82 research outputs found

    Optimal Binary Locally Repairable Codes via Anticodes

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    This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound

    A study of (xvt,xvt−1)-minihypers in PG(t,q)

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    AbstractWe study (xvt,xvt−1)-minihypers in PG(t,q), i.e. minihypers with the same parameters as a weighted sum of x hyperplanes. We characterize these minihypers as a nonnegative rational sum of hyperplanes and we use this characterization to extend and improve the main results of several papers which have appeared on the special case t=2. We establish a new link with coding theory and we use this link to construct several new infinite classes of (xvt,xvt−1)-minihypers in PG(t,q) that cannot be written as an integer sum of hyperplanes

    The second Feng-Rao number for codes coming from telescopic semigroups

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    In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Ap\'ery sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound

    Update-Efficiency and Local Repairability Limits for Capacity Approaching Codes

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    Motivated by distributed storage applications, we investigate the degree to which capacity achieving encodings can be efficiently updated when a single information bit changes, and the degree to which such encodings can be efficiently (i.e., locally) repaired when single encoded bit is lost. Specifically, we first develop conditions under which optimum error-correction and update-efficiency are possible, and establish that the number of encoded bits that must change in response to a change in a single information bit must scale logarithmically in the block-length of the code if we are to achieve any nontrivial rate with vanishing probability of error over the binary erasure or binary symmetric channels. Moreover, we show there exist capacity-achieving codes with this scaling. With respect to local repairability, we develop tight upper and lower bounds on the number of remaining encoded bits that are needed to recover a single lost bit of the encoding. In particular, we show that if the code-rate is ϵ\epsilon less than the capacity, then for optimal codes, the maximum number of codeword symbols required to recover one lost symbol must scale as log1/ϵ\log1/\epsilon. Several variations on---and extensions of---these results are also developed.Comment: Accepted to appear in JSA
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