDM) system. The existing constructions of complementary sets based on generalized Boolean functions are limited to lengths, which are powers of two. In this paper, we propose novel constructions of binary and nonbinary complementary sets of non-power-of-two length. Regardless of whether or not the length of the complementary set is a power of two, its PAPR is still upper bounded by the size of the complementary set. Therefore, the constructed complementary sets can be applied in practical OFDM systems where the number of used subcarriers is not a power of two. In addition, while the binary Golay complementary pairs exist only for limited lengths, the constructed binary complementary sets of size 4 exist for more lengths with PAPR at most 4. [ABSTRACT FROM PUBLISHER]

Fire codes are cyclic codes generated by the product of two polynomials: a binomial that characterizes the code’s guaranteed burst-correcting capability and an irreducible polynomial that characterizes the code length. However, the true burst-correcting capability of a Fire code may exceed its guaranteed burst-correcting capability, which can be thought of as the designed burst-correcting capability of the Fire code. The true burst-correcting capability of a Fire code depends on the irreducible polynomial used in code construction. In this paper, the maximum true burst-correcting capabilities of Fire codes are considered. Fire codes are classified based on three parameters: their designed burst-correcting capabilities, the least common multiple of the periods of the binomials, and the irreducible polynomials used in their constructions, and the ratios of the periods of primitive polynomials of the same degrees as the irreducible polynomials to the periods of the irreducible polynomials. It is shown that the maximum true burst-correcting capability of each class, which pertains to an infinite number of codes, can be determined by checking whether or not a finite number of incongruences have a solution. It is also shown that in each class, there is an infinite number of Fire codes, with increasing code lengths, that attain this maximum. The maximum true burst-correcting capabilities of several classes of Fire codes are determined. It is also shown that there are infinite sequences of irreducible polynomials that generate cyclic codes of rates approaching one and with burst-correcting capabilities that exceed any given number. [ABSTRACT FROM AUTHOR]

Maximum-distance separable (MDS) convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. There are very few general algebraic constructions of MDS convolutional codes. In this paper, we construct a large family of partial-unit-memory MDS convolutional codes over \mathbb Fq with flexible parameters. Compared with the previous work, the field size $q$ required to define these codes is much smaller. The construction also leads to many new strongly MDS convolutional codes, an important subclass of MDS convolutional codes. Some examples are presented at the end of this paper. [ABSTRACT FROM AUTHOR]

The theory of compressed sensing (CS) asserts that an unknown signal x\in \mathbb R ^p can be accurately recovered from an underdetermined set of n linear measurements with n\ll p provided that x is sufficiently sparse. However, in applications, the degree of sparsity is typically unknown, and the problem of directly estimating \|x\|_{0} has been a longstanding gap between theory and practice. A closely related issue is that \|x\|_{0} is a highly idealized measure of sparsity, and for real signals with entries not equal to 0, the value \|x\|0=p is not a useful description of compressibility. In our previous conference paper [1] that examined these problems, we considered an alternative measure of soft sparsity, \|x\|1^{2}/\|x\|2^{2} , and designed a procedure to estimate \|x\|1^{2}/\|x\|2^{2} that does not rely on sparsity assumptions. This paper offers a new deconvolution-based method for estimating unknown sparsity, which has wider applicability and sharper theoretical guarantees. In particular, we introduce a family of entropy-based sparsity measures sq(x)=\big ({\|x\|q}/{\|x\|1}\big )^{({q}/{1-q})} parameterized by q\in [0,\infty ] . This family interpolates between \|x\|_{0}=s_{0}(x) and \|x\|_{1}^{2}/\|x\|_{2}^{2}=s_{2}(x) as q ranges over [0, 2]. For any q\in (0,2]\setminus \{1\} , we propose an estimator \hat {s}_{q}(x) whose relative error converges at the dimension-free rate of 1/\sqrt n , even when p/n\to \infty . Our main results also describe the limiting distribution of \hat sq(x) , as well as some connections to basis pursuit denosing, the Lasso, deterministic measurement matrices, and inference problems in CS. [ABSTRACT FROM PUBLISHER]

Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in {−1, 1}). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $\text {ADF}(f)$ , is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $\text {CDF}(f,g)$ , is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of $\text {CDF}(f,g)$ and the geometric mean of $\text {ADF}(f)$ and $\text {ADF}{(g)}$ must be at least 1. For randomly selected pairs of long binary sequences, this quantity is typically around 2. In this paper, we show that Pursley and Sarwate’s bound is met for binary sequences precisely when $(f,g)$ is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite abelian group. If $f: G\rightarrow {\mathbf{C}} $ is a nonzero function with Fourier transform $\hat {f}$ , the Donoho-Stark uncertainty principle states that $| {\mathrm{ supp}}(f)|| {\mathrm{ supp}}(\hat {f})|\geq |G|$. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, if $f: G\rightarrow {\mathbf{F}} $ is a non-zero function from $G$ to a field ${\mathbf{F}}$ , and if $f$ has a Fourier transform $\hat {f}$ , the sharpened uncertainty principle states that $| {\mathrm{ supp}}(f)|| {\mathrm{ supp}}(\hat {f})|\geq |G|+| {\mathrm{ supp}}(f)|-|H({\mathrm{ supp}}(f))|$ , where $H({\mathrm{ supp}}(f))$ is the stabilizer of ${\mathrm{ supp}}(f)$ in $G$. [ABSTRACT FROM AUTHOR]

Recently, Etzion et al. introduced metrics on ${\mathbb{F}}_{2}^{n}$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs, which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between the weighted poset metrics and the directed graph-based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices, which admit the extended Hamming code $\tilde {\mathcal {H }}_{3}$ to be a two-perfect code. We also construct some families of such structures for any $k \geq 3$ , which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps. [ABSTRACT FROM AUTHOR]

Mathematical tools that are widely employed for the distribution-free analysis of the finite-sample performance of empirical estimators—like Efron-Stein’s inequality, or Azuma’s martingale inequality and derivatives—rely on the provision of tight bounds for the Lipschitz constants of the estimators. Lipschitz constants can be easily derived for simple functionals of the empirical measures, such as the Shannon or the Tsallis entropy. However, obtaining tight bounds for more generalized entropy functionals which cannot be decomposed into sums of identical terms is much more involved. The goal in this paper is the derivation of (empirical) Lipschitz constants for the maximum likelihood estimator of the family of Renyi entropies of order $\lambda >0$ , $\lambda \neq 1$. Analytic solutions for the optimal constants are obtained for the most important special cases; namely, for $0 < \lambda < 1$ , for $1 < \lambda < 2$ as well as for $\lambda =2$ (collision entropy) and $\lambda = 3$. For the remaining cases where no analytic solution is obtained (i.e., $\lambda > 2$ , $\lambda \neq 3$), an efficient way to compute the optimal constants is derived by reducing the complexity of the underlying optimization problem from exponential ${\varOmega}(n^{\| \mathcal {A} \|-1})$ to linear $\mathcal {O}(n)$ , where $n$ the number of available samples and $\| \mathcal {A} \| \geq 2$ the number of symbols of the source (with the assumption $\| \mathcal {A} \|$ does not grow with $n$). The optimal constants are subsequently used for the distribution-free performance analysis of the maximum likelihood estimator of Renyi entropy; this includes variance and concentration bounds, both under the assumption of independence as well as under strong mixing conditions. [ABSTRACT FROM AUTHOR]

The random variable simulation problem consists in using a $\boldsymbol {k}$ -dimensional i.i.d. random vector $\boldsymbol {X}^{\boldsymbol {k}}$ with distribution $\boldsymbol {P}_{\boldsymbol {X}}^{\boldsymbol {k}}$ to simulate an $\boldsymbol {n}$ -dimensional i.i.d. random vector $\boldsymbol {Y}^{\boldsymbol {n}}$ so that its distribution is approximately $\boldsymbol {Q}_{\boldsymbol {Y}}^{\boldsymbol {n}}$. In contrast to previous works, in this paper, we consider the standard Rényi divergence and two variants of all orders to measure the level of approximation. These two variants are the max-Rényi divergence $\boldsymbol {D}_{\boldsymbol {\alpha }}^{\mathsf {max}}(\boldsymbol {P},\boldsymbol {Q})$ and the sum-Rényi divergence $\boldsymbol {D}_{\boldsymbol {\alpha }}^{+}(\boldsymbol {P},\boldsymbol {Q})$. When $\boldsymbol {\alpha }=\infty $ , these two measures are strong because for any $\boldsymbol {\epsilon }\ge 0$ , $\boldsymbol {D}_{\infty }^{\mathsf {max}}(\boldsymbol {P},\boldsymbol {Q})\leq \boldsymbol {\epsilon }$ or $\boldsymbol {D}_{\infty }^{+}(\boldsymbol {P},\boldsymbol {Q})\leq \boldsymbol {\epsilon }$ implies $\boldsymbol {e}^{-\boldsymbol {\epsilon }}\leq ({\boldsymbol {P}(\boldsymbol {x})}/{\boldsymbol {Q}(\boldsymbol {x})})\leq \boldsymbol {e}^{\boldsymbol {\epsilon }}$ for all $\boldsymbol {x}$. Under these Rényi divergence measures, we characterize the asymptotics of normalized divergences as well as the Rényi conversion rates. The latter is defined as the supremum of ${\boldsymbol {n}}/{\boldsymbol {k}}$ such that the Rényi divergences vanish asymptotically. Our results show that, when the Rényi parameter is in the interval $(\textbf {0},\textbf {1})$ , the Rényi conversion rates equal the ratio of the Shannon entropies ${\boldsymbol {H}\left ({\boldsymbol {P}_{\boldsymbol {X}}}\right)}/{\boldsymbol {H}\left ({\boldsymbol {Q}_{\boldsymbol {Y}}}\right)}$ , which is consistent with traditional results in which the total variation measure was adopted. When the Rényi parameter is in the interval $(\textbf {1},\infty)$ , the Rényi conversion rates are, in general, smaller than ${\boldsymbol {H}\left ({\boldsymbol {P}_{\boldsymbol {X}}}\right)}/{\boldsymbol {H}\left ({\boldsymbol {Q}_{\boldsymbol {Y}}}\right)}$. When specialized to the case in which either $\boldsymbol {P}_{\boldsymbol {X}}$ or $\boldsymbol {Q}_{\boldsymbol {Y}}$ is uniform, the simulation problem reduces to the source resolvability and intrinsic randomness problems. The preceding results are used to characterize the asymptotics of Rényi divergences and the Rényi conversion rates for these two cases. [ABSTRACT FROM AUTHOR]

We consider the problem of single-round private information retrieval (PIR) from $N$ replicated databases. We consider the case when $B$ databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of $M$ messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the $T$ -privacy constraint in this paper, where any $T$ databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of correct symbols that can be retrieved privately (under the $T$ -privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be $C=(N-2B)/N \cdot (1-T/(N-2B))/(1-(T/(N-2B))^{M})$ , if $2B+T < N$. This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing $2B$ databases from the system, with a penalty factor of $(N-2B)/N$ , which signifies that even though the number of databases needed for PIR is effectively $N-2B$ , the user still needs to access the entire $N$ databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are mis-synchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the errors introduced by the Byzantine databases as opposed to erasures in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes. [ABSTRACT FROM AUTHOR]

This paper deals with lower bounds on aperiodic correlation of sequences. It intends to solve two open questions. The first one is on the validity of the Levenshtein bound for a set of sequences other than binary sequences or those over the roots of unity. Although this result could be a priori extended to polyphase sequences, a formal demonstration is presented here, proving that it does actually hold for these sequences. The second open question is on the possibility to find a bound tighter than Welch’s, in the case of a set consisting of two sequences $M = 2$. By including the specific structure of correlation sequences, a tighter lower bound is introduced for this case. Besides, this method also provides in the cases $M = 3$ and $M = 4$ a tighter bound than the up-to-now tightest bound provided by Liu et al. [ABSTRACT FROM AUTHOR]

This paper studies the problem of inferring whether an agent is directly influenced by another agent over a network. Agent $i$ influences agent $j$ if they are connected (according to the network topology), and if agent $j$ uses the data from agent $i$ to update its online learning algorithm. The solution of this inference task is challenging for two main reasons. First, only the output of the learning algorithm is available to the external observer that must perform the inference based on these indirect measurements. Second, only output measurements from a fraction of the network agents is available, with the total number of agents itself being also unknown. The main focus of this paper is ascertaining under these demanding conditions whether consistent tomography is possible, namely, whether it is possible to reconstruct the interaction profile of the observable portion of the network, with negligible error as the network size increases. We establish a critical achievability result, namely, that for symmetric combination policies and for any given fraction of observable agents, the interacting and non-interacting agent pairs split into two separate clusters as the network size increases. This remarkable property then enables the application of clustering algorithms to identify the interacting agents influencing the observations. We provide a set of numerical experiments that verify the results for finite network sizes and time horizons. The numerical experiments show that the results hold for asymmetric combination policies as well, which is particularly relevant in the context of causation. [ABSTRACT FROM AUTHOR]

This paper asks how much we can gain in terms of bandwidth and user satisfaction, if recommendation systems became bandwidth aware and took into account not only the user preferences, but also the fact that they may need to serve these users under bandwidth constraints, as is the case over wireless networks. We formulate this as a new problem in the context of index coding: we relax the index coding requirements to capture scenarios where each client has preferences associated with messages. The client is satisfied to receive any message she does not already have, with a satisfaction proportional to her preference for that message. We consistently find, over a number of scenarios we sample, that although the optimization problems are in general NP-hard, significant bandwidth savings are possible even when restricted to polynomial time algorithms. [ABSTRACT FROM AUTHOR]

This paper studies the fundamental limits of storage for structured data, where statistics and structure are both critical to the application. Accordingly, a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source (DMS). For the lossless subset-compression problem, it turns out that the optimal source code may not index the conventional source-typical sequences, but rather index certain subset-typical sequences consistent with the subset of interest. Building upon an achievability and a strong converse, an analytic expression is given, based on the Shannon entropy, relative entropy, and subset entropy, which identifies such subset-typical sequences for a broad class of subsets of a DMS. Intuitively, subset-typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy. For the lossy subset-compression problem, an upper bound is derived on the subset rate-distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset-typical distribution and over a set of certain auxiliary subsets. By proving a strong converse result, this upper bound is shown to be tight for a class of symmetric subsets. As shown in our numerical examples, more often than not, one achieves a gain in the fundamental limits, in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate-distortion function, respectively, although exceptions are also possible. [ABSTRACT FROM AUTHOR]

In this paper, we present a variable-length binary code for permutations of degree n, where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover’s enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary $n$ in a straightforward way. [ABSTRACT FROM AUTHOR]

The topological interference management (TIM) problem refers to the study of the $K$ -user partially connected interference networks with no channel state information at the transmitters (CSIT), except for the knowledge of network topology. In this paper, we study the TIM problem with confidential messages (TIM-CM), where message confidentiality must be satisfied in addition to reliability constraints. In particular, each transmitted message must be decodable at its intended receiver and remain confidential at the remaining $(K-1)$ receivers. Our main contribution is to present a comprehensive set of results for the TIM-CM problem by studying the symmetric secure degrees of freedom (SDoF). To this end, we first characterize necessary and sufficient conditions for feasibility of positive symmetric SDoF for any arbitrary topology. We next present two achievable schemes for the TIM-CM problem: For the first scheme, we use the concept of secure partition and, for the second one, we use the concept of secure independent sets. We also present outer bounds on symmetric SDoF for any arbitrary network topology. Using these bounds, we characterize the optimal symmetric SDoF of all $K=2$ -user and $K=3$ -user network topologies. [ABSTRACT FROM AUTHOR]

In this paper, we provide the left and right Parseval relations over the quaternions. Moreover, we construct a new infinite family of sequences almost over $Q_{8} $ with period $p$ by a generalization of Legendre mapping, where $p\equiv 1 \pmod {8}$ is prime and the sequences have ideal autocorrelation. The sequences are symmetric when $p\equiv 1 \pmod {16}$ , and antisymmetric when $p\equiv 9 \pmod {16} $. [ABSTRACT FROM AUTHOR]

Previous works on age of information and erasure channels have dealt with specific models and computed the average age or average peak age for certain settings. In this paper, given a source that produces a letter every $T_{s}$ seconds and an erasure channel that can be used every $T_{c}$ seconds, we ask what is the coding strategy that minimizes the time-average “age of information” that an observer of the channel output incurs. We first analyze the case where the source alphabet and the channel-input alphabet have the same size. We show that a trivial coding strategy is optimal and a closed form expression for the age can be derived. We then analyze the case where the alphabets have different sizes. We use a random coding argument to bound the average age and show that the average age achieved using random codes converges to the optimal average age of linear block codes as the source alphabet becomes large. [ABSTRACT FROM AUTHOR]

Some physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This motivated the application of the asymmetric limited magnitude error model in flash memory. In this paper, we present a new construction of quasi-perfect codes for such errors, and we also study the perfect codes with symmetric errors. Moreover, we show some nonexistence results on perfect codes for correcting single error of limited magnitude. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a model of a distributed storage system that is locally recoverable from any single server failure. Unlike the usual local recovery model of codes for distributed storage, this model accounts for the fact that each server or storage node in a network is connectable to only some, and not all other, nodes. This may happen for reasons such as physical separation, inhomogeneity in storage platforms, and so forth. We estimate the storage capacity of both undirected and directed networks under this model and propose some constructive schemes. From a coding theory point of view, we show that this model is approximately dual of the well-studied index coding problem. Furthermore, in this paper, we extend the above model to handle multiple server failures. Among other results, we provide an upper bound on the minimum pairwise distance of a set of words that can be stored in a graph with the local repair guarantee. The well-known impossibility bounds on the distance of locally recoverable codes follow from our result. [ABSTRACT FROM PUBLISHER]

The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. The main result of this paper gives a necessary and sufficient condition, stated in the language of graph limit theory, for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable. It is then shown that a small modification of the Candès–Recht nuclear norm minimization algorithm provides the required asymptotic solution whenever the sequence of problems is asymptotically solvable. The theory is fully deterministic, with no assumption of randomness. A number of open questions are listed. [ABSTRACT FROM AUTHOR]

A primitive $k$ -batch code encodes a string $\boldsymbol {x}$ of length $n$ into a string $\boldsymbol {y}$ of length $N$ , such that each multiset of $k$ symbols from $\boldsymbol {x}$ has $k$ mutually disjoint recovering sets from $\boldsymbol {y}$. In this paper, we discuss new constructions of binary primitive batch codes. First, we develop novel explicit and random coding constructions of linear primitive batch codes based on finite geometries. Second, a new explicit coding construction of binary primitive batch codes based on bivariate lifted multiplicity codes is provided. For any $k=n^ \varepsilon $ with $\varepsilon \in (0,0.47)\setminus \{1/5,1/4\}$ , our proposed codes have a better trade-off between the redundancy and the parameters $k,n$ than previously known batch codes. [ABSTRACT FROM AUTHOR]

We consider the problem of multiple-node repair in distributed storage systems under the cooperative model, where the repair bandwidth includes the amount of data exchanged between any two different storage nodes. Recently, explicit constructions of MDS codes with optimal cooperative repair bandwidth for all possible parameters were given by Ye and Barg (IEEE Transactions on Information Theory, 2019). The node size (or sub-packetization) in this construction scales as $\exp (\Theta (\text {n}^{\text {h}}))$ , where h is the number of failed nodes and n is the code length. In this paper, we give new explicit constructions of optimal MDS codes for all possible parameters under the cooperative model, and the node size of our new constructions only scales as $\exp (\text {O}(\text {n}))$ for any number of failed nodes. Furthermore, it is known that any optimal MDS code under the cooperative model (including, in particular, our new code construction) also achieves optimal repair bandwidth under the centralized model, where the amount of data exchanged between failed nodes is not included in the repair bandwidth. We further show that the node size of our new construction is also much smaller than that of the best known MDS code constructions for the centralized model. [ABSTRACT FROM AUTHOR]

High peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances. We continue the theoretical contributions made by Boche and Farell toward the understanding of peak value reduction, using the strategy known as tone reservation for orthogonal transmission schemes. There it was shown that for orthogonal frequency-division multiplexing (OFDM) systems, the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method. In this paper, we show that the combinatorial object introduced as perfect Walsh sum (PWS) plays a similar role for code-division multiple access (CDMA) systems as arithmetic progression for OFDM systems. By specific construction, we show that for a chosen numbers $m$ and $n$ , all subsets $\mathcal {I} $ of the set $[N]$ of the first $N=2^{n}$ natural numbers, which has the density in $[N]$ larger than a given $\delta \in (0,1)$ , i.e., $\left |{ \mathcal {I} }\right |/N\geq \delta $ , and which is sufficiently large enough, in the sense that $\left |{ I }\right |\geq 2(2/\delta)^{2^{m}-1}$ , contains a PWS of size $2^{m}$. By means of this result, and motivated by the previously mentioned connection between arithmetic progression and PWS, we show results for the PWS which are analogous to the famous Szemerédi theorem on arithmetic progressions, Conlon-Gower’s theorem on probabilistic construction of “sparse” sets containing an arithmetic progression, and even a solution of an analogon to the Erdős’ conjecture on arithmetic progressions. Those results give in particular an insight into the asymptotic limitations of tone reservation method for the CDMA systems. Besides, we show that a subset $I$ of $[N]$ is a PWS if and only if the embedding inequality of the subspace of $L^{1}([{0,1}])$ , containing linear combinations of Walsh functions indexed by elements of $\mathcal {I} $ , holds with the minimum possible embedding constant $\sqrt { \left |{ \mathcal {I} }\right |}$. The corresponding approach based in particular by the fact that the PWSs are the only Walsh sums having unit $L^{1}$ -norm, proven in this paper. By means of that results, we show that the minimum possible threshold constant for which the tone reservation method is applicable yields $\sqrt { \left |{ \mathcal {I} }\right |}$ if and only if the information set $\mathcal {I} $ is a PWS. [ABSTRACT FROM AUTHOR]

Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$ -EC-AUED codes, are studied in this paper. Given positive integers $q$ , $a$ , and $t$ , let $n_{q}(a,t+1)$ denote the length of the shortest $q$ -ary $t$ -EC-AUED code of size $a$. We introduce combinatorial constructions for $q$ -ary $t$ -EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of $n_{q}(a,t+1)$ for several new infinite families of $q,a$ , and $t$. [ABSTRACT FROM AUTHOR]

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for certain applications. We analyze both, the classical and the modified expansion complexity. Moreover, we also study the expansion complexity of the explicit inversive congruential generator. [ABSTRACT FROM AUTHOR]

Cyclic codes have been widely used in many applications of communication systems and data storage systems. This paper proposes a new procedure for decoding cyclic codes up to actual minimum distance. The decoding procedure consists of two steps: 1) computation of known syndromes and 2) computation of error positions and error values simultaneously. To do so, a matrix whose all entries are syndromes is called syndrome matrix. A matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix. In this paper, two novel methods are presented to determine error positions and error values simultaneously and directly. The first method uses a new partial syndrome matrix along with Gaussian elimination. The partial syndrome matrices for binary (respectively, ternary) cyclic codes of lengths from 69 to 99 (respectively, 16 to 37) are tabulated. For some cyclic codes, the partial syndrome matrices contain unknown syndromes; the second method constructs a matrix from a system of equations, which is generated by the determinants of different partial syndrome matrices and makes use of Gaussian elimination to determine its row rank. Many more cyclic codes beyond the Bose–Chaudhuri–Hocquenghem bound can be decoded with these methods. [ABSTRACT FROM AUTHOR]

In pliable index coding, we consider a server with m messages and n clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most \mathcal O(\log ^2(n)) broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case \mathcal O(n) transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by \mathcal O(\log ^2(n)) broadcast transmissions. We extend our algorithm to the t -requests case, where each client requires t unknown messages that she does not have, and show that our algorithm requires at most \mathcal O(t\log (n)+\log ^2(n)) broadcast transmissions. We construct lower bound instances that require at least \Omega (\log (n)) transmissions for linear pliable index coding and at least $\Omega (t+\log (n))$ transmissions for the t$ -requests case, indicating that both our upper and lower bounds are polynomials of \log (n)$ and differ within a factor of . We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely $\Theta (\log (n))$ , as compared with the $\Theta (n/\log (n))$ for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions. [ABSTRACT FROM AUTHOR]

In this paper, new index coding problems are studied, where each receiver has erroneous side information. Although side information is a crucial part of index coding, the existence of erroneous side information has not been considered yet. We study an index code with receivers that have erroneous side information symbols in the error-free broadcast channel, which is called an index code with side information errors (ICSIE). The encoding and decoding procedures of the ICSIE are proposed, based on the syndrome decoding. Then, we derive the bounds on the optimal codelength of the proposed index code with erroneous side information. Furthermore, we introduce a special graph for the proposed index coding problem, called a \delta s -cycle whose properties are similar to those of the cycle in the conventional index coding problem. Properties of the ICSIE are also discussed in the \delta s -cycle and clique. Finally, the proposed ICSIE is generalized to an index code for the scenario having both additive channel errors and side information errors, called a generalized error correcting index code. [ABSTRACT FROM AUTHOR]

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared with a random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with a crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this paper, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length. [ABSTRACT FROM PUBLISHER]

In high-dimensional sparse regression, would increasing the signal-to-noise ratio while fixing the sparsity level always lead to better model selection? For high-dimensional sparse regression problems, surprisingly, in this paper we answer this question in the negative in the regime of linear sparsity for the Lasso method, relying on a new concept we term effect size heterogeneity. Roughly speaking, a regression coefficient vector has high effect size heterogeneity if its nonzero entries have significantly different magnitudes. From the viewpoint of this new measure, we prove that the false and true positive rates achieve the optimal trade-off uniformly along the Lasso path when this measure is maximal in a certain sense, and the worst trade-off is achieved when it is minimal in the sense that all nonzero effect sizes are roughly equal. Moreover, we demonstrate that the first false selection occurs much earlier when effect size heterogeneity is minimal than when it is maximal. The underlying cause of these two phenomena is, metaphorically speaking, the “competition” among variables with effect sizes of the same magnitude in entering the model. Taken together, our findings suggest that effect size heterogeneity shall serve as an important complementary measure to the sparsity of regression coefficients in the analysis of high-dimensional regression problems. Our proofs use techniques from approximate message passing theory as well as a novel technique for estimating the rank of the first false variable. [ABSTRACT FROM AUTHOR]

Tsallis and Rényi entropies, which are monotone transformations of each other, are deformations of the celebrated Shannon entropy. Maximization of these deformed entropies, under suitable constraints, leads to the $q$ -exponential family which has applications in non-extensive statistical physics, information theory and statistics. In previous information-geometric studies, the $q$ -exponential family was analyzed using classical convex duality and Bregman divergence. In this paper, we show that a generalized $\lambda $ -duality, where $\lambda = 1 - q$ is to be interpreted as the constant information-geometric curvature, leads to a generalized exponential family which is essentially equivalent to the $q$ -exponential family and has deep connections with Rényi entropy and optimal transport. Using this generalized convex duality and its associated logarithmic divergence, we show that our $\lambda $ -exponential family satisfies properties that parallel and generalize those of the exponential family. Under our framework, the Rényi entropy and divergence arise naturally, and we give a new proof of the Tsallis/Rényi entropy maximizing property of the $q$ -exponential family. We also introduce a $\lambda $ -mixture family which may be regarded as the dual of the $\lambda $ -exponential family, and connect it with other mixture-type families. Finally, we discuss a duality between the $\lambda $ -exponential family and the $\lambda $ -logarithmic divergence, and study its statistical consequences. [ABSTRACT FROM AUTHOR]

In this paper, we study the duals of generalized bent functions $f: V_{n}\rightarrow \mathbb {Z}_{p^{k}}$ , where $V_{n}$ is an $n$ -dimensional vector space over $\mathbb {F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The duals of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of Çeşmelioğlu et al., 2016, we provide an explicit construction of generalized bent functions whose duals can be generalized bent or not generalized bent. We show that the well-known direct sum construction and the generalized indirect sum construction given in Wang and Fu, 2021. can provide secondary constructions of generalized bent functions whose duals can be generalized bent or not generalized bent. By using the knowledge on ideal decomposition in cyclotomic fields, we prove that $f^{**}(x)=f(-x)$ if $f$ is a generalized bent function and its dual $f^{*}$ is also a generalized bent function. For any non-weakly regular generalized bent function $f$ which satisfies that $f(x)=f(-x)$ and its dual $f^{*}$ is generalized bent, we give a property and as a consequence, we prove that there is no self-dual generalized bent function $f: V_{n}\rightarrow \mathbb {Z}_{p^{k}}$ if $p\equiv 3 ~(mod ~4)$ and $n$ is odd. When $p \equiv 1 ~(mod ~4)$ or $p\equiv 3 ~(mod ~4)$ and $n$ is even, we give a secondary construction of self-dual generalized bent functions. In the end, by the decomposition of generalized bent functions, we characterize the relations between the generalized bentness of the dual of a generalized bent function $f$ and the bentness of the duals of bent functions associated with the generalized bent function $f$ , as well as the relations of self-duality between a generalized bent function $f$ and bent functions associated with the generalized bent function $f$. [ABSTRACT FROM AUTHOR]

Motivated by the established notion of storage codes, we consider sets of infinite sequences over a finite alphabet such that every $k$ -tuple of consecutive entries is uniquely recoverable from its $l$ -neighborhood in the sequence. We address the problem of finding the maximum growth rate of the set, which we term capacity, as well as constructions of explicit families that approach the optimal rate. The techniques that we employ rely on the connection of this problem with constrained systems. In the second part of the paper we consider a modification of the problem wherein the entries in the sequence are viewed as random variables over a finite alphabet that follow some joint distribution, and the recovery condition requires that the Shannon entropy of the $k$ -tuple conditioned on its $l$ -neighborhood be bounded above by some $\epsilon >0$. We study properties of measures on infinite sequences that maximize the metric entropy under the recoverability condition. Drawing on tools from ergodic theory, we prove some properties of entropy-maximizing measures. We also suggest a procedure of constructing an $\epsilon $ -recoverable measure from a corresponding deterministic system. [ABSTRACT FROM AUTHOR]

The Langberg-Médard multiple unicast conjecture claims that for a strongly reachable $k$ -pair network, there exists a feasible multi-flow with rate $(1,1, {\dots },1)$. In this paper, we confirm the conjecture for $k=3$. [ABSTRACT FROM AUTHOR]

Since discovery of Goppa geometric codes, people have been asking the question: are there different constructions of block codes from algebraic curves that give the same parameters as Goppa geometric codes. Despite of great effort by researchers, no such constructions have been found so far. Although in literature, there are many constructions of block code from algebraic curves, most of them are quite different from the one by Goppa in nature and thus they have different parameters. Some of these constructions have the same parameters as Goppa geometric codes, however it was proved that these are essentially the same codes defined by Goppa. In this paper, we solve this question for the case where the characteristic of the ground field is 2, namely, we present a different construction of block codes from algebraic curves that give the same parameters as Goppa geometric codes for the characteristic 2 case. [ABSTRACT FROM PUBLISHER]

In this paper, the difference between the redundancy of the optimal asymmetric/symmetric fix-free code, and that of the optimal prefix-free code is considered as the penalty of benefiting from the desired properties of fix-free codes. This penalty is studied from different perspectives. In particular, it is shown that the average penalty of asymmetric fix-free codes is less than 0.21 bit per symbol. Moreover, it is proved that when the source alphabet size is sufficiently large, for almost all sources, the penalty is less than or equal to 0.182 bit per symbol. Regarding symmetric fix-free codes, it is shown that the average penalty tends to infinity as the source alphabet size increases. [ABSTRACT FROM AUTHOR]

In multiuser information theory, it is often assumed that every node in the network possesses all codebooks used in the network. This assumption may be impractical in distributed ad hoc, cognitive, or heterogeneous networks. This paper considers the two-user interference channel with one oblivious receiver (IC-OR), i.e., one receiver lacks knowledge of the interfering cookbook, whereas the other receiver knows both codebooks. This paper asks whether, and if so how much, the channel capacity of the IC-OR is reduced compared with that of the classical IC where both receivers know all codebooks. A novel outer bound is derived and shown to be achievable to within a gap for the class of injective semideterministic IC-ORs; the gap is shown to be zero for injective fully deterministic IC-ORs. An exact capacity result is shown for the general memoryless IC-OR when the nonoblivious receiver experiences very strong interference. For the linear deterministic IC-OR that models the Gaussian noise channel at high SNR, nonindependent identically distributed. Bernoulli(1/2) input bits are shown to achieve points not achievable by i.i.d. Bernoulli(1/2) input bits used in the same achievability scheme. For the real-valued Gaussian IC-OR, the gap is shown to be at most 1/2 bit per channel use, even though the set of optimal input distributions for the derived outer bound could not be determined. Toward understanding the Gaussian IC-OR, an achievability strategy is evaluated in which the input alphabets at the nonoblivious transmitter are a mixture of discrete and Gaussian random variables, where the cardinality of the discrete part is appropriately chosen as a function of the channel parameters. Surprisingly, as the oblivious receiver intuitively should not be able to jointly decode the intended and interfering messages (whose codebook is unavailable), it is shown that with this choice of input, the capacity region of the symmetric Gaussian IC-OR is to within 1/2\log \left (12\pi \mathrm e \right )\approx 3.34 bits (per channel use per user) of an outer bound for the classical Gaussian IC with full codebook knowledge at both receivers. [ABSTRACT FROM PUBLISHER]