arXiv subjects

Vitaly Skachek

Publications and source records attributed to Vitaly Skachek.

At least 37 records · Page 2Linked to original sources

arXiv2015

Data Dissemination Problem in Wireless Networks

In this work, we formulate and study a data dissemination problem, which can be viewed as a generalization of the index coding problem and of the data exchange problem to networks with an arbitrary topology. We define $r$-solvable networks, in which data dissemination can be achieved in $r > 0$ communications rounds. We show that the optimum number of transmissions for any one-round communications scheme is given by the minimum rank of a certain constrained family of matrices. For a special case of this problem, called bipartite data dissemination problem, we present lower and upper graph-theoretic bounds on the optimum number of transmissions. For general $r$-solvable networks, we derive an upper bound on the minimum number of transmissions in any scheme with $\geq r$ rounds. We experimentally compare the obtained upper bound to a simple lower bound.

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arXiv2015

New Bounds for Permutation Codes in Ulam Metric

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.

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arXiv2015

Refined Upper Bounds on Stopping Redundancy of Binary Linear Codes

The $l$-th stopping redundancy $ρ_l(\mathcal C)$ of the binary $[n, k, d]$ code $\mathcal C$, $1 \le l \le d$, is defined as the minimum number of rows in the parity-check matrix of $\mathcal C$, such that the smallest stopping set is of size at least $l$. The stopping redundancy $ρ(\mathcal C)$ is defined as $ρ_d(\mathcal C)$. In this work, we improve on the probabilistic analysis of stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best bounds known today. In our approach, we judiciously select the first few rows in the parity-check matrix, and then continue with the probabilistic method. By using similar techniques, we improve also on the best known bounds on $ρ_l(\mathcal C)$, for $1 \le l \le d$. Our approach is compared to the existing methods by numerical computations.

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arXiv2014

Linear Batch Codes

In an application, where a client wants to obtain many elements from a large database, it is often desirable to have some load balancing. Batch codes (introduced by Ishai et al. in STOC 2004) make it possible to do exactly that: the large database is divided between many servers, so that the client has to only make a small number of queries to every server to obtain sufficient information to reconstruct all desired elements. Other important parameters of the batch codes are total storage and the number of servers. Batch codes also have applications in cryptography (namely, in the construction of multi-query computationally-private information retrieval protocols). In this work, we initiate the study of linear batch codes. These codes, in particular, are of potential use in distributed storage systems. We show that a generator matrix of a binary linear batch code is also a generator matrix of classical binary linear error-correcting code. This immediately yields that a variety of upper bounds, which were developed for error-correcting codes, are applicable also to binary linear batch codes. We also propose new methods to construct large linear batch codes from the smaller ones.

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arXiv2013

Error-Correction in Flash Memories via Codes in the Ulam Metric

We consider rank modulation codes for flash memories that allow for handling arbitrary charge-drop errors. Unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements, the proposed \emph{translocation rank codes} account for more general forms of errors that arise in storage systems. Translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding. Our results include derivation of the asymptotic capacity of translocation rank codes, construction techniques for asymptotically good codes, as well as simple decoding methods for one class of constructed codes. As part of our exposition, we also highlight the close connections between the new code family and permutations with short common subsequences, deletion and insertion error-correcting codes for permutations, and permutation codes in the Hamming distance.

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arXiv2013

Optimal Index Codes with Near-Extreme Rates

The min-rank of a digraph was shown by Bar-Yossef et al. (2006) to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. In particular, it is shown that the decision problem whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. Additionally, a new upper bound on the min-rank of a digraph, the circuit-packing bound, is presented. This bound is often tighter than the previously known bounds. By employing this new bound, we present several families of digraphs whose min-ranks can be found in polynomial time.

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arXiv2012

Hybrid Noncoherent Network Coding

We describe a novel extension of subspace codes for noncoherent networks, suitable for use when the network is viewed as a communication system that introduces both dimension and symbol errors. We show that when symbol erasures occur in a significantly large number of different basis vectors transmitted through the network and when the min-cut of the networks is much smaller then the length of the transmitted codewords, the new family of codes outperforms their subspace code counterparts. For the proposed coding scheme, termed hybrid network coding, we derive two upper bounds on the size of the codes. These bounds represent a variation of the Singleton and of the sphere-packing bound. We show that a simple concatenated scheme that represents a combination of subspace codes and Reed-Solomon codes is asymptotically optimal with respect to the Singleton bound. Finally, we describe two efficient decoding algorithms for concatenated subspace codes that in certain cases have smaller complexity than subspace decoders.

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arXiv2012

Forwarding Without Repeating: Efficient Rumor Spreading in Bounded-Degree Graphs

We study a gossip protocol called forwarding without repeating (FWR). The objective is to spread multiple rumors over a graph as efficiently as possible. FWR accomplishes this by having nodes record which messages they have forwarded to each neighbor, so that each message is forwarded at most once to each neighbor. We prove that FWR spreads a rumor over a strongly connected digraph, with high probability, in time which is within a constant factor of optimal for digraphs with bounded out-degree. Moreover, on digraphs with bounded out-degree and bounded number of rumors, the number of transmissions required by FWR is arbitrarily better than that of existing approaches. Specifically, FWR requires O(n) messages on bounded-degree graphs with n nodes, whereas classical forwarding and an approach based on network coding both require ω(n) messages. Our results are obtained using combinatorial and probabilistic arguments. Notably, they do not depend on expansion properties of the underlying graph, and consequently the message complexity of FWR is arbitrarily better than classical forwarding even on constant-degree expander graphs, as n \rightarrow \infty. In resource-constrained applications, where each transmission consumes battery power and bandwidth, our results suggest that using a small amount of memory at each node leads to a significant savings.

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arXiv2012

On the Pseudocodeword Redundancy of Binary Linear Codes

The AWGNC, BSC, and max-fractional pseudocodeword redundancies of a binary linear code are defined to be the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code. It is shown that most codes do not have a finite pseudocodeword redundancy. Also, upper bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs, are provided. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Furthermore, comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel and Koetter is sharp.

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arXiv2011

Error Correction for Index Coding with Side Information

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $δ$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $α$-bound and the $κ$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another implicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced.

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arXiv2011

On the Security of Index Coding with Side Information

Security aspects of the Index Coding with Side Information (ICSI) problem are investigated. Building on the results of Bar-Yossef et al. (2006), the properties of linear index codes are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the linear index code based on a matrix $L$, whose column space code $C(L)$ has length $n$, minimum distance $d$ and dual distance $d^\perp$, is $(d-1-t)$-block secure (and hence also weakly secure) if the adversary knows in advance $t \leq d-2$ messages, and is completely insecure if the adversary knows in advance more than $n - d$ messages. Strong security is examined under the conditions that the adversary: (i) possesses $t$ messages in advance; (ii) eavesdrops at most $μ$ transmissions; (iii) corrupts at most $δ$ transmissions. We prove that for sufficiently large $q$, an optimal linear index code which is strongly secure against such an adversary has length $κ_q+μ+2δ$. Here $κ_q$ is a generalization of the min-rank over $F_q$ of the side information graph for the ICSI problem in its original formulation in the work of Bar- Yossef et al.

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arXiv2011

Index Coding and Error Correction

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $δ$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $α$-bound and the $κ$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Finally, the decoding of linear ECIC's is discussed. The syndrome decoding is shown to output the exact message if the weight of the error vector is less or equal to the error-correcting capability of the corresponding ECIC.

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arXiv2010

Secure Index Coding with Side Information

Security aspects of the Index Coding with Side Information (ICSI) problem are investigated. Building on the results of Bar-Yossef et al. (2006), the properties of linear coding schemes for the ICSI problem are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the coding scheme for the ICSI problem based on a linear code C of length n, minimum distance d and dual distance d^\perp, is (d-1-t)-block secure (and hence also weakly secure) if the adversary knows in advance t \le d - 2 messages, and is completely insecure if the adversary knows in advance more than n - d^\perp messages.

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arXiv2010

Correcting a Fraction of Errors in Nonbinary Expander Codes with Linear Programming

A linear-programming decoder for \emph{nonbinary} expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties. It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 δ_A δ_B (where δ_A and δ_B are the relative minimum distances of the constituent codes).

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arXiv2010

Characterization of Graph-cover Pseudocodewords of Codes over $F_3$

Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becom is more involved. In this work, a characterization of the fundamental cone for codes over $F_3$ is discussed.

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arXiv2010

Exploration of AWGNC and BSC Pseudocodeword Redundancy

The AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code is defined as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code. This paper provides new results on the AWGNC, BSC, and max-fractional pseudocodeword redundancies of codes. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Also, comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel and Koetter is sharp.

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arXiv2010

On the Pseudocodeword Redundancy

We define the AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance. We show that most codes do not have a finite pseudocodeword redundancy. We also provide bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs.

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arXiv2009

Recursive Code Construction for Random Networks

A modification of Koetter-Kschischang codes for random networks is presented (these codes were also studied by Wang et al. in the context of authentication problems). The new codes have higher information rate, while maintaining the same error-correcting capabilities. An efficient error-correcting algorithm is proposed for these codes.

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