Christian Deppe

Solution of Ulam's searching game with three lies or an optimal adaptive strategy for binary three-error-correcting codes

Abstract In this paper we determine the minimal number of yes–no queries that are needed to find an unknown integer between 1 and N, if at most three of the answers are lies. This strategy is also an optimal adaptive strategy for binary three-error-correcting codes.

Coding with Feedback and Searching with Lies

This paper gives a broad overview of the area of searching with errors and the related field of error-correcting coding. In the vast literature regarding this problem, many papers simultaneously deal with various sorts of restrictions on the searching protocol. We partition this survey into sections, choosing the most appropriate section for each topic.

Two Batch Search With Lie Cost

We consider the problem of searching for an unknown number in the search space U ={0,...,M-1}. q-ary questions can be asked and some of the answers may be wrong. An arbitrary integer weighted bipartite graph Gamma is given, stipulating the cost Gamma(i,j) of each answer jnei when the correct answer is i, i.e., the cost of a wrong answer. Correct answers are supposed to be cost-less. It is assumed that a maximum cost e for the sum of the cost of all wrong answers can be afforded by the responder during the whole search. We provide tight upper and lower bounds for the largest size M = M(q,e,Gamma,n) for which it is possible to find an unknown number x*isinU with n q-ary questions and maximum lie cost e. Our results improve the bounds of Cicalese et al. (2004) and Ahlswede et al. (2008). The questions in our strategies can be asked in two batches of nonadaptive questions. Finally, we remark that our results can be further generalized to a wider class of error models including also unidirectional errors.

Q-Ary Ulam-Rényi Game with Weighted Constrained Lies

The Ulam-Renyi game is a classical model for the problem of determining the minimum number of queries to find an unknown number in a finite set when up to a finite number of the answers may be erroneous/mendacious. In the variant considered in this paper, questions with q many possible answers are allowed, with q fixed and known beforehand; further, lies are constrained by a weighted bipartite graph (the “channel”). We provide a tight asymptotic estimate for the number of questions needed to solve the problem. Our results are constructive, and the appropriate searching strategies are actually provided. As an extra bonus, all our strategies use the minimum amount of adaptiveness: they ask a first batch of nonadaptive questions, and then, only depending on the answers to these questions, they ask a second nonadaptive batch.

Information Theory, Combinatorics, and Search Theory

We provide two new results for identification for sources. The first result is about block codes. In [Ahlswede and Cai, IEEE-IT, 52(9), 4198-4207, 2006] it is proven that the q-ary identification entropy HI,q(P ) is a lower bound for the average number L(P, P ) of expected checkings during the identification process. A necessary assumption for this proof is that the uniform distribution minimizes the symmetric running time LC(P, P ) for binary block codes C = {0, 1}. This assumption is proved in Sect. 2 not only for binary block codes but for any q-ary block code. The second result is about upper bounds for the worst-case running time. In [Ahlswede, Balkenhol and Kleinewchter, LNCS, 4123, 51-61, 2006] the authors proved in Theorem 3 that L(P ) < 3 by an inductive code construction. We discover an alteration of their scheme which strengthens this upper bound significantly.

Secrecy capacities of compound quantum wiretap channels and applications

We determine the secrecy capacity of the compound channel with quantumwiretapper and channel state information at the transmitter. Moreover, wederive a lower bound on the secrecy capacity of this channel without channelstate information and determine the secrecy capacity of the compoundclassical-quantum wiretap channel with channel state information at thetransmitter. We use this result to derive a new proof for a lower bound on theentanglement generating capacity of compound quantum channel. We also derive anew proof for the entanglement generating capacity of compound quantum channelwith channel state information at the encoder.

Classical-quantum arbitrarily varying wiretap channel—A capacity formula with Ahlswede Dichotomy—Resources

We establish the Ahlswede Dichotomy for arbitrarily varying classical-quantum wiretap channels, i.e., either the deterministic secrecy capacity of an arbitrarily varying classical-quantum wiretap channel is zero, or it equals its randomness assisted secrecy capacity. We analyze the secrecy capacity of arbitrarily varying classical-quantum wiretap channels when the sender and the receiver use various resources. It turns out that having randomness, common randomness, and correlation as resources are very helpful for achieving a positive deterministic secrecy capacity of arbitrarily varying classical-quantum wiretap channels. We prove the phenomenon “super-activation” for arbitrarily varying classical-quantum wiretap channels, i.e., two arbitrarily varying classical-quantum wiretap channels, both with zero deterministic secrecy capacity, if used together allow perfect secure transmission.

Quasi-Perfect Minimally Adaptive q-ary Search with Unreliable Tests

We consider the problem of determining the minimum number of queries to find an unknown number in a finite set when up to a finite number e of the answers may be erroneous. In the vast literature regarding this problem, the classical case of binary search is mostly considered, that is, when only yes-no questions are allowed. In this paper we consider the variant of the problem in which questions with q many possible answers are allowed. We prove that at most one question more than the information theoretic lower bound is sufficient to successfully find the unknown number. Moreover we prove that there are infinitely many cases when the information theoretic lower bound is exactly attained and the so called perfect strategies exist. Our strategies have the important feature that they use a minimum amount of adaptiveness, a relevant property in many practical situation.

Classical---quantum arbitrarily varying wiretap channel: Ahlswede dichotomy, positivity, resources, super-activation

We establish the Ahlswede dichotomy for arbitrarily varying classical–quantum wiretap channels, i.e., either the deterministic secrecy capacity of the channel is zero, or it equals its randomness-assisted secrecy capacity. We analyze the secrecy capacity of these channels when the sender and the receiver use various resources. It turns out that randomness, common randomness, and correlation as resources are very helpful for achieving a positive secrecy capacity. We prove the phenomenon “super-activation” for arbitrarily varying classical–quantum wiretap channels, i.e., two channels, both with zero deterministic secrecy capacity, if used together allow perfect secure transmission.

Bounds for threshold and majority group testing

We consider two generalizations of group testing: threshold group testing (introduced by Damaschke [8]) and majority group testing (a further generalization, including threshold group testing and a model introduced by Lebedev [15]).