146 research outputs found
Optimal Locally Repairable Systematic Codes Based on Packings
Locally repairable codes are desirable for distributed storage systems to
improve the repair efficiency. In this paper, we first build a bridge between
locally repairable code and packing. As an application of this bridge, some
optimal locally repairable codes can be obtained by packings, which gives
optimal locally repairable codes with flexible parameters.Comment: 13 page
Bounds and Constructions of Locally Repairable Codes: Parity-check Matrix Approach
A -ary locally repairable code (LRC) is an linear code
over such that every code symbol can be recovered by accessing
at most other code symbols. The well-known Singleton-like bound says that
and an LRC is said to be optimal if it attains
this bound. In this paper, we study the bounds and constructions of LRCs from
the view of parity-check matrices. Firstly, a simple and unified framework
based on parity-check matrix to analyze the bounds of LRCs is proposed. Several
useful structural properties on -ary optimal LRCs are obtained. We derive an
upper bound on the minimum distance of -ary optimal -LRCs in terms
of the field size . Then, we focus on constructions of optimal LRCs over
binary field. It is proved that there are only 5 classes of possible parameters
with which optimal binary -LRCs exist. Moreover, by employing the
proposed parity-check matrix approach, we completely enumerate all these 5
classes of possible optimal binary LRCs attaining the Singleton-like bound in
the sense of equivalence of linear codes.Comment: 18 page
Erasure Coding for Distributed Storage: An Overview
In a distributed storage system, code symbols are dispersed across space in
nodes or storage units as opposed to time. In settings such as that of a large
data center, an important consideration is the efficient repair of a failed
node. Efficient repair calls for erasure codes that in the face of node
failure, are efficient in terms of minimizing the amount of repair data
transferred over the network, the amount of data accessed at a helper node as
well as the number of helper nodes contacted. Coding theory has evolved to
handle these challenges by introducing two new classes of erasure codes, namely
regenerating codes and locally recoverable codes as well as by coming up with
novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an
overview of the efforts in this direction that have taken place over the past
decade.Comment: This survey article will appear in Science China Information Sciences
(SCIS) journa
Achieving Arbitrary Locality and Availability in Binary Codes
The th coordinate of an code is said to have locality and
availability if there exist disjoint groups, each containing at most
other coordinates that can together recover the value of the th
coordinate. This property is particularly useful for codes for distributed
storage systems because it permits local repair and parallel accesses of hot
data. In this paper, for any positive integers and , we construct a
binary linear code of length which has locality and
availability for all coordinates. The information rate of this code attains
, which is always higher than that of the direct product code,
the only known construction that can achieve arbitrary locality and
availability.Comment: 5 page
An Upper Bound On the Size of Locally Recoverable Codes
In a {\em locally recoverable} or {\em repairable} code, any symbol of a
codeword can be recovered by reading only a small (constant) number of other
symbols. The notion of local recoverability is important in the area of
distributed storage where a most frequent error-event is a single storage node
failure (erasure). A common objective is to repair the node by downloading data
from as few other storage node as possible. In this paper, we bound the minimum
distance of a code in terms of its length, size and locality. Unlike previous
bounds, our bound follows from a significantly simple analysis and depends on
the size of the alphabet being used. It turns out that the binary Simplex codes
satisfy our bound with equality; hence the Simplex codes are the first example
of a optimal binary locally repairable code family. We also provide
achievability results based on random coding and concatenated codes that are
numerically verified to be close to our bounds.Comment: A shorter version has appeared in IEEE NetCod, 201
Local Codes with Addition Based Repair
We consider the complexities of repair algorithms for locally repairable
codes and propose a class of codes that repair single node failures using
addition operations only, or codes with addition based repair. We construct two
families of codes with addition based repair. The first family attains distance
one less than the Singleton-like upper bound, while the second family attains
the Singleton-like upper bound
Applied Erasure Coding in Networks and Distributed Storage
The amount of digital data is rapidly growing. There is an increasing use of
a wide range of computer systems, from mobile devices to large-scale data
centers, and important for reliable operation of all computer systems is
mitigating the occurrence and the impact of errors in digital data. The demand
for new ultra-fast and highly reliable coding techniques for data at rest and
for data in transit is a major research challenge. Reliability is one of the
most important design requirements. The simplest way of providing a degree of
reliability is by using data replication techniques. However, replication is
highly inefficient in terms of capacity utilization. Erasure coding has
therefore become a viable alternative to replication since it provides the same
level of reliability as replication with significantly less storage overhead.
The present thesis investigates efficient constructions of erasure codes for
different applications. Methods from both coding and information theory have
been applied to network coding, Optical Packet Switching (OPS) networks and
distributed storage systems. The following four issues are addressed: -
Construction of binary and non-binary erasure codes; - Reduction of the header
overhead due to the encoding coefficients in network coding; - Construction and
implementation of new erasure codes for large-scale distributed storage systems
that provide savings in the storage and network resources compared to
state-of-the-art codes; and - Provision of a unified view on Quality of Service
(QoS) in OPS networks when erasure codes are used, with the focus on Packet
Loss Rate (PLR), survivability and secrecy
On Optimal Locally Repairable Codes with Super-Linear Length
Locally repairable codes which are optimal with respect to the bound
presented by Prakash et al. are considered. New upper bounds on the length of
such optimal codes are derived. The new bounds both improve and generalize
previously known bounds. Optimal codes are constructed, whose length is
order-optimal when compared with the new upper bounds. The length of the codes
is super-linear in the alphabet size
Codes with Unequal Disjoint Local Erasure Correction Constraints
Recently, locally repairable codes (LRCs) with local erasure correction
constraints that are unequal and disjoint have been proposed. In this work, we
study the same topic and provide some improved and additional results.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1701.0734
Locally Repairable Codes
Distributed storage systems for large-scale applications typically use
replication for reliability. Recently, erasure codes were used to reduce the
large storage overhead, while increasing data reliability. A main limitation of
off-the-shelf erasure codes is their high-repair cost during single node
failure events. A major open problem in this area has been the design of codes
that {\it i)} are repair efficient and {\it ii)} achieve arbitrarily high data
rates.
In this paper, we explore the repair metric of {\it locality}, which
corresponds to the number of disk accesses required during a
{\color{black}single} node repair. Under this metric we characterize an
information theoretic trade-off that binds together locality, code distance,
and the storage capacity of each node. We show the existence of optimal {\it
locally repairable codes} (LRCs) that achieve this trade-off. The achievability
proof uses a locality aware flow-graph gadget which leads to a randomized code
construction. Finally, we present an optimal and explicit LRC that achieves
arbitrarily high data-rates. Our locality optimal construction is based on
simple combinations of Reed-Solomon blocks.Comment: presented at ISIT 2012, accepted for publication in IEEE Trans. IT,
201
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