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Explicit maximally recoverable codes with locality, arXiv:1307.4150. Available online at http://arxiv.org
"... AbstractConsider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Such codes have been studied recently in the context of erasure coding for data storage, where the local parities facilit ..."
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AbstractConsider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Such codes have been studied recently in the context of erasure coding for data storage, where the local parities facilitate fast recovery of any single symbol when it is erased, while the heavy parities provide tolerance to a large number of simultaneous erasures. A code as above is maximally recoverable, if it corrects all erasure patterns which are information theoretically correctable given the prescribed dependency relations between data symbols and parity symbols. In this paper we present explicit families of maximally recoverable codes with locality. We also initiate the general study of the tradeoff between maximal recoverability and alphabet size.
Locality and availability in distributed storage
 IN PROCEEDINGS OF THE IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY (ISIT
, 2014
"... This paper studies the problem of code symbol availability: a code symbol is said to have (r, t)availability if it can be reconstructed from t disjoint groups of other symbols, each of size at most r. For example, 3replication supports (1, 2)availability as each symbol can be read from its t = 2 ..."
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This paper studies the problem of code symbol availability: a code symbol is said to have (r, t)availability if it can be reconstructed from t disjoint groups of other symbols, each of size at most r. For example, 3replication supports (1, 2)availability as each symbol can be read from its t = 2 other (disjoint) replicas, i.e., r = 1. However, the rate of replication must vanish like 1 t+1 as the availability increases. This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a tradeoff between minimum distance, availability and locality. Our codes match the aforementioned bound and their construction relies on combinatorial objects called resolvable designs. From a practical standpoint, our codes seem useful for distributed storage applications involving hot data, i.e., the information which is frequently accessed by multiple processes in parallel.
On Cooperative Local Repair in Distributed Storage
"... Abstract—Erasurecorrecting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. In this paper we study a generalization of the usual locally recoverable codes. We consider such codes that any small ..."
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Abstract—Erasurecorrecting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. In this paper we study a generalization of the usual locally recoverable codes. We consider such codes that any small set of codeword symbols is recoverable from a small number of other symbols. We call this cooperative local repair. We present bounds on the dimension of such codes as well as give explicit constructions of families of codes. Some other results regarding cooperative local repair are also presented, including an analysis for the Hadamard codes. I.
An Integer Programming Based Bound for Locally Repairable Codes
, 2014
"... The locally repairable code (LRC) studied in this paper is an [n, k] linear code of which the value at each coordinate can be recovered by a linear combination of at most r other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an i ..."
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The locally repairable code (LRC) studied in this paper is an [n, k] linear code of which the value at each coordinate can be recovered by a linear combination of at most r other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters n1> n2, where n1 = n r+1 and n2 = n1(r+1) − n. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field F2m, where m ≥ n1r. Based on these results, the largest possible minimum distance for all LRCs with r ≤ √n − 1 has been definitely determined, which is of great significance in practical use.