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In this paper, we study spatial degree-of-freedom (DOF) for two mutually interfering broadcast channels (IBC). As the demand for space-division multiple access (SDMA) increases, the IBC where each link has a single transmit node and multiple receive nodes becomes important. This paper presents the results of the lower and upper bounds on the DOF of the IBC. From the derived results, it is shown that for most cases, zero-forcing (ZF) beamforming can achieve the optimal DOF of the IBC except for some special cases. Also, we identify a condition that disabling receive cooperation in the multiple-input multiple-output interference channels causes no DOF loss. It is confirmed that we cannot expect a DOF improvement by enabling in-cell receive cooperation if in at least one of two BSs, the number of antennas is greater than or equal to that of users per cell. Furthermore, we observe a positive result that as the number of users goes to infinity, the total DOF of the IBC converges to the interference-free DOF, which is the maximum achievable DOF in the absence of the inter-cell interference. [ABSTRACT FROM PUBLISHER]

Hypergraphical sources are a natural class of sources for secret key generation, within which different subsets of terminals sharing secrets are allowed to discuss publicly in order to agree upon a global secret key. While their secrecy capacity, i.e., the maximum rate of a secret key that can be agreed upon by the entire set of terminals, is well-understood, what remains open is the maximum rate of a secret key that can be generated when there is a restriction on the overall rate of public discussion allowed. In this paper, we obtain a family of explicitly computable upper bounds on the number of bits of secret key that can be generated per bit of public discussion. These upper bounds are derived using a lamination technique based on the submodularity of the entropy function. In particular, a specific instance of these upper bounds, called the edge-partition bound, is shown to be tight for the pairwise independent network model, a special case of the hypergraphical source when the hypergraph is a graph. The secret key generation scheme achieving this upper bound is the tree-packing protocol of Nitinawarat et al., thereby resolving in the affirmative the discussion rate optimality of the tree-packing protocol. [ABSTRACT FROM AUTHOR]

The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the $(0,k)$ -runlength limited (RLL) constraint, i.e., no more than $k$ consecutive ‘0’s are allowed. The feedback capacity of this channel is derived for all $k\geq 1$ , and is given by $C^{\mathrm {fb}}_{(0,k)}(\varepsilon) = \max \frac {\overline {\varepsilon }H_{2}(\delta _{0})+\sum _{i=1}^{k-1}\left ({\overline {\varepsilon }^{i+1}H_{2}(\delta _{i})\prod _{m=0}^{i-1}\delta _{m}}\right)}{1+\sum _{i=0}^{k-1}\left ({\overline {\varepsilon }^{i+1} \prod _{m=0}^{i}\delta _{m}}\right)}$ , where $\varepsilon $ is the erasure probability, $\overline {\varepsilon }=1-\varepsilon $ and $H_{2}(\cdot)$ is the binary entropy function. The maximization is only over $\delta _{k-1}$ , while the parameters $\delta _{i}$ for $i\leq k-2$ are straightforward functions of $\delta _{k-1}$. The lower bound is obtained by constructing a simple coding for all $k\geq 1$. It is shown that the feedback capacity can be achieved using zero-error, variable length coding. For the converse, an upper bound on the non-causal setting, where the erasure is available to the encoder just prior to the transmission, is derived. This upper bound coincides with the lower bound and concludes the search for both the feedback capacity and the non-causal capacity. As a result, non-causal knowledge of the erasures at the encoder does not increase the feedback capacity for the $(0,k)$ -RLL input-constrained BEC. This property does not hold in general: the $(2,\infty)$ -RLL input-constrained BEC, where every ‘1’ is followed by at least two ‘0’s, is used to show that the feedback capacity can be strictly smaller than the non-causal capacity. [ABSTRACT FROM AUTHOR]

Grassmannian codes are known to be useful in error correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this paper, we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for the network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts, it is proved that the codes in the Hamming scheme form a subfamily of the Grassmannian codes. [ABSTRACT FROM AUTHOR]

New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint must only be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be an unbounded set. [ABSTRACT FROM AUTHOR]

For locally repairable codes (LRCs), Cadambe and Mazumdar derived the first field-dependent parameter bound, known as the C-M bound. However, the C-M bound depends on an undetermined parameter $ {k}^{({q})}_{{\textbf {opt}}}({n}, {d})$. In this paper, a sphere-packing approach is developed for upper bounding the parameter $ {k}$ for $[ {n}, {k}, {d}]$ linear LRCs with locality $ {r}$. When restricted to the binary field, three upper bounds (i.e., Bound A, Bound B, and Bound C) are derived in an explicit form. More specifically, Bound A holds under the hypothesis that the local repair groups are disjoint and of equal size. Comparing with previous bounds obtained under the same hypothesis, Bound A either covers them as special cases or has an advantage due to its explicit form. Then, the hypothesis is removed in Bound B and Bound C. As the price for explicit form, Bound B specially holds for $ {d}\geq \text {5}$ and Bound C for $ {r}=\textbf {2}$. Through specific comparisons, we show that Bound B and Bound C both tend to outperform the C-M bound, as $ {n}$ goes large. Moreover, a family of binary linear LRCs with $ {d}\geq \textbf {6}$ attaining Bound B are constructed and later extended to a wider range of parameters by a shortening technique. Lastly, most of the bounds and constructions are extended to $ {q}$ -ary LRCs. [ABSTRACT FROM AUTHOR]

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criterion for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring the subcodes of Gabidulin codes, if each of the rightmost $\delta -1$ columns in the Ferrers diagram $\cal F$ has at least $n-r$ dots, where $r$ is taken in a range, then the conditions that an FDRM code in $\cal F$ is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams. [ABSTRACT FROM AUTHOR]

Sparse representation over redundant dictionaries constitute a good model for many classes of signals (e.g., patches of natural images, segments of speech signals, and so on). However, despite its popularity, a very little are known about the representation capacity of this model. In this paper, we study how redundant a dictionary must be so as to allow any vector to admit a sparse approximation with a prescribed sparsity and a prescribed level of accuracy. We address this problem, both in a worst-case setting and in an average-case one. For each scenario, we derive lower and upper bounds on the minimal required overcompleteness. Our bounds have simple closed-form expressions that allow to easily deduce the asymptotic behavior in large dimensions. In particular, we find that the required overcompleteness grows exponentially with the sparsity level and polynomially with the allowed representation error. This implies that universal sparse representation is practical only at moderate sparsity levels, but can be achieved with a relatively high accuracy. As a side effect of our analysis, we obtain a tight lower bound on the regularized incomplete beta function, which may be interesting in its own right. We illustrate the validity of our results through numerical simulations, which support our findings. [ABSTRACT FROM AUTHOR]

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a family of easily encodable rate-compatible low-density parity-check (LDPC) codes. PBRL codes have an excellent performance across all design rates. Quasi-cyclic (QC) PBRL code families permit high-speed decoder implementations. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the PBRL family at various design rates. This paper introduces a design method to obtain better QC PBRL code families at short block-lengths (of a few hundred bits) for low frame error rate (FER) requirements. We first select a protomatrix for the highest design rate. To add a new row to lower the rate, we keep all the previously obtained rows of the PBRL protomatrix fixed and select the new row that maximizes an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protomatrix. The new QC PBRL code families outperform the original PBRL codes at short block-lengths by providing a significantly better low-FER performance. The standard approach to computing the aforementioned upper bounds requires complexity that grows exponentially with the size of the protomatrix. However, we show that the structure of the PBRL protomatrix lets us obtain the upper bounds with complexity that grows only linearly with the size of the PBRL protomatrix. Using the complexity reduction results, we also establish an equivalence between the exhaustive search to design a new row for the PBRL protomatrix according to the new design method and an integer linear program. [ABSTRACT FROM AUTHOR]

A binary matrix is called an ${s}$ -separable code for the disjunctive multiple-access channel (disj-MAC) if Boolean sums of sets of ${s}$ columns are all distinct. The well-known issue of the combinatorial coding theory is to obtain upper and lower bounds on the rate of ${s}$ -separable codes for the ${disj}$ -MAC. In our paper, we generalize the problem and discuss upper and lower bounds on the rate of ${q}$ -ary ${s}$ -separable codes for the models of noiseless symmetric MAC, i.e., at each time instant the output signal of MAC is a symmetric function of its ${s}$ input signals. [ABSTRACT FROM AUTHOR]

Mutually uncorrelated (MU) codes are a class of codes in which no proper prefix of one codeword is a suffix of another codeword. These codes were originally studied for synchronization purposes and recently, Yazdi et al. showed their applicability to enable random access in DNA storage. In this paper,we follow the research of Yazdi et al. and study MU codes along with their extensions to correct errors and balanced codes. We first review a well-known construction of MU codes and study the asymptotic behavior of its cardinality. This task is accomplished by studying a special class of run-length limited codes that impose the longest run of zeros to be at most some function of the codewords length. We also present an efficient algorithm for this class of constrained codes and show how to use this analysis for MU codes. Next, we extend the results on the run-length limited codes in order to study $(d_{h},d_{m})$ -MU codes that impose a minimum Hamming distance of $d_{h}$ between different codewords and $d_{m}$ between prefixes and suffixes. In particular, we show an efficient construction of these codes with nearly optimal redundancy. We also provide similar results for the edit distance and balanced MU codes. Last, we draw connections to the problems of comma-free and prefix synchronized codes. [ABSTRACT FROM AUTHOR]

Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing. In this paper, we give the first quantum algorithms for estimating $\alpha $ -Rényi entropies (Shannon entropy being 1-Rényi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for $\alpha $ -Rényi entropy estimation for all $\alpha \geq 0$ values, including tight bounds for the Shannon entropy, the Hartley entropy ($\alpha =0$), and the collision entropy ($\alpha =2$). We also provide quantum upper bounds for estimating min-entropy ($\alpha =+\infty $) as well as the Kullback–Leibler divergence. We complement our results with quantum lower bounds on $\alpha $ -Rényi entropy estimation for all $\alpha \geq 0$ values. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH); however, with many new technical ingredients: 1) we improve the error dependence of the BHH framework by a fine-tuned error analysis together with Montanaro’s approach to estimating the expected output of quantum subroutines for $\alpha =0,1$ ; 2) we develop a procedure, similar to cooling schedules in simulated annealing, for general $\alpha \geq 0$ , and 3) in the cases of integer $\alpha \geq 2$ and $\alpha =+\infty $ , we reduce the entropy estimation problem to the $\alpha $ -distinctness and $\lceil \log n\rceil $ -distinctness problems, respectively. [ABSTRACT FROM AUTHOR]

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth. [ABSTRACT FROM AUTHOR]

Gaussian thermal loss channels are of particular importance to quantum communication theory since they model realistic optical communication channels. Except for special cases, the quantum capacity of Gaussian thermal loss channels is not yet quantified completely. In this paper, we provide improved upper bounds of the Gaussian thermal loss channel capacity, both in energy-constrained and unconstrained scenarios. We briefly review Gottesman-Kitaev-Preskill (GKP) codes and discuss their experimental implementation. We then prove, in the energy-unconstrained case, that a family of GKP codes achieves the quantum capacity of Gaussian thermal loss channels up to at most a constant gap from the improved upper bound. In the energy-constrained case, we formulate a biconvex encoding and decoding optimization problem to maximize entanglement fidelity. Then, we solve the biconvex optimization heuristically by an alternating semi-definite programming method and report that, starting from Haar random initial codes, our numerical optimization yields a hexagonal GKP code as an optimal encoding in a practically relevant regime. [ABSTRACT FROM AUTHOR]

Uncertainty quantification of a technical system can be done using density estimation. We usually start with a stochastic model, which is fitted to the technical system, and the density estimation is done using data from this stochastic model. However, in any application, such a stochastic model will not be perfect, and the estimation of the density should take into account the inadequacy of the stochastic model. In this paper, we show how observed data of the real system together with an imperfect simulation model can be used to derive confidence bands for the density of the technical system. Our main result is that the newly introduced confidence bands allow to derive lower and upper bounds on the probability of intervals in the technical system. Furthermore, we present an upper bound on the area of the confidence band in case of a smooth density. The results are illustrated by applying the estimates to simulated and real data. [ABSTRACT FROM AUTHOR]

The Korkine–Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around 7/8 times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is polynomially smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even polynomially and exponentially smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr–Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang et al. Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang et al., especially when the basis matrix is ill conditioned. [ABSTRACT FROM AUTHOR]

In this paper, we propose a characterization for non-elementary trapping sets (NETSs) of low-density parity-check (LDPC) codes. The characterization is based on viewing an NETS as a hierarchy of embedded graphs starting from an ETS. The characterization corresponds to an efficient search algorithm that under certain conditions is exhaustive. As an application of the proposed characterization/search, we obtain lower and upper bounds on the stopping distance $s_{\min }$ of LDPC codes. We examine a large number of regular and irregular LDPC codes and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $s_{\min }$. Finding $s_{\min }$ , or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm. For a constant degree distribution and range of search, the worst case computational complexity of the proposed search algorithms for finding NETSs and stopping sets is linear in the code’s block length $n$. The average search complexity for stopping sets, however, is constant in $n$ , if the simple cycles required as input to the search algorithm are available. [ABSTRACT FROM AUTHOR]

Synthesis of DNA molecules offers unprecedented advances in storage technology. Yet, the microscopic world in which these molecules reside induces error patterns that are fundamentally different from their digital counterparts. Hence, to maintain reliability in reading and writing, new coding schemes must be developed. In a reading technique called shotgun sequencing, a long DNA string is read in a sliding window fashion, and a profile vector is produced. It was recently suggested by Kiah et al. that such a vector can represent the permutation which is induced by its entries, and hence a rank-modulation scheme arises. Although this interpretation suggests high error tolerance, it is unclear which permutations are feasible and how to produce a DNA string whose profile vector induces a given permutation. In this paper, by observing some necessary conditions, an upper bound for the number of feasible permutations is given. Furthermore, a technique for deciding the feasibility of a permutation is devised. By using insights from this technique, an algorithm for producing a considerable number of feasible permutations is given, which applies to any alphabet size and any window length. [ABSTRACT FROM AUTHOR]

In this paper, we study upper and lower bounds on the error exponents for joint source-channel coding with decoder side-information. The results in the paper are nontrivial extensions of Csiszár's classical paper “Joint Source-Channel Error Exponent”, Problems of Control and Information Theory, 1980. Unlike the joint source-channel coding result in Csiszár's paper, it is not obvious whether the lower bound and the upper bound are equivalent even if the channel coding error exponent is known. For a class of channels, including symmetric channels, we apply a game-theoretic result to establish the existence of a saddle point and, hence, prove that the lower and upper bounds are the same if the channel coding error exponent is known. More interestingly, we show that encoder side-information does not increase the error exponents in this case. [ABSTRACT FROM AUTHOR]

The polarization process of polar codes over a prime $q$ -ary alphabet is studied. Recently, it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this paper, a different approach to computing the degree of this polynomial for any prime alphabet size is shown. This approach yields a lower degree polynomial for various values of the alphabet size that were examined. It is also shown that even lower degree polynomial can be computed with an additional numerical effort. [ABSTRACT FROM AUTHOR]

When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), at the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have arbitrary distance requirements. Unlike existing methods, where both the locality and the local distance requirements imposed on every node are identical, we allow the requirements to vary arbitrarily from node to node. Such a property can be an advantage for distributed storage systems with non-homogeneous characteristics. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by dimension upper bounds that do not depend on the distance. [ABSTRACT FROM AUTHOR]

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

New lower and upper bounds on the reliability function of typewriter channels are given. Our lower bounds improve upon the (multiletter) expurgated bound of Gallager, furnishing a new and simple counterexample to a conjecture made in 1967 by Shannon, Gallager and Berlekamp on its tightness. The only other known counter example is due to Katsman, Tsfasman and Vlăduţ who used algebraic-geometric codes on a $q$ -ary symmetric channels, $q\geq 49$. Here we prove, by introducing dependence between codewords of a random ensemble, that the conjecture is false even for a typewriter channel with $q=4$ inputs. In the process, we also demonstrate that Lovász’s proof of the capacity of the pentagon was implicitly contained (but unnoticed!) in the works of Jelinek and Gallager on the expurgated bound done at least ten years before Lovász. In the opposite direction, new upper bounds on the reliability function are derived for channels with an odd number of inputs by using an adaptation of Delsarte’s linear programming bound. First, we derive a bound based on the minimum distance, which combines Lovász’s construction for bounding the graph capacity with the McEliece-Rodemich-Rumsey-Welch construction for bounding the minimum distance of codes in the Hamming space. Then, for the particular case of cross-over probability 1/2, we derive an improved bound by also using the method of Kalai and Linial to study the spectrum distribution of codes. [ABSTRACT FROM AUTHOR]

The expansion complexity is a new figure of merit for cryptographic sequences. In this paper, we present an explicit formula of the (irreducible) expansion complexity of ultimately periodic sequences over finite fields. We also provide improved upper and lower bounds on the $N$ th irreducible expansion complexity when they are not explicitly determined. In addition, for some infinite sequences with given nonlinear complexity, a tighter upper bound of their $N$ th expansion complexity is given. [ABSTRACT FROM AUTHOR]

A complete characterization for the restricted isometry constant (RIC) bounds on $\delta _{{{ tk}}}$ for all $ {t}>0$ is an important problem on recovery of sparse signals with prior support information via weighted $\ell _{{p}}$ -minimization ($0 < {p} \leqslant 1$). In this paper, new bounds on the restricted isometry constants $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$), where $d$ is a key constant determined by prior support information, are established to guarantee the sparse signal recovery via the weighted $\ell _{{p}}$ minimization in both noiseless and noisy settings. This result fills a vacancy on $\delta _{{{ tk}}}$ with $0 < {t} < \frac {4}{3}{d}$ , compared with previous works on $\delta _{{{ tk}}}$ (${t} \geqslant \frac {4}3{d}$). We show that, when the accuracy of prior support estimate is at least 50%, the new recovery condition in terms of $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$) via weighted $\ell _{1}$ minimization is weaker than the condition required by classical $\ell _{1}$ minimization without weighting. Our weighted $\ell _{1}$ minimization gives better recovery error bounds in noisy setting. Similarly, the new recovery condition in terms of $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$) is extended to weighted $\ell _{{p}}$ ($0 < {p} < 1$) minimization, and it is also weaker than the condition obtained by standard non-convex $\ell _{{p}}$ ($0 < {p} < 1$) minimization without weighting. Numerical illustrations are provided to demonstrate our new theoretical results. [ABSTRACT FROM AUTHOR]

The study of constant-weight codes in $\ell _{1}$ -metric was motivated by the duplication-correcting problem for data storage in live DNA. It is interesting to determine the maximum size of a code given the length ${n}$ , weight ${w}$ , minimum distance ${d}$ and the alphabet size ${q}$. In this paper, based on graph decompositions, we determine the maximum size of ternary codes with constant weight w and distance $2{w}-2$ for all sufficiently large length ${n}$. Previously, this was known only for a very sparse family ${n}$ of density $4/{w}({w}-1)$. [ABSTRACT FROM AUTHOR]

The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell $ defective items, where $0 \leq \ell < u$ , and arbitrary otherwise. This is called threshold group testing. The parameter $g = u - \ell - 1$ is called the gap. In this paper, we focus on the case $g > 0$ , i.e., threshold group testing with a gap. Note that the results presented here are also applicable to the case $g = 0$ ; however, the results are not as efficient as those in related work. Currently, a few reported studies have investigated test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a more accurate theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing and other tasks such as constructing cover-free families or learning hidden graphs. Specifically, we present a better exact upper bound on the number of tests for disjunct matrices compared with that in related work. The third and fourth contributions are a reduced exact upper bound on the number of tests and a reduced asymptotic bound on the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a simulation on the number of tests of the resulting improvements for previous work and the proposed theorems. [ABSTRACT FROM AUTHOR]

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients $a_{1},a_{2},a_{3}\in {\mathbb F} _{2^{n}}$ with $n=2m$ such that $f(x) = {x}^{3\cdot 2^{m}} + a_{1}x^{2^{m+1}+1} + a_{2} x^{2^{m}+2} + a_{3}x^{3}$ is an APN function over ${\mathbb F}_{2^{n}}$. Our work together with the follow-up work by Chase and Lisoněk indicates that all such APN quadrinomials $f(x)$ are affine equivalent to two instances of Gold functions, which resolves the first half of an open problem by Carlet at the International Workshop on the Arithmetic of Finite Fields, 83-107, 2014. [ABSTRACT FROM AUTHOR]

This paper investigates a novel offline change-point detection problem from an information-theoretic perspective. In contrast to most related works, we assume that the knowledge of the underlying pre- and post-change distributions are not known and can only be learned from the training sequences which are available. We further require the probability of the estimation error to decay either exponentially or sub-exponentially fast (corresponding respectively to the large and moderate deviations regimes in information theory parlance). Based on the training sequences as well as the test sequence consisting of a single change-point, we design a change-point estimator and further show that this estimator is optimal by establishing matching (strong) converses. This leads to a full characterization of the optimal confidence width (i.e., half the width of the confidence interval within which the true change-point is located at with high probability) as a function of the undetected error, under both the large and moderate deviations regimes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the list decoding property of codes under insertion and deletion errors (insdel for short). Firstly, we analyse the list decodability of random insdel codes. Our result provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of insdel codes along with their efficient encoding and decoding algorithms through concatenation method which provides a Zyablov-type bound for insdel metric codes. [ABSTRACT FROM AUTHOR]

We consider the use of the well-known dual capacity bounding technique for deriving upper bounds on the capacity of indecomposable finite-state channels (FSCs) with finite input and output alphabets. In this technique, capacity upper bounds are obtained by choosing suitable test distributions on the sequence of channel outputs. We propose test distributions that arise from certain graphical structures called Q-graphs. As we show in this paper, the advantage of this choice of test distribution is that, for the important sub-classes of unifilar and input-driven FSCs, the resulting upper bounds can be formulated as a dynamic programming (DP) problem, which makes the bounds tractable. We illustrate this for several examples of FSCs, where we are able to solve the associated DP problems explicitly to obtain capacity upper bounds that either match or beat the best previously reported bounds. For instance, for the classical trapdoor channel, we improve the best known upper bound of 0.661 (due to Lutz (2014)) to 0.584, shrinking the gap to the best known lower bound of 0.572, all bounds being in units of bits per channel use. [ABSTRACT FROM AUTHOR]

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality r if, for every coordinate, its value at a codeword can be deduced from the value of (certain) r other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods. [ABSTRACT FROM AUTHOR]

Compressed sensing deals with the recovery of sparse signals from linear measurements. Without any additional information, it is possible to recover an s -sparse signal using m \gtrsim s \log (d/s) measurements in a robust and stable way. Some applications provide additional information, such as on the location the support of the signal. Using this information, it is conceivable that the threshold amount of measurements can be lowered. A proposed algorithm for this task is weighted \ell 1 -minimization. Put shortly, one modifies standard \ell 1 -minimization by assigning different weights to different parts of the index set $[1, \ldots d]$ . The task of choosing the weights is, however, non-trivial. This paper provides a complete answer to the question of an optimal choice of the weights. In fact, it is shown that it is possible to directly calculate unique weights that are optimal in the sense that the threshold amount of measurements needed for exact recovery is minimized. The proof uses recent results about the connection between convex geometry and compressed sensing-type algorithms. [ABSTRACT FROM PUBLISHER]

We revisit the problem of secret key agreement using interactive public communication for two parties and propose a new secret key agreement protocol. The protocol attains the secret key capacity for general observations and attains the second-order asymptotic term in the maximum length of a secret key for independent and identically distributed observations. In contrast to the previously suggested secret key agreement protocols, the proposed protocol uses interactive communication. In fact, the standard one-way communication protocol used prior to this paper fails to attain the asymptotic results above. Our converse proofs rely on a recently established upper bound for secret key lengths. Both our lower and upper bounds are derived in a single-shot setup and the asymptotic results are obtained as corollaries. [ABSTRACT FROM PUBLISHER]

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert–Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets. [ABSTRACT FROM AUTHOR]

Gabidulin codes can be seen as the rank-metric equivalent of Reed–Solomon codes. It was recently proved, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially sized lists exist for codes with specific parameters. The second bound presents exponentially sized lists explicitly for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes. [ABSTRACT FROM AUTHOR]

An upper bound on the error probability of specific lattices, based on their distance spectrum, is constructed. The derivation is accomplished using a simple alternative to the Minkowski–Hlawka mean-value theorem of the geometry of numbers. In many ways, the new bound greatly resembles the Shulman–Feder bound for linear codes. Based on the new bound, error-exponent and channel-dispersion expressions are derived for specific lattice sequences (of increasing dimension) over the AWGN channel. Measuring a sequence’s gap to capacity, using the new asymptotics, is demonstrated. Additional finite dimension results, encountered along the way, are presented. [ABSTRACT FROM PUBLISHER]

The $K$ -user multiple-input multiple-output (MIMO) multi-way relay channel (Y-channel) consisting of $K$ users with $M$ antennas each and a common relay node with $N$ antennas is studied in this paper. Each user wants to exchange messages with all the other users through the relay. A transmission strategy is proposed for this channel. The proposed strategy is based on two steps: 1) channel diagonalization and 2) cyclic communication. The channel diagonalization is applied by using zero-forcing beam-forming. After the channel diagonalization, the channel is decomposed into parallel sub-channels. Cyclic communication is then applied, where signal-space alignment for network-coding is used over each sub-channel. The proposed strategy achieves the optimal degree-of-freedom (DoF) region of the channel if $N\leq M$ . To prove this, a new DoFs outer bound is derived. As a by-product, we conclude that the MIMO Y-channel is not separable, i.e., independent coding on separate sub-channels is not enough, and one has to code jointly over several sub-channels. [ABSTRACT FROM AUTHOR]

Separable codes were defined by Cheng and Miao in 2011, motivated by applications to the identification of pirates in a multimedia setting. Combinatorially, \overline t -separable codes lie somewhere between t -frameproof and (t-1) -frameproof codes: all t -frameproof codes are (t-1) -frameproof. Results for frameproof codes show that (when q is large) there are q codewords, and that no q -ary n can have more than approximately codewords. This paper provides improved probabilistic existence results for t\geq 3 . More precisely, for all t\geq 3 and all n\geq 3 , there exists a constant \kappa (depending only on t and n ), such that there exists a codewords for all sufficiently large integers q . This shows, in particular, that the upper bound [derived from the bound on (t-1) -frameproof codes] on the number of codewords in a t=2 . Results due to Gao and Ge show that a q -separable code of length n can contain at most \frac {3}{2}q^{2\lceil n/3\rceil }-\frac {1}{2}q^{\lceil n/3\rceil } codewords, and that codes with at least \kappa q^{2n/3} codewords exist. Thus, optimal \overline 2 -separable codes behave neither like two-frameproof nor one-frameproof codes. This paper also observes that the bound of Gao and Ge can be strengthened to show that the number of codewords of a q -ary \overline 2 -separable code of length $n$ is at most q^{\lceil 2n/3\rceil }+\tfrac {1}{2}q^{\lfloor n/3\rfloor }(q^{\lfloor n/3\rfloor }-1) . [ABSTRACT FROM PUBLISHER]

In this paper, we study the capacity of the diamond channel. We focus on the special case where the channel between the source node and the two relay nodes are two separate links with finite capacities and the link from the two relay nodes to the destination node is a Gaussian multiple access channel. We call this model the Gaussian multiple access diamond channel. We first propose an upper bound on the capacity. This upper bound is a single-letterization of an $n$ -letter upper bound proposed by Traskov and Kramer, and is tighter than the cut-set bound. As for the lower bound, we propose an achievability scheme based on sending correlated codes through the multiple access channel with superposition structure. We then specialize this achievable rate to the Gaussian multiple access diamond channel. Noting the similarity between the upper and lower bounds, we provide sufficient and necessary conditions that a Gaussian multiple access diamond channel has to satisfy such that the proposed upper and lower bounds meet. Thus, for a Gaussian multiple access diamond channel that satisfies these conditions, we have found its capacity. [ABSTRACT FROM PUBLISHER]

Communication systems are traditionally designed to have tight transmitter–receiver synchronization. This requirement has negligible overhead in the high-signal-to-noise ratio (SNR) regime. However, in many applications, such as wireless sensor networks, communication needs to happen primarily in the energy-efficient regime of low SNR, where requiring tight synchronization can be highly suboptimal. In this paper, we model the noisy channel with synchronization errors as a duplication/deletion/substitution channel. For this channel, we propose a new communication scheme that requires only loose transmitter–receiver synchronization. We show that the proposed scheme is asymptotically optimal for the Gaussian channel with synchronization errors in terms of energy efficiency as measured by the rate per unit energy. In the process, we also establish that the lack of synchronization causes negligible loss in energy efficiency. We further show that, for a general discrete memoryless channel with synchronization errors and a general input cost function admitting a zero-cost symbol, the rate per unit cost achieved by the proposed scheme is within a factor two of the information-theoretic optimum. [ABSTRACT FROM PUBLISHER]

A well-known technique in estimating the probabilities of rare events in general and in information theory in particular (used, for example, in the sphere–packing bound) is that of finding a reference probability measure under which the event of interest has the probability of order one and estimating the probability in question by means of the Kullback–Leibler divergence. A method has recently been proposed in [2] that can be viewed as an extension of this idea in which the probability under the reference measure may itself be decaying exponentially, and the Rényi divergence is used instead. The purpose of this paper is to demonstrate the usefulness of this approach in various information–theoretic settings. For the problem of channel coding, we provide a general methodology for obtaining matched, mismatched, and robust error exponent bounds, as well as new results in a variety of particular channel models. Other applications we address include rate-distortion coding and the problem of guessing. [ABSTRACT FROM PUBLISHER]

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length nr from cyclic codes of length n and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length n with a primitive n-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of n. This means that we shall not expect good cyclic codes of length n in general if, for example, the length n is divisible by a high power of a prime. [ABSTRACT FROM AUTHOR]

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

The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel’s ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes. [ABSTRACT FROM AUTHOR]

The paper addresses a joint sequential changepoint detection and identification/isolation problem for a general stochastic model, assuming that the observed data may be dependent and non-identically distributed, the prior distribution of the change point is arbitrary, and the post-change hypotheses are composite. The developed detection–identification theory generalizes the changepoint detection theory developed by Tartakovsky (2019) to the case of multiple composite post-change hypotheses when one has not only to detect a change as quickly as possible but also to identify (or isolate) the true post-change distribution. We propose a multi-hypothesis change detection–identification rule and show that it is nearly optimal, minimizing moments of the delay to detection as the probability of a false alarm and the probabilities of misidentification go to zero. [ABSTRACT FROM AUTHOR]

The Demirci-Selçuk’s meet-in-the-middle attack is one of the most important methods among all the cryptanalytic vectors, which gives the best result against the round-reduced AES with respect to the rounds, and tradeoffs between data, time and memory. While we have already built provable security models against the differential cryptanalysis, linear cryptanalysis cryptanalysis, impossible differential and zero-correlation linear cryptanalysis, the provable security against the meet-in-the-middle attack is missing. In this paper, we propose the subset representation of function based on which we could give an algorithm to compute the exact number of parameters of the Demirci-Selçuk’s distinguisher given the input and output, respectively. Experiments show that this algorithm can be more efficient than the automatical tool presented by Shi et al. at Asiacrypt 2018. We further extract a formula based on this algorithm and show an upper bound for the length of the Demirci-Selçuk’s distinguisher of an iterative SPN cipher. We prove that for an SPN block cipher whose block size equals the key size, an effective Demirci-Selçuk-type meet-in-the-middle distinguisher covers at most twice the maximum of the primitive indexes of the linear layer and its inverse. As a result, we show that the known length of the Demirci-Selçuk’s distinguisher of the AES-128 cannot be improved unless the details of the S-boxes are exploited, which demonstrates that the AES has a provable security against the Demirci-Selçuk’s meet-in-the-middle attack. [ABSTRACT FROM AUTHOR]

Log-rank conjecture is one of challenging problems in communication complexity. It is known to hold for some special function classes such as XOR functions ƒ(x ⊕ y) when the outer function ƒ is monotone, symmetric, AC0, low F2-degree, linear threshold or read-k. However, for AND functions ƒ(x ∧ y), Log-rank conjecture is only known to hold when the outer function is a monotone function or a function with small alternating numbers. In this paper, we show that Log-rank conjecture holds for AND functions whose outer functions are read-once F2-degree polynomials, constant-F2-degree polynomials, and symmetric functions. For read-once and symmetric AND functions, their proof are based on the elementary analysis of Möbius sparsity of the outer functions and designing efficient protocols for computing the corresponding AND functions in terms of the Möbius sparsity of their outer functions. For a constant-F2-degree AND function ƒ(x ∧ y, we use the concept of the variable rank to design a decision tree for computing f where the tree depth is bounded above by a polynomial in the logarithm of the Möbius sparsity of ƒ. The communication protocol can be obtained easily from the constructed decision tree. Our results deepen the understanding of the log-rank conjecture for AND functions. [ABSTRACT FROM AUTHOR]

In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string x given random “traces” of x where each trace is generated by deleting each coordinate of x independently with probability p < 1. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and generalizations of this problem in an effort to attain a deeper and more comprehensive understanding. Perhaps our most surprising results are: 1) We prove that exp (O(n1/4 √log n)) traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown √n × √n matrix is deleted independently with probability p. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is exp(O(n1/3)). 2) An optimal result for random matrix reconstruction: we show that Θ(log n) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is exp(O(log1/3n)). 3) We show that exp (O(k1/3log2/3n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/log2n). 4) We show that poly(n) traces suffice if x is k-sparse and we additionally have a “separation” promise, specifically that the indices of 1’s in x all differ by Ω (k log n). [ABSTRACT FROM AUTHOR]

2-to-1 mappings over finite fields play an important role in symmetric cryptography, particularly in the constructions of APN functions, bent functions, and semi-bent functions. Very recently, Mesnager and Qu [IEEE Trans. Inf. Theory 65 (12): 7884-7895] provided a systematic study of 2-to-1 mappings over finite fields. In particular, they determined all 2-to-1 mappings of degree at most 4 over any finite field. Besides, another research direction is to consider 2-to-1 polynomials with few terms. Some results about 2-to-1 monomials and binomials have been obtained in [IEEE Trans. Inf. Theory 65 (12): 7884-7895]. Motivated by their work, in this present paper, we push further the study of 2-to-1 mappings, particularly over finite fields with characteristic 2 (binary case being the most interesting for applications). Firstly, we completely determine 2-to-1 polynomials with degree 5 over F2n using the well-known Hasse-Weil bound. Besides, we consider 2-to-1 mappings with few terms, mainly trinomials and quadrinomials. Using the multivariate method and the resultant of two polynomials, we present two classes of 2-to-1 trinomials, which explain all the examples of 2-to-1 trinomials of the form xk + β xℓ + α ∈ F2n[x] with n ≤ 7. We derive twelve classes of 2-to-1 quadrinomials with trivial coefficients over F2n. [ABSTRACT FROM AUTHOR]