Rudolf Ahlswede

Bielefeld University, Bielefeld, North Rhine-Westphalia, Germany

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Publications (279)144.23 Total impact

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    R. Ahlswede
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    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+loglog|X|}+1.\exp\{C\cdot n\}\leq c(n)\leq\exp\{Cn+\log n+\log\log|X|\}+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 926-937;
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    R. Ahlswede, N. Cai
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    ABSTRACT: We analyze wire-tape 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) wire-tape channel and also for the reversely degraded wire-tape 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 non-secure common randomness.
    08/2006: pages 258-275;
  • R. Ahlswede, I. Karapetyan
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    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 1064-1065;
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    R. Ahlswede, L. Khachatrian
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    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 [9].
    08/2006: pages 955-970;
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    R. Ahlswede, C. Mauduit, A. Sárközy
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    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 308-325;
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    R. Ahlswede, N. Cai
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    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 \binomWk-1C\in\binom{\Omega}{k-1} 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 979-1005;
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    R. Ahlswede
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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,...,q-1}, 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
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    Rudolf Ahlswede, Christian Deppe, Vladimir Lebedev
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    ABSTRACT: We investigate non--binary 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 = {Non--binary 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 = {1862-4405}, 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: Error-correcting codes, localized errors, feedback, variable length codes} }
    01/2006;
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    Rudolf Ahlswede, Ning Cai
    General Theory of Information Transfer and Combinatorics; 01/2006
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    Rudolf Ahlswede, János Körner
    General Theory of Information Transfer and Combinatorics; 01/2006
  • Rudolf Ahlswede, Vladimir Blinovsky
    General Theory of Information Transfer and Combinatorics; 01/2006
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    General Theory of Information Transfer and Combinatorics; 01/2006
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    Rudolf Ahlswede, Evgueni A. Haroutunian
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    ABSTRACT: We introduce a new aspect of the influence of the information-theoretical 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. N-sequences 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
  • 01/2006;
  • 01/2006;
  • Rudolf Ahlswede, Vladimir Blinovsky
    J. Comb. Theory, Ser. A. 01/2006; 113:1621-1628.
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    ABSTRACT: We consider codes over the alphabet Q={0,1,..,q-1}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 q-ary 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 Varshamov-Tenengolts codes for correcting binary asymmetric errors). We finally consider the problem of detecting all unidirectional errors of level l.
    CoRR. 01/2006; abs/cs/0607132.
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    ABSTRACT: We study a model for language evolution which was intro- duced by Nowak and Krakauer ((12)). We analyze discrete distance spaces and prove a conjecture of Nowak for all metrics with a positive semidefi- nite associated matrix. This natural class of metrics includes all metrics studied by different authors in this connection. In particular it includes all ultra-metric spaces. Furthermore, the role of feedback is explored and multi-user scenarios are studied. In all models we give lower and upper bounds for the fitness.
    General Theory of Information Transfer and Combinatorics; 01/2006
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    Søren Riis, Rudolf Ahlswede
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    ABSTRACT: In most of todays information networks messages are send in packets of information that is not modified or mixed with the content of other packets during transmission. This holds on macro level (e.g. the internet, wireless communications) as well as on micro level (e.g. communication within processors, communication between a processor and external devises). Today messages in wireless communication are sent in a manner where each active communication channel carries exactly one “conversation”. This approach can be improved considerably by a cleverly designed but sometimes rather complicated channel sharing scheme (network coding). The approach is very new and is still in its pioneering phase. Worldwide only a handful of papers in network coding were published year 2001 or before, 8 papers in 2002, 23 papers in 2003 and over 25 papers already in the first half of 2004; (according to the database developed by R. Koetters). The first conference on Network Coding and applications is scheduled for Trento, Italy April 2005. Research into network coding is growing fast, and Microsoft, IBM and other companies have research teams who are researching this new field. A few American universities (Princeton, MIT, Caltech and Berkeley) have also established research groups in network coding.
    General Theory of Information Transfer and Combinatorics; 01/2006
  • General Theory of Information Transfer and Combinatorics; 01/2006

Publication Stats

6k Citations
144.23 Total Impact Points

Institutions

  • 1976–2013
    • Bielefeld University
      • • Faculty of Mathematics
      • • Institute of Mathematical Economics
      Bielefeld, North Rhine-Westphalia, Germany
  • 2007
    • Alfréd Rényi Institute of Mathematics
      Budapeŝto, Budapest, Hungary
  • 2003
    • Eötvös Loránd University
      • Department of Algebra and Number Theory
      Budapest, Budapest fovaros, Hungary
  • 2002
    • University of Rostock
      Rostock, Mecklenburg-Vorpommern, Germany
  • 1968–1976
    • The Ohio State University
      • Department of Mathematics
      Columbus, Ohio, United States
  • 1969
    • Cornell University
      Ithaca, New York, United States