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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]

In this paper, we put forth a new joint sparse recovery algorithm called signal space matching pursuit (SSMP). The key idea of the proposed SSMP algorithm is to sequentially investigate the support of jointly sparse vectors to minimize the subspace distance to the residual space. Our performance guarantee analysis indicates that SSMP accurately reconstructs any row K-sparse matrix of rank r in the full row rank scenario if the sampling matrix $\mathbf {A}$ satisfies $\mathop{\mathrm{krank}}\limits (\mathbf {A}) \ge {K}+1$ , which meets the fundamental minimum requirement on $\mathbf {A}$ to ensure exact recovery. We also show that SSMP guarantees exact reconstruction in at most ${K}-{r}+\lceil \frac {{r}}{{L}} \rceil $ iterations, provided that $\mathbf {A}$ satisfies the restricted isometry property (RIP) of order ${L}({K}-{r})+{r}+1$ with $\delta _{{L}({K}-{r})+{r}+1} < \max \left \{{ \frac {\sqrt {{r}}}{\sqrt {{K}+\frac {{r}}{4}}+\sqrt {\frac {{r}}{4}}}, \frac {\sqrt {{L}}}{\sqrt {K}+1.15 \sqrt {{L}}} }\right \}$ , where L is the number of indices chosen in each iteration. This implies that the requirement on the RIP constant becomes less restrictive when r increases. Such behavior seems to be natural but has not been reported for most of conventional methods. We also show that if ${r}=1$ , then by running more than K iterations, the performance guarantee of SSMP can be improved to $\delta _{\lfloor 7.8{K} \rfloor } \le 0.155$. Furthermore, we show that under a suitable RIP condition, the reconstruction error of SSMP is upper bounded by a constant multiple of the noise power, which demonstrates the robustness of SSMP to measurement noise. Finally, from extensive numerical experiments, we show that SSMP outperforms conventional joint sparse recovery algorithms both in noiseless and noisy scenarios. [ABSTRACT FROM AUTHOR]

In index coding, a server broadcasts multiple messages to their respective receivers, each with some side information that can be utilized to reduce the amount of communication from the server. Distributed index coding is an extension of index coding in which the messages are broadcast from multiple servers, each storing different subsets of the messages. In this paper, the optimal tradeoff among the message rates and the server broadcast rates, which is defined formally as the capacity region, is studied for a general distributed index coding problem. Inner and outer bounds on the capacity region are established that have matching sum-rates for all 218 non-isomorphic four-message problems with equal link capacities for all the links from servers to receivers. The proposed inner bound is built on a distributed composite coding scheme that outperforms the existing schemes by incorporating more flexible decoding configurations and enhanced fractional rate allocations into two-stage composite coding, a scheme that was originally introduced for centralized index coding. The proposed outer bound is built on the polymatroidal axioms of entropy, as well as functional dependences such as the fd-separation introduced by the multi-server nature of the problem. This outer bound utilizes general groupings of servers with different levels of granularity, which allows a natural tradeoff between computational complexity and tightness of the bound, and includes and improves upon all existing outer bounds for distributed index coding. Specific features of the proposed inner and outer bounds are demonstrated through concrete examples with four or five messages. [ABSTRACT FROM AUTHOR]

In Eurocrypt’18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over $\mathbb {F}_{2^{2m}}$ with $m$ odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined. [ABSTRACT FROM AUTHOR]

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

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length $ {n} > {q}+1$ are also obtained. [ABSTRACT FROM AUTHOR]

Hermitian functional and differential codes are AG-codes defined on a Hermitian curve. To ensure good performance, the divisors defining such AG-codes have to be carefully chosen, exploiting the rich combinatorial and algebraic properties of the Hermitian curves. In this paper, the case of differential codes $C_{\Omega }(\mathtt {D},m\mathtt {T})$ on the Hermitian curve $\mathscr {H}_{q^{3}}$ defined over $\mathbb {F}_{q^{6}}$ is worked out where $\mathop {\mathrm {supp}}\nolimits (\mathtt {T}):= \mathscr {H}_{q^{3}}(\mathbb {F}_{q^{2}})$ , the set of all $\mathbb {F}_{q^{2}}$ -rational points of $\mathscr {H}_{q^{3}}$ , while $\mathtt {D}$ is taken, as usual, to be the sum of the points in the complementary set $D = \mathscr {H}_{q^{3}}(\mathbb {F}_{q^{6}})\setminus \mathscr {H} _{q^{3}}(\mathbb {F}_{q^{2}})$. For certain values of $m$ , such codes $C_{\Omega }(\mathtt {D},m\mathtt {T})$ have better minimum distance compared with true values of 1-point Hermitian codes. The automorphism group of $C_{L}(\mathtt {D},m\mathtt {T})$ , $m\leq q^{3}-2$ , is isomorphic to $PGU(3,q)$. [ABSTRACT FROM AUTHOR]

In the traditional index coding problem, a server employs coding to send messages to a set of clients within the same broadcast domain. Each client already has some messages as side information and requests a particular unknown message from the server. All clients learn the coding matrix so that they can decode and retrieve their requested data. Our starting observation comes from the work by Karmoose et al., which shows that learning the coding matrix can pose privacy concerns: it may enable a client to infer information about the requests and side information of other clients. In this paper, we mitigate this privacy concern by allowing each client to have limited access to the coding matrix. In particular, we design coding matrices so that each client needs only to learn some of (and not all) the rows to decode her requested message. We start by showing that this approach can indeed help mitigate that privacy concern. We do so by considering two different privacy metrics. The first one shows the attained privacy benefits based on a geometric interpretation of the problem. Differently, the second metric, referred to as maximal information leakage, provides upper bounds on: (i) the guessing power of the adversaries (i.e., curious clients) when our proposed approach is employed, and (ii) the effect of decreasing the number of accessible rows on the attained privacy. Then, we propose the use of $k$ -limited-access schemes: given an index coding scheme that employs $T$ transmissions, we create a $k$ -limited-access scheme with $T_{k}\geq T$ transmissions, and with the property that each client needs at most $k$ transmissions to decode her message. We derive upper and lower bounds on $T_{k}$ for all values of $k$ , and develop deterministic designs for these schemes, which are universal, i.e., independent of the coding matrix. We show that our schemes are order-optimal for some parameter regimes, and we propose heuristics that complement the universal schemes for the remaining regimes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the partition inequality, joint convexity, and Pinsker’s inequality, for a divergence that generalizes the Tsallis Relative Entropy and Kullback–Leibler divergence. The generalized divergence is defined in terms of a deformed exponential function, which replaces the Tsallis $q$ -exponential. We also constructed a family of probability distributions related to the generalized divergence. We found necessary and sufficient conditions for the partition inequality to be satisfied. A sufficient condition for the joint convexity was established. We proved that the generalized divergence satisfies the partition inequality, and is jointly convex, if, and only if, it coincides with the Tsallis relative entropy. As an application of partition inequality, a criterion for the Pinsker’s inequality was found. [ABSTRACT FROM AUTHOR]

Splitter sets have been widely studied due to their applications in flash memories, and their close relations with lattice tilings and conflict avoiding codes. In this paper, we give necessary and sufficient conditions for the existence of nonsingular perfect splitter sets, ${B}[-{k}_{1},{k}_{2}]({p})$ sets, where $0\le {k}_{1}\leq {k}_{2}=4$. Meanwhile, constructions of nonsingular perfect splitter sets are given. When perfect splitter sets do not exist, we present four new constructions of quasi-perfect splitter sets. Finally, we give a connection between nonsingular splitter sets and Cayley graphs, and as a byproduct, a general lower bound on the maximum size of nonsingular splitter sets is given. [ABSTRACT FROM AUTHOR]

In this paper we investigate the construction and the properties of spatially inhomogeneous divergences, functionals arising from optimal Entropy-Transport problems that are computed in terms of an entropy function $F$ and a cost function. Starting from the power-like entropy $F(s)=(s^{p}-p(s-1)-1)/ (p(p-1))$ and a suitable cost depending on a metric $\mathsf {d}$ on a space $X$ , our main result ensures that for every $p>1$ the related inhomogeneous divergence induces a distance on the space of finite measures over $X$. We also study in detail the pure entropic setting, that can be recovered as a particular case when the transport is forbidden. In this situation, corresponding to the classical theory of $F$ -divergences, we show that the construction naturally produces a symmetric divergence and we highlight the important role played by the class of Matusita’s divergences. [ABSTRACT FROM AUTHOR]

Let $\Bbb F_{q}$ be a finite field with $q$ elements and $p$ be a prime with $\gcd (p,q)=1$. Let $G$ be a finite abelian $p$ -group and $\Bbb F_{q}(G)$ be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra $\Bbb F_{q}(G)$. Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of $\Bbb F_{q}(G)$. [ABSTRACT FROM AUTHOR]

Permutation codes and multipermutation codes are widely studied due to various applications in information theory. Designing codes correcting deletion errors has been the main subject of works in the literature and to the best of our knowledge, there exist only optimal codes capable of correcting a single deletion in a permutation. In this paper, we construct several classes of permutation and multipermutation codes that are capable of correcting a burst deletion of length $s \geq 2$ , for both stable and unstable models. Efficient error decoders are provided to show the correctness of our constructions. [ABSTRACT FROM AUTHOR]

Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of $x^{r}h(x^{(q-1)/d})$ , those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results. [ABSTRACT FROM AUTHOR]

Offset codebook or ${\mathsf {OCB}}$ mode is a popular block cipher mode of operation for authenticated encryption. The latest version of this cipher, called ${\mathsf {OCB3}}$ , is one of the finalists in CAESAR. In this paper, we explore the scope of random read access and out-of-sequence decryption in ${\mathsf {OCB}}$. We observe that the current versions of ${\mathsf {OCB}}$ are inefficient in this respect owing to the ineptness of the underlying mask generating function (MGF). We propose new candidates for MGF based on ${\mathsf {AES}}$ round function, which are efficient in direct computation and provide comparable performance in the usual setting. Our schemes are not the obvious choices for MGF in conventional sense as they do not have optimal almost XOR universal (AXU) bound. In existing ${\mathsf {OCB}}$ designs, the MGFs are required to have $ 2^{-n} $ , i.e. optimal, AXU bound in order to upper bound the distinguishing advantage to $ O(\sigma ^{2}/2^{n}) $ , where $ n $ is the block size of the underlying block cipher and $ \sigma $ is the total number of blocks among all queries. We find this specific requirement too restrictive. We abstract the ${\mathsf {OCB}}$ design, termed as ${\mathsf {GOCB}}$ , to look into the universal notion required from the underlying MGF. We propose a relaxed notion of AXU, called locally imperfect XOR universal (LIXU) hash, which can be of independent interest. Using LIXU as the underlying MGF, we recover reasonable security bounds for our schemes. [ABSTRACT FROM AUTHOR]

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

In EUROCRYPT 2018, Cid et al. introduced a new concept on the cryptographic property of S-boxes: boomerang connectivity table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut. In this paper, aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment. First, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition by Cid et al. Second, thanks to Carlet’s idea, we give a characterization of functions $f$ from $ {\mathbb F}_{2}^{n}$ to itself with boomerang uniformity $\delta _{f}$ by means of the Walsh transform. Third, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of 4-uniform BCT permutation polynomials over $ {\mathbb F}_{2^{n}}$. [ABSTRACT FROM AUTHOR]

Minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two-party computation. In this paper, we focus on constructing minimal linear codes with $\frac {w_{\min }}{w_{\max }}\leq \frac {p-1}{p}$ for any odd prime $p$ based on a generic construction of linear codes, where $w_{\min }$ and $w_{\max }$ denote the minimum and maximum nonzero weights in a code, respectively. First, we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal. Second, we also present an infinite family of minimal linear codes by employing partial spreads, which can be viewed as a generalization of the construction of Ding et al. In addition, we determine the weight distributions of all these minimal linear codes. [ABSTRACT FROM AUTHOR]

Parent-identifying schemes provide a way to identify causes from effects for some information systems, such as digital fingerprinting and group testing. In this paper, we consider the combinatorial structures for parent-identifying schemes. First, we establish an equivalent relationship between the parent-identifying schemes and forbidden configurations. Based on this relationship, we derive the probabilistic existence lower bounds for two related combinatorial structures, that is, $t$ -parent-identifying set systems ($t$ -IPPS) and $t$ -multimedia parent-identifying codes ($t$ -MIPPC), which are used in broadcast encryption and multimedia fingerprinting, respectively. The probabilistic lower bound for the maximum size of a $t$ -IPPS has the asymptotically optimal order of magnitude in many cases, and that for $t$ -MIPPC provides the asymptotically optimal code rate when $t=2$ and the best known asymptotic code rate when $t\ge 3$. Furthermore, we analyze the structure of 2-IPPS and prove some bounds for certain cases. [ABSTRACT FROM AUTHOR]

The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some substructures of the code’s Tanner graph, collectively referred to as trapping sets (TSs). In this paper, we study the asymptotic average number of different types of trapping sets such as elementary TSs (ETS), leafless ETSs (LETS), absorbing sets (ABS), elementary ABSs (EABS), and stopping sets (SS), in random variable-regular and irregular LDPC code ensembles. We demonstrate that, regardless of the type of the TS, as the code’s length tends to infinity, the average number of a given structure tends to infinity, to a positive constant, or to zero, if the structure contains no cycle, only one cycle, or more than one cycle, respectively. For the case where the structure contains a single cycle, we derive the asymptotic expected multiplicity of the structure by counting the average number of its constituent cycles and all the possible ways that the structure can be constructed from the cycle. This, in general, involves computing the expected number of cycles of a certain length with a certain given combination of node degrees, or computing the expected number of cycles of a certain length expanded to the desired structure by the connection of trees to its nodes. The asymptotic results obtained in this work, which are independent of the block length and only depend on the code’s degree distributions, are shown to be accurate even for finite-length codes. [ABSTRACT FROM AUTHOR]