1,293 research outputs found
Optimal Linear and Cyclic Locally Repairable Codes over Small Fields
We consider locally repairable codes over small fields and propose
constructions of optimal cyclic and linear codes in terms of the dimension for
a given distance and length. Four new constructions of optimal linear codes
over small fields with locality properties are developed. The first two
approaches give binary cyclic codes with locality two. While the first
construction has availability one, the second binary code is characterized by
multiple available repair sets based on a binary Simplex code. The third
approach extends the first one to q-ary cyclic codes including (binary)
extension fields, where the locality property is determined by the properties
of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear
codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem,
Israe
Cooperative Local Repair in Distributed Storage
Erasure-correcting codes, that support local repair of codeword symbols, have
attracted substantial attention recently for their application in distributed
storage systems. This paper investigates a generalization of the usual locally
repairable codes. In particular, this paper studies a class of codes with the
following property: any small set of codeword symbols can be reconstructed
(repaired) from a small number of other symbols. This is referred to as
cooperative local repair. The main contribution of this paper is bounds on the
trade-off of the minimum distance and the dimension of such codes, as well as
explicit constructions of families of codes that enable cooperative local
repair. Some other results regarding cooperative local repair are also
presented, including an analysis for the well-known Hadamard/Simplex codes.Comment: Fixed some minor issues in Theorem 1, EURASIP Journal on Advances in
Signal Processing, December 201
Locality and Availability in Distributed Storage
This paper studies the problem of code symbol availability: a code symbol is
said to have -availability if it can be reconstructed from disjoint
groups of other symbols, each of size at most . For example, -replication
supports -availability as each symbol can be read from its other
(disjoint) replicas, i.e., . However, the rate of replication must vanish
like 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 trade-off 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.Comment: Submitted to ISIT 201
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