Search

It is well known that cooperation between users in a communication network can lead to significant performance gains. A common assumption in past works is that all the users are aware of the resources available for cooperation, and know exactly to what extent these resources can be used. Unfortunately, in many modern communication networks, the availability of cooperation links cannot be guaranteed a priori, due to the dynamic nature of the network. In this paper, a family of models is suggested where the cooperation links may or may not be present. Coding schemes are devised that exploit the cooperation links if they are present, and can still operate (although at reduced rates) if cooperation is not possible. [ABSTRACT FROM AUTHOR]

Caching is a promising solution to satisfy the ever-increasing demands for the multi-media traffics. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains over the uncoded caching schemes. However, to implement the coded caching schemes, each file has to be split into F packets, which usually increases exponentially with the number of users K . Thus, designing caching schemes that decrease the order of F is meaningful for practical implementations. In this paper, by reviewing the Ali-Niesen caching scheme, the placement delivery array (PDA) design problem is first formulated to characterize the placement issue and the delivery issue with a single array. Moreover, we show that, through designing appropriate PDA, new centralized coded caching schemes can be discovered. Second, it is shown that the Ali-Niesen scheme corresponds to a special class of PDA, which realizes the best coding gain with the least F . Third, we present a new construction of PDA for the centralized coded caching system, wherein the cache size M at each user (identical cache size is assumed at all users) and the number of files N ( q is an integer, such that q\geq 2 ). The new construction can decrease the required F of Ali-Niesen scheme to O(e^{K\cdot ({M}/{N})\ln \!\,({N}/\!{M})}) or O(e^{K\cdot (1-({M}/\!{N}))\ln \!\,(N/(N-M))})$ , respectively, while the coding gain loss is only 1. [ABSTRACT FROM AUTHOR]

The approximation of stable linear time-invariant (LTI) systems is studied for the Paley–Wiener space \mathcal PW\pi ^{1} of bandlimited functions with absolutely integrable Fourier transform. For pointwise sampling, it is known that there exist stable LTI systems and functions such that the approximation process diverges, regardless of the oversampling factor. Recently, it was shown that the divergence can be overcome by using more general measurement functionals that are based on a complete orthonormal system. However, this approach requires the approximation process to have an increased bandwidth. In this paper, a two channel approximation process is presented that is uniformly convergent for all stable LTI systems and all functions in \mathcal PW\pi ^{1} . An advantage of the two channel approach compared with the one channel approach is the reduction of the approximation bandwidth, which can be exactly the same as the input function bandwidth. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm referred to as multipath matching pursuit (MMP) that investigates multiple promising candidates to recover sparse signals from compressed measurements. Our method is inspired by the fact that the problem to find the candidate that minimizes the residual is readily modeled as a combinatoric tree search problem and the greedy search strategy is a good fit for solving this problem. In the empirical results as well as the restricted isometry property-based performance guarantee, we show that the proposed MMP algorithm is effective in reconstructing original sparse signals for both noiseless and noisy scenarios. [ABSTRACT FROM AUTHOR]

A (layered) broadcast approach is studied for the fading wiretap channel without the channel state information (CSI) at the transmitter. Two broadcast schemes, based on superposition coding and embedded coding, respectively, are developed to encode information into a number of layers and use stochastic encoding to keep the corresponding information secret from an eavesdropper. The layers that can be successfully and securely transmitted are determined by the channel states to the legitimate receiver and the eavesdropper. The advantage of these broadcast approaches is that the transmitter does not need to know the CSI to the legitimate receiver and the eavesdropper, but the scheme still adapts to the channel states of the legitimate receiver and the eavesdropper. Three scenarios of block fading wiretap channels with stringent delay constraints are studied, in which either the legitimate receiver's channel, the eavesdropper's channel, or both channels are fading. For each scenario, the secrecy rate that can be achieved via the broadcast approach developed in this paper is derived, and the optimal power allocation over the layers (or the conditions on the optimal power allocation) is also characterized. A notion of probabilistic secrecy, which characterizes the probability that a certain secrecy rate of decoded messages is achieved during one block, is also introduced and studied for scenarios when the eavesdropper's channel is fading. Numerical examples are provided to demonstrate the impact of the CSI at the transmitter and the channel fluctuations of the eavesdropper on the average secrecy rate. These examples also demonstrate the advantage of the proposed broadcast approach over the compound channel approach. [ABSTRACT FROM AUTHOR]

One of the most important challenges in the context of fix-free codes is proving the 3\over 4-conjecture, which guarantees the existence of fix-free codewords of lengths \ell 1,\ell 2,\ldots, \ell N if \sum i=1^{N}2^{-\ell i}\leq {{ 3}\over { 4}}. Although this conjecture has not become a theorem yet, some researchers have proved the problem with some extra constraints on the codelengths. One of those, we call it Yekhanin's constraint, is \sum i:\ell i-\lambda \leq 12^{-\ell i}\geq {{ 1}\over { 2}}, where \lambda =\min k\ell k. In this paper, it is shown that such a constraint is not so restrictive. We prove that almost 93.8% of the N-tuple codelength vectors with Kraft sum 3\over 4 do satisfy Yekhanin's constraint. One can optimistically interpret this result as almost 93.8% of the road of proving the 3\over 4-conjecture is paved. [ABSTRACT FROM PUBLISHER]

Sparse subspace clustering (SSC) relies on sparse regression for accurate neighbor identification. Inspired by recent progress in compressive sensing, this paper proposes a new sparse regression scheme for SSC via two-step reweighted $\ell _{1} $ -minimization, which also generalizes a two-step $\ell _{1} $ -minimization algorithm introduced by E. J. Candès et al. in [The Annals of Statistics, vol. 42, no. 2, pp. 669–699, 2014] without incurring extra algorithmic complexity. To fully exploit the prior information offered by the computed sparse representation vector in the first step, our approach places a weight on each component of the regression vector, and solves a weighted LASSO in the second step. We propose a data weighting rule suitable for enhancing neighbor identification accuracy. Then, under the formulation of the dual problem of weighted LASSO, we study in depth the theoretical neighbor recovery rates of the proposed scheme. Specifically, an interesting connection between the locations of nonzeros of the optimal sparse solution to the weighted LASSO and the indexes of the active constraints of the dual problem is established. Afterwards, under the semi-random model, analytic probability lower/upper bounds for various neighbor recovery events are derived. Our analytic results confirm that, with the aid of data weighting and if the prior neighbor information is accurate enough, the proposed scheme with a higher probability can produce many correct neighbors and few incorrect neighbors as compared to the solution without data weighting. Computer simulations are provided to validate our analytic study and evidence the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The Bayesian predictive density has complex representation and does not belong to any finite-dimensional statistical model except for in limited situations. In this paper, we introduce its simple approximate representation employing its projection onto a finite-dimensional exponential family. Its theoretical properties are established parallelly to those of the Bayesian predictive density when the model belongs to curved exponential families. It is also demonstrated that the projection asymptotically coincides with the plugin density with the posterior mean of the expectation parameter of the exponential family, which we refer to as the Bayes extended estimator. Information-geometric correspondence indicates that the Bayesian predictive density can be represented as the posterior mean of the infinite-dimensional exponential family. The Kullback–Leibler risk performance of the approximation is demonstrated by numerical simulations and it indicates that the posterior mean of the expectation parameter approaches the Bayesian predictive density as the dimension of the exponential family increases. It also suggests that approximation by projection onto an exponential family of reasonable size is practically advantageous with respect to risk performance and computational cost. [ABSTRACT FROM AUTHOR]

Locally recoverable (LRC) codes and their variants have been extensively studied in recent years. In this paper we focus on cyclic constructions of LRC codes and derive conditions on the zeros of the code that support the property of hierarchical locality. As a result, we obtain a general family of hierarchical LRC codes for a new range of code parameters. We also observe that our approach enables one to represent an LRC code in quasicyclic form, and use this representation to construct tail-biting convolutional LRC codes with locality. Among other results, we extend the general approach to cyclic codes with locality to multidimensional cyclic codes, yielding new families of LRC codes with availability, and construct a family of $q$ -ary cyclic hierarchical LRC codes of unbounded length. [ABSTRACT FROM AUTHOR]

A Grain-like structure is a cascade connection of a primitive LFSR into an NFSR. It is well known that such structures generate sequences with periods multiples of the period of primitive LFSR sequences. In this paper, we study weak Grain-like structures, i.e., Grain-like structures generating at least one sequence with minimum period. Assume the orders of LFSR and NFSR are $n$ and $m$ respectively. We prove that weak Grain-like structures always exist when $m>n$. For $m=n$ , we give three classes of weak Grain-like structures. Then we extend the method in the second class to prove that weak Grain-like structures exist when $m\ge n-\lg n+4$. Moreover, our experimental data shows that, the ratio of weak Grain-like structures approximates a value which is a little more than 63% for small $m=n$ , and weak Grain-like structures exist with $m=3$ or 4 when $n\le 18$. [ABSTRACT FROM AUTHOR]

An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we refer to as locality, could be large. In this paper, we pose the index coding problem as that of minimizing the broadcast rate for a given value of locality (or vice versa) and designing codes that achieve the optimal trade-off between locality and rate. We identify the optimal broadcast rate corresponding to the minimum possible value of locality for all single unicast problems. We present new structural properties of index codes which allow us to characterize the optimal trade-off achieved by: vector linear codes when the side information graph is a directed cycle; and scalar linear codes when the minrank of the side information graph is one less than the order of the problem. We also identify the optimal trade-off among all codes, including non-linear codes, when the side information graph is a directed 3-cycle. Finally, we present techniques to design locally decodable index codes for arbitrary single unicast problems and arbitrary values of locality. [ABSTRACT FROM AUTHOR]

The problem of $X$ -secure $T$ -private information retrieval from MDS coded storage is studied in this paper, where the user wishes to privately retrieve one out of $K$ independent messages that are distributed over $N$ servers according to an MDS code. It is guaranteed that any group of up to $X$ colluding servers learn nothing about the messages and that any group of up to $T$ colluding servers learn nothing about the identity of desired message. A lower bound of achievable rates is proved by presenting a novel scheme based on cross-subspace alignment and a successive decoding with interference cancellation strategy. For large number of messages $(K\rightarrow \infty)$ the achieved rate, which we conjecture to be optimal, improves upon the best known rates previously reported in the literature by Raviv and Karpuk, and generalizes an achievable rate for MDS-TPIR previously found by Freij-Hollanti et al. that is also conjectured to be asymptotically optimal. The setting is then expanded to allow unresponsive and Byzantine servers. Finally, the scheme is applied to find a new lower convex hull of (download, upload) pairs of secure and private distributed matrix multiplication that generalizes, and in certain asymptotic settings strictly improves upon the best known previous results. [ABSTRACT FROM AUTHOR]

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315–6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for $(4\delta,2\delta,2\delta)_{q}$ -CDCs is calculated. It is greater than 0.99926 for any prime power $q$ and any $\delta \geq 3$. [ABSTRACT FROM AUTHOR]

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an $\omega $ th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators, where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes. [ABSTRACT FROM AUTHOR]

In the index coding problem a sender holds a message $x \in \{0,1\}^{n}$ and wishes to broadcast information to $n$ receivers in a way that enables the $i$ th receiver to retrieve the $i$ th bit $x_{i}$. Every receiver has prior side information comprising a subset of the bits of $x$ , and the goal is to minimize the length of the information sent via the broadcast channel. Porter and Wootters have recently introduced the model of embedded index coding, where the receivers also play the role of the sender and the goal is to minimize the total length of their broadcast information. An embedded index code is said to be task-based if every receiver retrieves its bit based only on the information provided by one of the receivers. This paper studies the effect of the task-based restriction on linear embedded index coding. It is shown that for certain side information maps there exists a linear embedded index code of length quadratically smaller than that of any task-based embedded index code. The result attains, up to a multiplicative constant, the largest possible gap between the two quantities. The proof is by an explicit construction and the analysis involves spectral techniques. [ABSTRACT FROM AUTHOR]

The paper proposes an improved difference-of-syndromes decoding algorithm for decoding quadratic residue codes up to half the minimum distance. Therein, soft-decision information is not essential, but it can be used to speed up the computations. The improvement over the prior art is demonstrated by simulations. Moreover, the proposed algorithm is also compared with the classic Berlekamp-Massey algorithm by taking two BCH codes as examples. [ABSTRACT FROM AUTHOR]

The analysis of nonconvex matrix completion has recently attracted much attention in the community of machine learning thanks to its computational convenience. Existing analysis on this problem, however, usually relies on $\ell _{2,\infty }$ projection or regularization that involves unknown model parameters, although they are observed to be unnecessary in numerical simulations. In this paper, we extend the analysis of the vanilla gradient descent for positive semidefinite matrix completion in the literature to the rectangular case, and more significantly, improve the required sampling rate from $O(\mathrm {poly}(\kappa)\mu ^{3}~r^{3} \log ^{3}~n/n)$ to $O(\mu ^{2}~r^{2} \kappa ^{14} \log n/n)$. Our technical ideas and contributions are potentially useful in improving the leave-one-out analysis in other related problems. [ABSTRACT FROM AUTHOR]

An $[{n}, {k}, {n}-{k}+1]$ linear code is called an MDS code. An $[{n}, {k}, {n}-{k}]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system ${S}(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for $ {t}\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $ {GF}(3^{ {s}})$ holding an infinite family of 3-designs and an infinite family of near MDS codes over $ {GF}(2^{2 {s}})$ holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal. [ABSTRACT FROM AUTHOR]

Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb {Z}_{2^{k}r}$ which is based on so called $B_{1}[{4}](2^{k}r)$ set. In fact, we reduce the construction of a maximal size $B_{1}[{4}](2^{k}r)$ set for $k\geq 3$ to the construction of a maximal size $B_{1}[{4}](2^{k-3}r)$ set. Finally, we give an explicit formula of a maximal size $B_{1}[{4}](4r)$ set and some lower bounds of a maximal size $B_{1}[{4}](2r)$ set. By computer searching up to $q\leq 106$ , we conjecture that those lower bounds are tight. [ABSTRACT FROM AUTHOR]

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotic constructions of locally repairable codes in the literature. In this paper, we provide a new explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furthermore, we show that the Gilbert-Varshamov type bound of locally repairable codes over non-prime finite fields can be improved for sufficiently large alphabet sizes. [ABSTRACT FROM AUTHOR]

The binary Hamming codes with parameters $[{2^{m}-1, 2^{m}-1-m, 3}]$ are perfect. Their extended codes have parameters $[{2^{m}, 2^{m}-1-m, 4}]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[{2^{m+s}+2^{s}-2^{m},2^{m+s}+2^{s}-2^{m}-2m-2,4}]$ , which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters $[{2^{m}+2, 2^{m}-2m, 6}]$. Both classes of binary linear codes have new parameters. [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]

Our first goal in this paper is to investigate stationarization and asymptotic decorrelation for fractional Brownian motion (fBm) using wavelet packet transform. The wavelet packets are generated by the N^\mathrm th -order Daubechies scaling function and wavelet. To decorrelate the wavelet packet coefficients asymptotically, we take two strategies: fix $N$ and let absolute scale difference or absolute difference between time shifts get large; and fix scale level and time shift and let $N$ get large. Our second goal is to present the asymptotic properties of the entropy-like cost functional and denoising cost functional and their impact on the selection of best wavelet packet bases when used for fBm plus or not plus independent Gaussian white noise. [ABSTRACT FROM PUBLISHER]

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]

Suppose Y^n is obtained by observing a uniform Bernoulli random vector X^n through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Y^n and a Boolean function \mathsf b(X^n) could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of Y^n under logarithmic loss, given \mathsf b(X^n) . In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function—the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than a dictator, and that for a strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels. [ABSTRACT FROM PUBLISHER]

A long-standing conjecture of Golomb and Welch, raised in 1970, states that there is no perfect r error correcting Lee code of length n for n\geq 3 and r>1 under the l_{p} metric, where 1\leq p<\infty . We show some nonexistence results of linear perfect lp codes for p=1 and 2\leq p<\infty , r=2^{1/p},3^{1/p} . We also give an algebraic construction of quasi-perfect l_{p}$ codes for p=1, r=2$ , and 2\leq p<\infty , r=2^{1/p} . [ABSTRACT FROM PUBLISHER]

Network switches and routers scale in rate by distributing the packet read/write operations across multiple memory banks. Rate scaling is achieved so long as sufficiently many packets can be written and read in parallel. However, due to the non-determinism of the read process, parallel pending read requests may contend on memory banks, and thus significantly lower the switching rate. In this paper, we provide a constructive study of codes that guarantee fully parallel data reconstruction without contention. We call these codes “switch codes,” and construct three optimal switch-code families with different parameters. All the constructions use only simple XOR-based encoding and decoding operations, an important advantage when operated in ultra-high speeds. Switch codes achieve their good performance by spanning simultaneous disjoint local-decoding sets for all their information symbols. Switch codes may be regarded as an extreme version of the previously studied batch codes, where the switch version requires parallel reconstruction of all the information symbols. [ABSTRACT FROM PUBLISHER]

Broadcasting $K$ independent messages to multiple users where each user demands all the messages and already has a subset of the messages as side information is studied. Recently, Natarajan et al. proposed a novel broadcasting strategy called lattice index coding, which adopts lattices constructed over some principal ideal domains (PID) for transmission. Using the structure of lattices over PID, they showed that this scheme provides uniform side information gain for any side information configuration, a desired property which essentially guarantees a fair signal-to-noise ratio gain when normalized by the amount of information contained in side information. In this paper, we generalize this strategy to a general ring of algebraic integers, which may not be a PID. Upper and lower bounds on the side information gains for the proposed scheme constructed over some interesting classes of number fields are provided and are shown to coincide asymptotically in message rate. This generalization substantially enlarges the design space and partially includes the scheme by Natarajan et al. as a special case. Perhaps more importantly, in addition to side information gains, the proposed lattice index codes benefit from diversity gains inherent in constellations carved from number fields when used over Rayleigh fading channel. Some interesting examples are provided for which the proposed scheme allows the messages to be from the same field. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce the so-called hierarchical interaction models, where we assume that the computation of the value of a function m: \mathbb R^d\rightarrow \mathbb R is done in several layers, where in each layer a function of at most d^* inputs computed by the previous layer is evaluated. We investigate two different regression estimates based on polynomial splines and on neural networks, and show that if the regression function satisfies a hierarchical interaction model and all occurring functions in the model are smooth, the rate of convergence of these estimates depends on d^* (and not on $d$ ). Hence, in this case, the estimates can achieve good rate of convergence even for large $d$ , and are in this sense able to circumvent the so-called curse of dimensionality. [ABSTRACT FROM PUBLISHER]

Both Maximum Distance Separable and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining the existence of q -ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where q is even. This paper focuses on the case where q is odd. We construct a few classes of new MDS self-dual codes through generalized Reed–Solomon codes. More precisely, we show that for any given even length n , we have a q and q is sufficiently large (say q\ge 4^{n}\times n^{2}) . Furthermore, we prove that there exists a q -ary MDS self-dual code of length n if q=r^{2} and n$ satisfies one of the three conditions: 1) n\le r$ and n$ is even; 2) q$ is odd and n-1$ is an odd divisor of and $n=2tr$ for any $t\le (r-1)/2$ . [ABSTRACT FROM PUBLISHER]

We consider the problem of recovering a complete (i.e., square and invertible) matrix A0 , from Y \in \mathbb R ^n \times p with Y = A0 X0 , provided X0 is sufficiently sparse. This recovery problem is central to theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers A0 when X0 has O \left ( n \right ) nonzeros per column, under suitable probability model for X0 . In contrast, prior results based on efficient algorithms either only guarantee recovery when X0 has O(\sqrt n) zeros per column, or require multiple rounds of semidefinite programming relaxation to work when X0 has $O(n)$ nonzeros per column. Our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint. In this paper, we provide a geometric characterization of the objective landscape. In particular, we show that the problem is highly structured with high probability: 1) there are no “spurious” local minimizers and 2) around all saddle points the objective has a negative directional curvature. This distinctive structure makes the problem amenable to efficient optimization algorithms. In a companion paper, we design a second-order trust-region algorithm over the sphere that provably converges to a local minimizer from arbitrary initializations, despite the presence of saddle points. [ABSTRACT FROM AUTHOR]

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 this paper, let n=2k and d=3\cdot 2^{k}-2 with k\geq 3 and \gcd (d,2^n-1)=1 . Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary m -sequence of period 2^n-1 and its $d$ -decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al. [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]

When the information source is a continuous distribution and the rate–distortion function is strictly larger than the Shannon lower bound, the explicit evaluation of the rate–distortion function is not straightforward. We evaluate the rate–distortion function for an independent identically distributed gamma source with respect to the absolute-log distortion measure. The logarithmic transformation reduces this rate–distortion problem to that under the absolute distortion measure. Extending the explicit evaluation of the rate–distortion function for the Gaussian sources, we obtain the parametric form of the rate–distortion function. We show that the optimal distribution of reconstruction consists of a continuous component enclosed by left and right discrete components, and the left discrete component vanishes when the acceptable distortion is small. We further extend the result for a wider class of source distributions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel “genie chains” approach to obtain information theoretic degrees of freedom (DoF) outer bounds for MIMO wireless interference networks. This new approach creates a chain of mappings from genie signals provided to a receiver to the exposed signal spaces at that receiver, and then the exposed signal spaces serve as the genie signals for the next receiver in the chain subject to certain linear independence requirements. Our approach essentially converts an information theoretic DoF outer bound problem into a linear algebra problem. Several applications of the genie chains approach are presented. [ABSTRACT FROM AUTHOR]

It is shown that the pairs of maximal linear recursive sequences ( $m$ -sequences) typically have mean square aperiodic crosscorrelation on par with that of random sequences, but that if one takes a pair of $m$ -sequences where one is the reverse of the other, and shifts them appropriately, one can get significantly lower mean square aperiodic crosscorrelation. Sequence pairs with even lower mean square aperiodic crosscorrelation are constructed by taking a Legendre sequence, cyclically shifting it, and then cutting it (approximately) in half and using the halves as the sequences of the pair. In some of these constructions, the mean square aperiodic crosscorrelation can be lowered further if one truncates or periodically extends (appends) the sequences. Exact asymptotic formulas for mean squared aperiodic crosscorrelation are proved for sequences derived from additive characters (including $m$ -sequences and modified versions thereof) and multiplicative characters (including Legendre sequences and their relatives). Data are presented that show that the sequences of modest length have performance that closely approximates the asymptotic formulas. [ABSTRACT FROM PUBLISHER]

This paper studies an almost-lossless source-channel coding scheme in which source messages are assigned to different classes and encoded with a channel code that depends on the class index. The code performance is analyzed by means of random-coding error exponents and validated by simulation of a low-complexity implementation using existing source and channel codes. While each class code can be seen as a concatenation of a source code and a channel code, the overall performance improves on that of separate source-channel coding and approaches that of joint source-channel coding when the number of classes increases. [ABSTRACT FROM PUBLISHER]

The Tate pairing computation consists of two parts: Miller step and final exponentiation step. In this paper, we investigate the structure of the final exponentiation step. Consider an order r subgroup of an elliptic curve defined over \BBF q with embedding degree k. The final exponentiation in the Tate pairing is an exponentiation of an element in \BBF q^{k} by (q^k-1)/r. The hardest part of this computation is to raise to the power \lambda :=\Phi k(q)/r, where \Phi k(\cdot) denotes the kth cyclotomic polynomial. Write it as \lambda =\lambda 0+\lambda 1q+\cdots +\lambda \varphi (k)-1q^{\varphi (k)-1} in the q-ary representation. The final exponentiation cost mostly depends on \kappa (\lambda), the size of the maximum of \vert \lambda i\vert . In many parameterized pairing-friendly curves, the value \kappa is about \left (1- 1\over \rho \varphi (k)\right)\log 2 q where \rho =\log 2 q/\log 2 r, while random curves will have \kappa \approx \log 2 q. We investigate how this small \kappa is obtained for parameterized pairing-friendly elliptic curves, and show that \left (1- 1\over \rho \varphi (k)\right)\log 2 q is the lower bound for all known construction methods of parameterized pairing-friendly curves. In the second part of our paper, we propose a method to obtain a modified Tate pairing with small \kappa for any pairing-friendly elliptic curves including those not belonging to parameterized families. More precisely, our method finds an integer m using the lattice basis reduction such that \kappa (m\lambda)=\left (1- 1\over \rho \varphi (k)\right)\log 2 q. Using this modified Tate pairing, we can reduce the number of squarings in the final exponentiation by a factor of \left (1- 1\over \rho \varphi (k)\right) from the usual Tate pairing. We apply our method to several known pairing-friendly curves to verify the expected speedup. [ABSTRACT FROM PUBLISHER]

In this paper, a new generalized cyclotomy with respect to a positive odd integer is introduced, and a construction of frequency hopping sequence sets and two constructions of frequency hopping sequences are proposed as its applications. The frequency hopping sequence sets and frequency hopping sequences obtained in this paper can be optimal with respect to the Peng–Fan bound and Lempel–Greenberger bound, respectively. Further, the length of sequences in the optimal frequency hopping sequence sets can be any odd integer larger than 3. Some of them have new parameters. [ABSTRACT FROM PUBLISHER]

This paper studies networks where all nodes are distributed on a unit square A\buildrel\triangle\over= [- 1\over 2, 1\over 2]^2 following a Poisson distribution with known density \rho and a pair of nodes separated by an Euclidean distance x are directly connected with probability gr\rho(x){\buildrel{\triangle}\over{=}} g(x/r\rho), independent of the event that any other pair of nodes are directly connected. Here, g:[0,\infty)\rightarrow [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g\left (x\right)=o(1/(x^2\log^2x)); further, r\rho=\sqrt {(\log \rho +b)/(C\rho)} where C=\int\Re^{2}g(\left \Vert {\mmb {x}}\right \Vert)d {\mmb {x}} and b is a constant. Denote the aforementioned network by \cal G\left (\cal X\rho,gr\rho,A\right). We show that as \rho \rightarrow \infty, 1) the distribution of the number of isolated nodes in \cal G\left (\cal X\rho,gr\rho,A\right) converges to a Poisson distribution with mean e^-b; 2) asymptotically almost surely (a.a.s.) there is no component in \cal G\left (\cal X\rho,gr\rho,A\right) of fixed and finite order k> 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as \rho \rightarrow \infty, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for \cal G\left (\cal X\rho,gr\rho,A\right) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b\rightarrow \infty as \rho \rightarrow \infty. These results expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more general and more practical random connection model. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Blahut–Arimoto algorithm for maximizing Massey's directed information. The algorithm can be used for estimating the capacity of channels with delayed feedback, where the feedback is a deterministic function of the output. In order to maximize the directed information, we apply the ideas from the regular Blahut–Arimoto algorithm, i.e., the alternating maximization procedure, to our new problem. We provide both upper and lower bound sequences that converge to the optimum global value. Our main insight in this paper is that in order to find the maximum of the directed information over a causal conditioning probability mass function, one can use a backward index time maximization combined with the alternating maximization procedure. We give a detailed description of the algorithm, showing its complexity and the memory needed, and present several numerical examples. [ABSTRACT FROM AUTHOR]

Let M be a square-free odd integer and \bf Z/(M) the integer residue ring modulo M. This paper studies the distinctness of primitive sequences over \bf Z/(M) modulo 2. Recently, for the case of M=pq, a product of two distinct prime numbers p and q, the problem has been almost completely solved. As for the case that M is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2n^\prime+1 over \bf Z/(M) is distinct modulo 2, where n^\prime is a positive integer. Besides as an independent interest, this paper also involves two distribution properties of primitive sequences over \bf Z/(M), which are related closely to our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the redundancy of Huffman codes. In particular, we consider sources for which the probability of one of the source symbols is known. We prove a conjecture of Ye and Yeung regarding the upper bound on the redundancy of such Huffman codes, which yields in a tight upper bound. We also derive a tight lower bound for the redundancy under the same assumption. We further apply the method introduced in this paper to other related problems. It is shown that several other previously known bounds with different constraints follow immediately from our results. [ABSTRACT FROM PUBLISHER]

In 1986, D. V. Chudnovsky and G. V. Chudnovsky first employed algebraic curves over finite fields to construct bilinear multiplication algorithms implicitly through supercodes introduced by Shparlinski–Tsfasman–Vladuţ, or equivalently, multiplication-friendly codes that we will introduce in this paper. This idea was further developed by Shparlinski–Tsfasman–Vladuţ in order to study the asymptotic behavior of multiplication complexity in extension fields. Later on, Ballet et al. further investigated the method and obtained some improvements. Recently, Ballet and Pieltant made use of curves over an extension field of \BBF2 to obtain an improvement on the complexity of multiplications in extensions of the binary field. In this paper, we develop the multiplication-friendly splitting technique and then apply this technique to study asymptotic behavior of multiplications in extension fields. By combining this with the idea of using algebraic function fields, we are able to improve further the asymptotic results of multiplication complexity. In particular, the improvement for small fields such as the binary and ternary fields is substantial. [ABSTRACT FROM PUBLISHER]

The fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests (measurements). The key contribution of this paper is in adopting a new information-theoretic perspective on group testing problems. We formulate the group testing problem as a channel coding/decoding problem and derive a single-letter characterization for the total number of tests used to identify the defective set. Although the focus of this paper is primarily on group testing, our main result is generally applicable to other compressive sensing models. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the pairing inversion problem of Atei pairing can be solved under the assumption that we have a generic algorithm for solving “exponentiation inversion” problem. With such an algorithm, the inversion problem of Atei pairing can be solved without solving the Miller inversion. Thus, the pairing inversion problem of Atei pairing is reduced to the exponentiation inversion problem. [ABSTRACT FROM AUTHOR]

In an earlier transactions paper, Rosnes and Ytrehus presented an efficient algorithm for determining all stopping sets of low-density parity-check (LDPC) codes, up to a specified weight, and also gave results for a number of well-known codes including the family of IEEE 802.16e LDPC codes, commonly referred to as the WiMax codes. It is the purpose of this short paper to review the algorithm for determining the initial part of the stopping set weight spectrum (which includes the codeword weight spectrum), and to provide some improvements to the algorithm. As a consequence, complete stopping set weight spectra up to weight 32 (for selected IEEE 802.16e LDPC codes) can be provided, while in previous work only stopping set weights up to 28 are reported. In the published standard for the IEEE 802.16e codes there are two methods of construction presented, depending upon the code rate and the code length. We compare the stopping sets of the resulting codes and provide complete stopping set weight spectra (up to five terms) for all IEEE 802.16e LDPC codes using both construction methods. [ABSTRACT FROM PUBLISHER]

It is known that a high-dimensional sparse vector \bf x^* in \cal R^n can be recovered from low-dimensional measurements \bf y=A\bf x^* where A^m \times n(m < n) is the measurement matrix. In this paper, with A being a random Gaussian matrix, we investigate the recovering ability of \ell_p-minimization (0\leq p \leq 1) as p varies, where \ell_p-minimization returns a vector with the least \ell_p quasi-norm among all the vectors \bf x satisfying A\bf x=\bf y. Besides analyzing the performance of strong recovery where \ell_p-minimization is required to recover all the sparse vectors up to certain sparsity, we also for the first time analyze the performance of “weak” recovery of \ell_p-minimization (0\leq p < 1) where the aim is to recover all the sparse vectors on one support with a fixed sign pattern. When \alpha (:=m \over n) \rightarrow 1, we provide sharp thresholds of the sparsity ratio (i.e., percentage of nonzero entries of a vector) that differentiates the success and failure via \ell_p-minimization. For strong recovery, the threshold strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. Surprisingly, for weak recovery, the threshold is 2/3 for all p in [0,1), while the threshold is 1 for \ell_1-minimization. We also explicitly demonstrate that \ell_p-minimization (p < 1) can return a denser solution than \ell_1-minimization. For any \alpha \in (0,1), we provide bounds of the sparsity ratio for strong recovery and weak recovery, respectively, below which \ell_p-minimization succeeds. Our bound of strong recovery improves on the existing bounds when \alpha is large. In particular, regarding the recovery threshold, this paper argues that \ell_p-minimization has a higher threshold with smaller p for strong recovery; the threshold is the same for all p for sectional recovery; and \ell_1-minimization can outperform \ell_p-minimization for weak recovery. These are in contrast to traditional wisdom that \ell_p-minimization, though computationally more expensive, always has better sparse recovery ability than \ell_1-minimization since it is closer to \ell_0-minimization. Finally, we provide an intuitive explanation to our findings. Numerical examples are also used to unambiguously confirm and illustrate the theoretical predictions. [ABSTRACT FROM AUTHOR]