Publications (304)184.71 Total impact
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ABSTRACT: An (n, N)–connector of depth d is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length at most d joining each input to its corresponding output. In this paper we consider the problem of construction of sparse depth two connectors with n ≪ N . We use posets of star products and their matching properties to construct such connectors. In particular this gives a simple explicit construction for connectors of size O(N log n/ log log n). Thus our earlier idea to use other posets than the family of subsets of a finite set was successful.Journal of Combinatorial Theory Series A 11/2006; 113(8). DOI:10.1016/j.jcta.2006.03.009 · 0.78 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We find the formula for the cardinality of a maximal set of integers from {1,…,n} which does not contain k+1 pairwise coprimes and each integer has a divisor from a specified set of r primes. We also find the explicit formula for this set when r=k+1.Journal of Combinatorial Theory Series A 11/2006; 113(8):16211628. DOI:10.1016/j.jcta.2006.03.015 · 0.78 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: After Ahlswede introduced identification for source coding he discovered identification entropy and demonstrated that it plays a role analogously to classical entropy in Shannon's noiseless source coding. We give now even more insight into this functional interpreting its two factorsIEEE Transactions on Information Theory 10/2006; 52(952):4198  4207. DOI:10.1109/TIT.2006.879972 · 2.33 Impact Factor 
Conference Paper: Capacity of Quantum Arbitrarily Varying Channels
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ABSTRACT: We prove that the average error capacity C<sub>q</sub> of a quantum arbitrarily varying channel (QAVC) equals 0 or else the random code capacity lowbarC (Ahlswede's dichotomy). We also establish a necessary and sufficient condition for C<sub>q</sub> > 0Information Theory, 2006 IEEE International Symposium on; 08/2006 
Conference Paper: Nonbinary error correcting codes with noiseless feedback, localized errors, or both
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ABSTRACT: The two models described in this paper having as ingredients feedback resp. localized errors give possibilities for code constructions not available in the standard model of error correction and also for probabilistic channel models. For the feedback model we present here a coding scheme, which we call the rubber method, because it is based on erasing letters. It is the first scheme achieving the capacity curve for q ges 3. It could be discovered only in the gary case for q ges 3, because the letter zero is not used as an information symbol, but solely for error correction. However an extension of the method from using single zeros to blocks of zeros also gives Berlekamp's result  by a different scheme. In the model with feedback and localized errors the help of feedback is addressed. We give an optimal construction for oneerror correcting codes with feedback and localized errorsInformation Theory, 2006 IEEE International Symposium on; 08/2006  [Show abstract] [Hide abstract]
ABSTRACT: We continue the investigation of Part I, keep its terminology, and also continue the numbering of sections, equations, theorems etc. Consequently we start here with Section 6. As mentioned in Section 4 we present now criteria for a triple (r,t,p) to be k–admissible. Then we consider the f–complexity (extended now to k–ary alphabets) Gk(F)\Gamma_k(\mathcal{F}) of a family F\mathcal{F}. It serves again as a performance parameter of key spaces in cryptography. We give a lower bound for the f–complexity for a family of the type constructed in Part I. In the last sections we explain what can be said about the theoretically best families F\mathcal{F} with respect to their f–complexity Gk(F)\Gamma_k(\mathcal{F}). We begin with straightforward extensions of the results of [4] for k=2 to general k by using the same Covering Lemma as in [1].08/2006: pages 308325;  [Show abstract] [Hide abstract]
ABSTRACT: Let F be a finite family of sets and G(F) be the intersection graph of F (the vertices of G(F) are the sets of family F and the edges of G(F) correspond to intersecting pairs of sets). The transversal number τ(F) is the minimum number of points meeting all sets of F. The independent (stability) number α(F) is the maximum number of pairwise disjoint sets in F. The clique number ω(F) is the maximum number of pairwise intersecting sets in F. The coloring number q(F) is the minimum number of classes in a partition of F into pairwise disjoint sets.08/2006: pages 10641065;  [Show abstract] [Hide abstract]
ABSTRACT: Consider (X,E)(X,{\mathcal E}), where X is a finite set and E{\mathcal E} is a system of subsets whose union equals X. For every natural number n ∈ℕ define the cartesian products X n =∏1 n X and En=Õ1nE{\mathcal E}_n=\prod_1^n{\mathcal E}. The following problem is investigated: how many sets of En{\mathcal E}_n are needed to cover X n ? Let this number be denoted by c(n). It is proved that for all n ∈ℕ exp{Cn} £ c(n) £ exp{Cn+logn+loglogX}+1.\exp\{C\cdot n\}\leq c(n)\leq\exp\{Cn+\log n+\log\logX\}+1. A formula for C is given. The result generalizes to the case where X and E{\mathcal E} are not necessarily finite and also to the case of non–identical factors in the product. As applications one obtains estimates on the minimal size of an externally stable set in cartesian product graphs and also estimates on the minimal number of cliques needed to cover such graphs.08/2006: pages 926937;  [Show abstract] [Hide abstract]
ABSTRACT: Denote by Ω={1,...,n} an n–element set. For all A,B Î \binomWkA,B\in\binom{\Omega}k, the k–element subsets of Ω, define the relation ~ as follows: A~B iff A and B have a common shadow, i.e. there is a C Î \binomWk1C\in\binom{\Omega}{k1} with C ⊂A and C ⊂B. For fixed integer α, our goal is to find a family A{\mathcal A} of k–subsets with size α, having as many as possible ~–relations for all pairs of its elements. For k=2 this was achieved by Ahlswede and Katona [2] many years ago.08/2006: pages 9791005;  [Show abstract] [Hide abstract]
ABSTRACT: To a large extent the present work is far from being conclusive, instead, new directions of research in combinatorial extremal theory are started. Also questions concerning generalizations are immediately noticeable. The incentive came from problems in several fields such as Algebra, Geometry, Probability, Information and Complexity Theory. Like several basic combinatorial problems they may play a role in other fields. For scenarios of interplay we refer also to [R. Ahlswede, “Advances on extremal problems in number theory and combinatorics”, in: C. Casacuberta et al. (eds.), 3rd European congress of mathematics (ECM) Volume I. Basel: Birkhäuser. Prog. Math. 201, 147–175 (2001; Zbl 1094.11001)].08/2006: pages 955970;  [Show abstract] [Hide abstract]
ABSTRACT: We analyze wiretape channels with secure feedback from the legitimate receiver. We present a lower bound on the transmission capacity (Theorem 1), which we conjecture to be tight and which is proved to be tight (Corollary 1) for Wyner’s original (degraded) wiretape channel and also for the reversely degraded wiretape channel for which the legitimate receiver gets a degraded version from the enemy (Corollary 2). Somewhat surprisingly we completely determine the capacities of secure common randomness (Theorem 2) and secure identification (Theorem 3 and Corollary 3). Unlike for the DMC, these quantities are different here, because identification is linked to nonsecure common randomness.08/2006: pages 258275;  [Show abstract] [Hide abstract]
ABSTRACT: We investigate nonbinary error correcting codes with noiseless feedback, localized errors, or both. It turns out that the Hamming bound is a central concept. For block codes with feedback we present here a coding scheme based on an idea of erasions, which we call the {bf rubber method}. It gives an optimal rate for big error correcting fraction $tau$ ($>{1over q}$) and infinitely many points on the Hamming bound for small $tau$. We also consider variable length codes with all lengths bounded from above by $n$ and the end of a word carries the symbol $Box$ and is thus recognizable by the decoder. For both, the $Box$model with feedback and the $Box$model with localized errors, the Hamming bound is the exact capacity curve for $tau <1/2.$ Somewhat surprisingly, whereas with feedback the capacity curve coincides with the Hamming bound also for $1/2leq tau leq 1$, in this range for localized errors the capacity curve equals 0. Also we give constructions for the models with both, feedback and localized errors. @InProceedings{ahlswede_et_al:DSP:2006:784, author = {Rudolf Ahlswede and Christian Deppe and Vladimir Lebedev}, title = {Nonbinary error correcting codes with noiseless feedback, localized errors, or both}, booktitle = {Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, year = {2006}, editor = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, number = {06201}, series = {Dagstuhl Seminar Proceedings}, ISSN = {18624405}, publisher = {Internationales Begegnungs und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2006/784}, annote = {Keywords: Errorcorrecting codes, localized errors, feedback, variable length codes} } 
Conference Paper: Another diametric theorem in Hamming spaces: Optimal group anticodes
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ABSTRACT: In the last century together with Levon Khachatrian we established a diametric theorem in Hamming space H<sup>n</sup>=(X<sup>n</sup>,d H ). Now we contribute a diametric theorem for such spaces, if they are endowed with the group structure G<sup>n</sup>=<sup>n</sup>Σ 1 G, the direct sum of a group G on X={0,1,...,q1}, and as candidates are considered subgroups of G<sup>n</sup>. For all finite groups G, every permitted distance d, and all n≥d subgroups of G<sup>n</sup>with diameter d have maximal cardinality q<sup>d</sup>. Other extremal problems can also be studied in this setting.Information Theory Workshop, 2006. ITW '06 Punta del Este. IEEE; 04/2006  [Show abstract] [Hide abstract]
ABSTRACT: In Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0⩽t, k⩽n determine or estimate the maximum number of (0,1)vectors in a kdimensional subspace of the Euclidean nspace Rn, such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the (0,1)vectors of the same weight ω.The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used.In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21]. We also consider a diametric problem under dimension constraint.In Part II, we study the restricted case and solve the problem for t=1 and k<2ω, and also for any fixed 1⩽t⩽ω and k large.Journal of Combinatorial Theory Series A 04/2006; 113(3113):483519. DOI:10.1016/j.jcta.2005.04.009 · 0.78 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The goals of this seminar have been (1) to identify and match recently developed methods to specific tasks and data sets in a core of application areas; next, based on feedback from the specific applied domain, (2) to fine tune and personalize those applications, and finally (3) to identify and tackle novel combinatorial and algorithmic problems, in some cases all the way to the development of novel software tools. @InProceedings{ahlswede_et_al:DSP:2006:792, author = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, title = {06201 Executive Summary  Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, booktitle = {Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, year = {2006}, editor = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, number = {06201}, series = {Dagstuhl Seminar Proceedings}, ISSN = {18624405}, publisher = {Internationales Begegnungs und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2006/792}, annote = {Keywords: Data compression, pattern matching, pattern discovery, search, sorting, molecular biology, reconstruction, genome rearrangements} } 
Conference Paper: On Logarithmically Asymptotically Optimal Testing of Hypotheses and Identification
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ABSTRACT: We introduce a new aspect of the influence of the informationtheoretical methods on the statistical theory. The procedures of the probability distributions identification for K(≥1) random objects each having one from the known set of M(≥2) distributions are studied. Nsequences of discrete independent random variables represent results of N observations for each of K objects. On the base of such samples decisions must be made concerning probability distributions of the objects. For N ® ¥N \longrightarrow \infty the exponential decrease of the test’s error probabilities is considered. The reliability matrices of logarithmically asymptotically optimal procedures are investigated for some models and formulations of the identification problems. The optimal subsets of reliabilities which values may be given beforehand and conditions guaranteeing positiveness of all the reliabilities are investigated.General Theory of Information Transfer and Combinatorics; 01/2006 
Conference Paper: Estimating with Randomized Encoding the Joint Empirical Distribution in a Correlated Source.
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ABSTRACT: In order to put the present model and our results into the right perspectives we describe first key steps in multiuser source coding theory.General Theory of Information Transfer and Combinatorics; 01/2006 
Conference Paper: Correlation Inequalities in Function Spaces
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ABSTRACT: We give a condition for a Borel measure on R [0,1] which is sufficient for the validity of an ADtype correlation inequality in the function space.General Theory of Information Transfer and Combinatorics; 01/2006 
Article: Combinatorial and Algorithmic Foundations of Pattern and Association Discovery, 14.05.  19.05.2006
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ABSTRACT: We consider codes over the alphabet Q={0,1,..,q1}intended for the control of unidirectional errors of level l. That is, the transmission channel is such that the received word cannot contain both a component larger than the transmitted one and a component smaller than the transmitted one. Moreover, the absolute value of the difference between a transmitted component and its received version is at most l. We introduce and study qary codes capable of correcting all unidirectional errors of level l. Lower and upper bounds for the maximal size of those codes are presented. We also study codes for this aim that are defined by a single equation on the codeword coordinates(similar to the VarshamovTenengolts codes for correcting binary asymmetric errors). We finally consider the problem of detecting all unidirectional errors of level l.
Publication Stats
9k  Citations  
184.71  Total Impact Points  
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Institutions

20122013

The Institute for Information Transmission Problems
Moskva, Moscow, Russia


19762013

Bielefeld University
 Faculty of Mathematics
Bielefeld, North RhineWestphalia, Germany


1978

Hungarian Academy of Sciences
 MTA Computer and Automation Research Institute
Budapeŝto, Budapest, Hungary


19681976

The Ohio State University
 Department of Mathematics
Columbus, Ohio, United States


1969

Cornell University
Ithaca, New York, United States
