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  1. Book ; Online: Fast Decoding of Union-free Codes

    Vorobyev, Ilya

    2021  

    Abstract: Union-free codes and disjunctive codes are two combinatorial structures, which are used in nonadaptive group testing to find a set of $d$ defective elements among $n$ samples by carrying out the minimal number of tests $t$. It is known that union-free ... ...

    Abstract Union-free codes and disjunctive codes are two combinatorial structures, which are used in nonadaptive group testing to find a set of $d$ defective elements among $n$ samples by carrying out the minimal number of tests $t$. It is known that union-free codes have a larger rate, whereas disjunctive codes provide a more efficient decoding algorithm. In this paper we introduce a new family of codes for nonadaptive group testing with fast decoding. The rate of these codes is larger than the rate of disjunctive codes, while the decoding algorithm has the same complexity. In addition, we derive a lower bound on the rate of new codes for the case of $d=2$ defectives, which is significantly better than the bound for disjunctive codes and almost as good as the bound for union-free codes.
    Keywords Computer Science - Information Theory ; Mathematics - Combinatorics
    Publishing date 2021-08-30
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  2. Book ; Online: Complete Traceability Multimedia Fingerprinting Codes Resistant to Averaging Attack and Adversarial Noise with Optimal Rate

    Vorobyev, Ilya

    2021  

    Abstract: In this paper we consider complete traceability multimedia fingerprinting codes resistant to averaging attacks and adversarial noise. Recently it was shown that there are no such codes for the case of an arbitrary linear attack. However, for the case of ... ...

    Abstract In this paper we consider complete traceability multimedia fingerprinting codes resistant to averaging attacks and adversarial noise. Recently it was shown that there are no such codes for the case of an arbitrary linear attack. However, for the case of averaging attacks complete traceability multimedia fingerprinting codes of exponential cardinality resistant to constant adversarial noise were constructed in 2020 by Egorova et al. We continue this work and provide an improved lower bound on the rate of these codes.
    Keywords Computer Science - Information Theory
    Publishing date 2021-08-20
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  3. Book ; Online: Optimal Multistage Group Testing Algorithm for 3 Defectives

    Vorobyev, Ilya

    2020  

    Abstract: Group testing is a well-known search problem that consists in detecting of $s$ defective members of a set of $t$ samples by carrying out tests on properly chosen subsets of samples. In classical group testing the goal is to find all defective elements by ...

    Abstract Group testing is a well-known search problem that consists in detecting of $s$ defective members of a set of $t$ samples by carrying out tests on properly chosen subsets of samples. In classical group testing the goal is to find all defective elements by using the minimal possible number of tests in the worst case. In this work, a multistage group testing problem is considered. Our goal is to construct a multistage search procedure, having asymptotically the same number of tests as the optimal adaptive algorithm. We propose a new approach to designing multistage algorithms, which allows us to construct a 5-stage algorithm for finding 3 defectives with the optimal number $3\log_2t(1+o(1))$ of tests.
    Keywords Computer Science - Information Theory ; Mathematics - Combinatorics ; 68R05
    Publishing date 2020-01-22
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  4. Book ; Online: A New Algorithm for Two-Stage Group Testing

    Vorobyev, Ilya

    2019  

    Abstract: Group testing is a well-known search problem that consists in detecting of $s$ defective members of a set of $t$ samples by carrying out tests on properly chosen subsets of samples. In classical group testing the goal is to find all defective elements by ...

    Abstract Group testing is a well-known search problem that consists in detecting of $s$ defective members of a set of $t$ samples by carrying out tests on properly chosen subsets of samples. In classical group testing the goal is to find all defective elements by using the minimal possible number of tests in the worst case. In this work, two-stage group testing is considered. Using the hypergraph approach we design a new search algorithm, which allows improving the known results for fixed $s$ and $t\to\infty$. For the case $s=2$ this algorithm achieves information-theoretic lower bound $2\log_2t(1+o(1))$ on the number of tests in the worst case. Also, the problem of finding $m$ out of $s$ defectives is considered.

    Comment: 7 pages, 1 table
    Keywords Computer Science - Information Theory
    Publishing date 2019-01-20
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  5. Book ; Online: Efficient Combinatorial Group Testing

    Goshkoder, Daniil / Polyanskii, Nikita / Vorobyev, Ilya

    Bridging the Gap between Union-Free and Disjunctive Codes

    2024  

    Abstract: This work focuses on non-adaptive group testing, with a primary goal of efficiently identifying a set of at most $d$ defective elements among a given set of elements using the fewest possible number of tests. Non-adaptive combinatorial group testing ... ...

    Abstract This work focuses on non-adaptive group testing, with a primary goal of efficiently identifying a set of at most $d$ defective elements among a given set of elements using the fewest possible number of tests. Non-adaptive combinatorial group testing often employs disjunctive codes and union-free codes. This paper discusses union-free codes with fast decoding (UFFD codes), a recently introduced class of union-free codes that combine the best of both worlds -- the linear complexity decoding of disjunctive codes and the fewest number of tests of union-free codes. In our study, we distinguish two subclasses of these codes -- one subclass, denoted as $(=d)$-UFFD codes, can be used when the number of defectives $d$ is a priori known, whereas $(\le d)$-UFFD codes works for any subset of at most $d$ defectives. Previous studies have established a lower bound on the rate of these codes for $d=2$. Our contribution lies in deriving new lower bounds on the rate for both $(=d)$- and $(\le d)$-UFFD codes for an arbitrary number $d \ge 2$ of defectives. Our results show that for $d\to\infty$, the rate of $(=d)$-UFFD codes is twice as large as the best-known lower bound on the rate of $d$-disjunctive codes. In addition, the rate of $(\le d)$-UFFD code is shown to be better than the known lower bound on the rate of $d$-disjunctive codes for small values of $d$.
    Keywords Computer Science - Information Theory
    Subject code 003
    Publishing date 2024-01-29
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  6. Book ; Online: Secure Codes with List Decoding

    Gu, Yujie / Vorobyev, Ilya / Miao, Ying

    2023  

    Abstract: In this paper we consider combinatorial secure codes in traitor tracing for protecting copyright of multimedia content. First, we introduce a new notion of secure codes with list decoding (SCLDs) for collusion-resistant multimedia fingerprinting, which ... ...

    Abstract In this paper we consider combinatorial secure codes in traitor tracing for protecting copyright of multimedia content. First, we introduce a new notion of secure codes with list decoding (SCLDs) for collusion-resistant multimedia fingerprinting, which includes many existing types of fingerprinting codes as special cases. Next, we build efficient identifying algorithms for SCLDs with complete traceability and establish bounds on its largest possible code rate. In comparison with the existing fingerprinting codes, it is shown that SCLDs have not only much more efficient traceability than separable codes but also a much larger code rate than frameproof codes. As a byproduct, new bounds on the largest code rate of binary separable codes are established as well. Furthermore, a two-stage dynamic traitor tracing framework is proposed for multimedia fingerprinting in the dynamic scenario, which could not only efficiently achieve the complete traceability but also provide a much larger capacity than the static scenario.

    Comment: 20 pages
    Keywords Computer Science - Information Theory ; Computer Science - Cryptography and Security ; Computer Science - Discrete Mathematics ; Mathematics - Combinatorics
    Subject code 003
    Publishing date 2023-02-05
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  7. Book ; Online: Correcting one error in channels with feedback

    Vorobyev, Ilya / Lebedev, Alexey / Lebedev, Vladimir / Deppe, Christian

    2023  

    Abstract: We address the problem of correcting a single error in an arbitrary discrete memoryless channel with error-free instantaneous feedback. For the case of a one-time feedback, we propose a method for constructing optimal transmission strategies. The ... ...

    Abstract We address the problem of correcting a single error in an arbitrary discrete memoryless channel with error-free instantaneous feedback. For the case of a one-time feedback, we propose a method for constructing optimal transmission strategies. The obtained result allows us to prove that for a binary channel, two feedbacks are sufficient to transmit the same number of messages as in the case of complete feedback. We also apply the developed techniques to a binary asymmetric channel to construct transmission strategies for small lengths.
    Keywords Computer Science - Information Theory ; Mathematics - Combinatorics
    Publishing date 2023-01-04
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  8. Book ; Online: Codes Correcting a Single Long Duplication Error

    Goshkoder, Daniil / Polyanskii, Nikita / Vorobyev, Ilya

    2023  

    Abstract: We consider the problem of constructing a code capable of correcting a single long tandem duplication error of variable length. As the main contribution of this paper, we present a $q$-ary efficiently encodable code of length $n+1$ and redundancy $1$ ... ...

    Abstract We consider the problem of constructing a code capable of correcting a single long tandem duplication error of variable length. As the main contribution of this paper, we present a $q$-ary efficiently encodable code of length $n+1$ and redundancy $1$ that can correct a single duplication of length at least $K=4\cdot\lceil \log_q n\rceil +1$. The complexity of encoding is $O(\frac{n^2}{\log n})$ and the complexity of decoding is $O(n)$. We also present a $q$-ary non-efficient code of length $n+1$ correcting single long duplication of length at least $K = \lceil \log_q n\rceil +\phi(n)$, where $\phi(n)\rightarrow{\infty}$ as $n\rightarrow{\infty}$. This code has redundancy less than $1$ for sufficiently large $n$. Moreover, we show that in the class of codes correcting a single long duplication with redundancy $1$, the value $K$ in our constructions is order-optimal.
    Keywords Computer Science - Information Theory
    Publishing date 2023-04-24
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  9. Book ; Online: Correcting One Error in Non-Binary Channels with Feedback

    Vorobyev, Ilya / Lebedev, Vladimir / Lebedev, Alexey

    2023  

    Abstract: In this paper, the problem of correction of a single error in $q$-ary symmetric channel with noiseless feedback is considered. We propose an algorithm to construct codes with feedback inductively. For all prime power $q$ we prove that two instances of ... ...

    Abstract In this paper, the problem of correction of a single error in $q$-ary symmetric channel with noiseless feedback is considered. We propose an algorithm to construct codes with feedback inductively. For all prime power $q$ we prove that two instances of feedback are sufficient to transmit over the $q$-ary symmetric channel the same number of messages as in the case of complete feedback. Our other contribution is the construction of codes with one-time feedback with the same parameters as Hamming codes for $q$ that is not a prime power. We also construct single-error-correcting codes with one-time feedback of size $q^{n-2}$ for arbitrary $q$ and $n\leq q+1$, which can be seen as an analog for Reed-Solomon codes.
    Keywords Computer Science - Information Theory ; Mathematics - Combinatorics
    Publishing date 2023-05-11
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    Document type Book ; Online
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  10. Book ; Online: Feedback Insertion-Deletion Codes

    Maringer, Georg / Polyanskii, Nikita / Vorobyev, Ilya / Welter, Lorenz

    2020  

    Abstract: In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that we can transmit $n$ binary symbols one-by-one over the channel, in which some symbols can be deleted and ... ...

    Abstract In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that we can transmit $n$ binary symbols one-by-one over the channel, in which some symbols can be deleted and some additional symbols can be inserted. After each transmission, the encoder is notified about the insertions or deletions that have occurred within the previous transmission and the encoding strategy can be adapted accordingly. The goal is to design an encoder which is able to transmit as much information as possible under the assumption that the total number of deletions and insertions is limited by $\tau n$, $0<\tau<1$. We show how this problem can be reduced to the problem of transmitting messages over the substitution channel. Thereby, the maximal asymptotic rate of feedback insertion-deletion codes is completely established. For the substitution channel with random errors, an optimal algorithm with a nice intuition behind it was presented in 1963 by Horstein. However, the analysis of this algorithm for the adversarial substitution channel is quite complicated and was completed only 13 years later by Zigangirov. We revisit Zigangirov's result and present a more elaborate version of his proof.

    Comment: 34 pages, 9 figures
    Keywords Computer Science - Information Theory ; Computer Science - Discrete Mathematics
    Subject code 003
    Publishing date 2020-10-29
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