This paper is motivated by the problem of error-free communication through the i.i.d. duplication channel which acts on the transmitted sequence by inserting a random number of duplicates of each symbol next to the original symbol. A construction of optimal codes correcting all patterns of errors of this type is described, and the zero-error capacity of the duplication channel is determined. A generalization of these results to channel models in which blocks of @math consecutive symbols are being duplicated, and which are inspired by DNA-based data storage systems where the stored molecules are subject to tandem-duplication mutations, is also given.
The binary @math - @math -insertion channel was first studied in @cite_16 , where a construction of codes correcting @math such errors was given, and asymptotic bounds on the cardinality of optimal codes derived for any fixed @math and @math . Another construction for same model was later given in @cite_9 . Generalizations of the constructions from @cite_16 and @cite_9 for the @math -duplication channel were given in @cite_22 and @cite_5 , respectively. The best known asymptotic bounds for the @math -duplication channel in the regime of fixed @math and @math , were reported in @cite_22 , where the upper bound from @cite_16 (for @math ) was improved for every @math , and the bounds were also generalized to the case @math .
@cite_5 @cite_9 @cite_16 @cite_22
2858362441 2082173153 2888898008
A (tandem) duplication of length @math is an insertion of an exact copy of a substring of length @math next to its original position. This and related types of impairments are of relevance in modeling communication in the presence of synchronization errors, as well as in several information storage applications. We demonstrate that Levenshtein’s construction of binary codes correcting insertions of zeros is, with minor modifications, applicable also to channels with arbitrary alphabets and with duplication errors of arbitrary (but fixed) length @math . Furthermore, we derive bounds on the cardinality of optimal @math -ary codes correcting up to @math duplications of length @math and establish the following corollaries in the asymptotic regime of growing block length: 1) the presented family of codes is optimal for every @math , in the sense of the asymptotic scaling of code redundancy; 2) the upper bound, when specialized to @math , @math , improves upon Levenshtein’s bound for every @math ; and 3) the bounds coincide for @math , thus yielding the exact asymptotic behavior of the size of optimal single-duplication-correcting codes. In this paper, we study codes correcting @math duplications of @math consecutive symbols. These errors are known as tandem duplication errors, where a sequence of symbols is repeated and inserted directly after its original occurrence. Using sphere packing arguments, we derive non-asymptotic upper bounds on the cardinality of codes that correct such errors for any choice of parameters. Based on the fact that a code correcting insertions of @math zero-blocks can be used to correct @math tandem duplications, we construct codes for tandem duplication errors. We compare the cardinalities of these codes with their sphere packing upper bounds. Finally, we discuss the asymptotic behavior of the derived codes and bounds, which yields insights about the tandem duplication channel. In this paper we study the problem of finding maximally sized subsets of binary strings (codes) of equal length that are immune to a given number @math of repetitions, in the sense that no two strings in the code can give rise to the same string after @math repetitions. We propose explicit number theoretic constructions of such subsets. In the case of @math repetition, the proposed construction is asymptotically optimal. For @math , the proposed construction is within a constant factor of the best known upper bound on the cardinality of a set of strings immune to @math repetitions. Inspired by these constructions, we then develop a prefixing method for correcting any prescribed number @math of repetition errors in an arbitrary binary linear block code. The proposed method constructs for each string in the given code a carefully chosen prefix such that the resulting strings are all of the same length and such that despite up to any @math repetitions in the concatenation of the prefix and the codeword, the original codeword can be recovered. In this construction, the prefix length is made to scale logarithmically with the length of strings in the original code. As a result, the guaranteed immunity to repetition errors is achieved while the added redundancy is asymptotically negligible. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We consider codes over fixed alphabets against worst-case symbol deletions. For any fixed @math , we construct a family of codes over alphabet of size @math with positive rate, which allow efficient recovery from a worst-case deletion fraction approaching @math . In particular, for binary codes, we are able to recover a fraction of deletions approaching @math . Previously, even non-constructively the largest deletion fraction known to be correctable with positive rate was @math , and around @math for the binary case. Our result pins down the largest fraction of correctable deletions for @math -ary codes as @math , since @math is an upper bound even for the simpler model of erasures where the locations of the missing symbols are known. Closing the gap between @math and @math for the limit of worst-case deletions correctable by binary codes remains a tantalizing open question. We derive a new upper bound on the diameter of a polyhedron @math P = x ? R n : A x ≤ b , where @math A ? Z m × n . The bound is polynomial in @math n and the largest absolute value of a sub-determinant of @math A , denoted by @math Δ . More precisely, we show that the diameter of @math P is bounded by @math O ( Δ 2 n 4 log n Δ ) . If @math P is bounded, then we show that the diameter of @math P is at most @math O ( Δ 2 n 3.5 log n Δ ) . For the special case in which @math A is a totally unimodular matrix, the bounds are @math O ( n 4 log n ) and @math O ( n 3.5 log n ) respectively. This improves over the previous best bound of @math O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1---16, 1994). In this paper, we study the asymptotic throughput capacity of a static multi-channel multi-interface infrastructure wireless mesh network (InfWMN) wherein each infrastructure node has m interfaces and c channels of unequal bandwidth are available. First, an upper bound on the InfWMN per-user capacity is established. Then, the feasible lower bound is derived by construction. We prove that both lower and upper bounds are tight. We limit our analysis for more practical case of @math . However, for the asymptotic upper bound, our analysis can be used for the general case in which there is no constraint on m and c. Our study shows that in such a network with Nc randomly distributed mesh clients, Nr regularly placed mesh routers, and Ng gateways, the asymptotic per-client throughput capacity has different bounds, which depend on the ratio between the total available bandwidth for the network and the sum of m first greatest data rates of c available channels, i.e., @math . The results of this paper are more general compared to the existing published researches. In addition, in the case that @math , our results reduce to the previously reported studies. This implies that our study is comprehensive compared to the formerly published researches. In an index coding with side-information (ICSI) problem there is a sender with a set of independent messages @math and there are receivers @math , each identified with ( @math ), where @math is the set of messages wanted by receiver @math and @math is the set of messages known a priori to receiver @math . We call a ICSI problem to be single-prior if @math In addition, if @math it is called a single-uniprior problem and has been studied in detail by Ong, Ho and Lim OHL . Error Correcting index codes (ECIC) have been studied by Dau, Skachek and Chee DSC and lower and upper bounds on the optimal length of ECICs have been reported. In this paper we show that for single-prior ICSI problems the lower and upper bounds on the optimal ECICs coincide. This leads to construction of optimal length linear ECICs. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes ( , 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property (also known as minimum storage regenerating codes, or MSR codes). Explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math , where @math . The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math . Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure-tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes (Dimakis et. al., 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property. Explicit constructions of such codes in the literature are only available for the cases where there are at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math where @math The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math . We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of @math nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level @math . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. The ability to store data in the DNA of a living organism has applications in a variety of areas including synthetic biology and watermarking of patented genetically modified organisms. Data stored in this medium are subject to errors arising from various mutations, such as point mutations, indels, and tandem duplication, which need to be corrected to maintain data integrity. In this paper, we provide error-correcting codes for errors caused by tandem duplications, which create a copy of a block of the sequence and insert it in a tandem manner, i.e., next to the original. In particular, we present two families of codes for correcting errors due to tandem duplications of a fixed length: the first family can correct any number of errors, while the second corrects a bounded number of errors. We also study codes for correcting tandem duplications of length up to a given constant @math , where we are primarily focused on the cases of @math . Finally, we provide a full classification of the sets of lengths allowed in tandem duplication that result in a unique root for all sequences. We consider the problem of estimating the unknown response function in the multichannel deconvolution model with a boxcar-like kernel which is of particular interest in signal processing. It is known that, when the number of channels is finite, the precision of reconstruction of the response function increases as the number of channels @math grow (even when the total number of observations @math for all channels @math remains constant) and this requires that the parameter of the channels form a Badly Approximable @math -tuple. Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels @math grow with the total number of observations @math . However, in real-life situations, the number of channels @math usually refers to the number of physical devices and, consequently, may grow to infinity only at a slow rate as @math . When @math grows slowly as @math increases, we develop a procedure for the construction of a Badly Approximable @math -tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this @math -tuple, which explicitly shows its dependence on @math as @math is growing. This result is further used for the evaluation of the @math -risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels @math which minimizes the @math -risk. In this paper, we study sequential testing problems with overlapping hypotheses. We first focus on the simple problem of assessing if the mean µ of a Gaussian distribution is @math , both answers are considered to be correct. Then, we consider PAC-best arm identification in a bandit model: given K probability distributions on R with means @math , we derive the asymptotic complexity of identifying, with risk at most @math , an index @math such that @math . We provide non asymptotic bounds on the error of a parallel General Likelihood Ratio Test, which can also be used for more general testing problems. We further propose lower bound on the number of observation needed to identify a correct hypothesis. Those lower bounds rely on information-theoretic arguments, and specifically on two versions of a change of measure lemma (a high-level form, and a low-level form) whose relative merits are discussed. This paper considers the problem of variable-length coding over a discrete memoryless channel with noiseless feedback. This paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence, termed extrinsic Jensen–Shannon (EJS) divergence. It is shown that strictly positive lower bounds on EJS divergence provide nonasymptotic upper bounds on the expected code length. This paper presents strictly positive lower bounds on EJS divergence, and hence nonasymptotic upper bounds on the expected code length, for the following two coding schemes: 1) variable-length posterior matching and 2) MaxEJS coding scheme that is based on a greedy maximization of the EJS divergence. As an asymptotic corollary of the main results, this paper also provides a rate–reliability test. Variable-length coding schemes that satisfy the condition(s) of the test for parameters @math and @math are guaranteed to achieve a rate @math and an error exponent @math . The results are specialized for posterior matching and MaxEJS to obtain deterministic one-phase coding schemes achieving capacity and optimal error exponent. For the special case of symmetric binary-input channels, simpler deterministic schemes of optimal performance are proposed and analyzed. We study the degrees of freedom (DoF) of the @math -user interference channel with coordinated multipoint (CoMP) transmission and reception. Each message is jointly transmitted by @math successive transmitters, and is jointly received by @math successive receivers. We refer to this channel as the CoMP channel with a transmit cooperation order of @math and receive cooperation order of @math . Since the channel has a total of @math transmit antennas and @math receive antennas, the maximum possible DoF is equal to @math . We show that the CoMP channel has @math DoF if and only if @math . The key idea is that the zero forcing of the interference corresponding to the @math message at the decoder of the @math message, where @math , can be viewed as a shared responsibility between the @math transmitters carrying the @math message, and the @math receivers decoding the @math message. For the general case, we derive an outer bound that states that the DoF is bounded above by @math . For the special case with only CoMP transmission, i.e, @math , we propose a scheme that can achieve @math DoF for all @math K @math M_ t @math M_ t -1$ receivers, thereby allowing each of these receivers to enjoy 1 DoF, and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space. The achievability proofs are based on the notion of algebraic independence from algebraic geometry. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math Q @math $ . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. In this paper, we study online algorithms that schedule malleable jobs, i.e., jobs that can be parallelized on any subset of the available @math m identical machines. We study a model that accounts for the tradeoff between multiprocessor speedup and overhead time, namely, if job @math j has processing requirement @math pj and is assigned to run on @math kj machines, then @math j's execution time becomes @math pj kj+(kj-1)c, where @math c is a constant parameter to the problem. For @math m=2, we present an online algorithm OCS that has a strong competitive ratio of 3 2, matching a previously established lower bound. We also present an online algorithm ASYM2 that is asymptotically @math ((4-∈) (3-∈))-competitive when @math m=2, where @math 0<∈≤2 is a parameter to the algorithm, improving upon an existing asymptotically (3 2)-competitive algorithm. Finally, we present an online algorithm OTO that is strongly @math 2-competitive when @math m=3, improving upon the previous best upper bound of @math 9 4. Index Coding has received considerable attention recently motivated in part by real-world applications and in part by its connection to Network Coding. The basic setting of Index Coding encodes the problem input as an undirected graph and the fundamental parameter is the broadcast rate @math , the average communication cost per bit for sufficiently long messages (i.e. the non-linear vector capacity). Recent nontrivial bounds on @math were derived from the study of other Index Coding capacities (e.g. the scalar capacity @math ) by Bar- (2006), Lubetzky and Stav (2007) and (2008). However, these indirect bounds shed little light on the behavior of @math : there was no known polynomial-time algorithm for approximating @math in a general network to within a nontrivial (i.e. @math ) factor, and the exact value of @math remained unknown for any graph where Index Coding is nontrivial. Our main contribution is a direct information-theoretic analysis of the broadcast rate @math using linear programs, in contrast to previous approaches that compared @math with graph-theoretic parameters. This allows us to resolve the aforementioned two open questions. We provide a polynomial-time algorithm with a nontrivial approximation ratio for computing @math in a general network along with a polynomial-time decision procedure for recognizing instances with @math . In addition, we pinpoint @math precisely for various classes of graphs (e.g. for various Cayley graphs of cyclic groups) thereby simultaneously improving the previously known upper and lower bounds for these graphs. Via this approach we construct graphs where the difference between @math and its trivial lower bound is linear in the number of vertices and ones where @math is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse. Given an @math matrix @math , its Schatten- @math norm, @math , is defined as @math , where @math is the @math -th largest singular value of @math . These norms have been studied in functional analysis in the context of non-commutative @math -spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embeddability, are still open. Specifically, given a set of matrices @math , suppose we want to construct a linear map @math such that @math for each @math , where @math and @math , and further, @math for a given approximation factor @math and real number @math . Then how large do @math and @math need to be as a function of @math ? We nearly resolve this question for every @math , for the case where @math can be expressed as @math , where @math and @math are arbitrary matrices that are allowed to depend on @math , that is, @math can be implemented by left and right matrix multiplication. Namely, for every @math , we provide nearly matching upper and lower bounds on the size of @math and @math as a function of @math . Importantly, our upper bounds are oblivious , meaning that @math and @math do not depend on the @math , while our lower bounds hold even if @math and @math depend on the @math . As an application of our upper bounds, we answer a recent open question of about space-approximation trade-offs for the Schatten @math -norm, showing in a data stream it is possible to estimate the Schatten- @math norm up to a factor of @math using @math space.