51 research outputs found

    Locally Encodable and Decodable Codes for Distributed Storage Systems

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    We consider the locality of encoding and decoding operations in distributed storage systems (DSS), and propose a new class of codes, called locally encodable and decodable codes (LEDC), that provides a higher degree of operational locality compared to currently known codes. For a given locality structure, we derive an upper bound on the global distance and demonstrate the existence of an optimal LEDC for sufficiently large field size. In addition, we also construct two families of optimal LEDC for fields with size linear in code length.Comment: 7 page

    Matrix Multiplication Verification Using Coding Theory

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    We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n×nn \times n matrices AA, BB, and CC as input, to decide whether AB=CAB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in O~(n2)\widetilde{O}(n^2) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(nω)o(n^{\omega}) time). To that end, we give two algorithms for MMV in the case where ABCAB - C is sparse. Specifically, when ABCAB - C has at most O(nδ)O(n^{\delta}) non-zero entries for a constant 0δ<20 \leq \delta < 2, we give (1) a deterministic O(nωε)O(n^{\omega - \varepsilon})-time algorithm for constant ε=ε(δ)>0\varepsilon = \varepsilon(\delta) > 0, and (2) a randomized O~(n2)\widetilde{O}(n^2)-time algorithm using δ/2log2n+O(1)\delta/2 \cdot \log_2 n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA, 2018) when δ1.056\delta \geq 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log2n+O(1)\log_2 n + O(1) random bits (in turn fewer than Freivalds's algorithm). We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(nω)\Omega(n^{\omega}) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic O~(n2)\widetilde{O}(n^2)-time reductions)

    Network Coding for Wireless and Wired Networks: Design, Performance and Achievable Rates

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    Many point-to-point communication problems are relatively well understood in the literature. For example, in addition to knowing what the capacity of a point-to-point channel is, we also know how to construct codes that will come arbitrarily close to the capacity of these channels. However, we know very little about networks. For example, we do not know the capacity of the two-way relay channel which consists of only three transmitters. The situation is not so different in the wired networks except special cases like multicasting. To understand networks better, in this thesis we study network coding which is considered to be a promising technique since the time it was shown to achieve the single-source multicast capacity. First we design and analyze deterministic and random network coding schemes for a cooperative communication setup with multiple sources and destinations. We show that our schemes outperform conventional cooperation in terms of the diversity-multiplexing tradeoff (DMT). Specifically, it can offer the maximum diversity order at the expense of a slightly reduced multiplexing rate. We derive the necessary and sufficient conditions to achieve the maximum diversity order. We show that when the parity-check matrix for a systematic maximum distance separable (MDS) code is used as the network coding matrix, the maximum diversity is achieved. We present two ways to generate full-diversity network coding matrices: namely using the Cauchy matrices and the Vandermonde matrices. We also analyze a selection relaying scheme and prove that a multiplicative diversity order is possible with enough number of relay selection rounds. In addition to the above scheme for wireless networks, we also study wired networks, and apply network coding together with interference alignment. We consider networks with KK source nodes and JJ destination nodes with arbitrary message demands. We first consider a simple network consisting of three source nodes and four destination nodes and show that each user can achieve a rate of one half. Then we extend the result for the general case which states that when the min-cut between each source-destination pair is one, it is possible to achieve a sum rate that is arbitrarily close to the min-cut between the source nodes whose messages are demanded and the destination node where the sum rate is the summation of all the demanded source message rates plus the biggest interferer\u27s rate

    A construction of MDS array codes

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    In this paper a new construction of MDS array codes is introduced. In order to obtain a code with this property, the parity-check matrix is constructed just using a superregular matrix of blocks composed of powers of the companion matrix of a primitive polynomial. Also a decoding algorithm for these codes is introduced.The work of the first and the second authors was partially supported by Spanish grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España. The work of first author was also partially supported by a grant for research students from the Generalitat Valenciana with reference BFPI/2008/138. The work of the third author was partially supported by the research project UMH-Bancaja with reference IPZS01
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