172 research outputs found
Exact Regeneration Codes for Distributed Storage Repair Using Interference Alignment
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes
has recently motivated a new class of codes, called Regenerating Codes, that
optimally trade off storage cost for repair bandwidth. On one end of this
spectrum of Regenerating Codes are Minimum Storage Regenerating (MSR) codes
that can match the minimum storage cost of MDS codes while also significantly
reducing repair bandwidth. In this paper, we describe Exact-MSR codes which
allow for any failed nodes (whether they are systematic or parity nodes) to be
regenerated exactly rather than only functionally or information-equivalently.
We show that Exact-MSR codes come with no loss of optimality with respect to
random-network-coding based MSR codes (matching the cutset-based lower bound on
repair bandwidth) for the cases of: (a) k/n <= 1/2; and (b) k <= 3. Our
constructive approach is based on interference alignment techniques, and,
unlike the previous class of random-network-coding based approaches, we provide
explicit and deterministic coding schemes that require a finite-field size of
at most 2(n-k).Comment: to be submitted to IEEE Transactions on Information Theor
Access vs. Bandwidth in Codes for Storage
Maximum distance separable (MDS) codes are widely used in storage systems to
protect against disk (node) failures. A node is said to have capacity over
some field , if it can store that amount of symbols of the field.
An MDS code uses nodes of capacity to store information
nodes. The MDS property guarantees the resiliency to any node failures.
An \emph{optimal bandwidth} (resp. \emph{optimal access}) MDS code communicates
(resp. accesses) the minimum amount of data during the repair process of a
single failed node. It was shown that this amount equals a fraction of
of data stored in each node. In previous optimal bandwidth
constructions, scaled polynomially with in codes with asymptotic rate
. Moreover, in constructions with a constant number of parities, i.e. rate
approaches 1, is scaled exponentially w.r.t. . In this paper, we focus
on the later case of constant number of parities , and ask the following
question: Given the capacity of a node what is the largest number of
information disks in an optimal bandwidth (resp. access) MDS
code. We give an upper bound for the general case, and two tight bounds in the
special cases of two important families of codes. Moreover, the bounds show
that in some cases optimal-bandwidth code has larger than optimal-access
code, and therefore these two measures are not equivalent.Comment: This paper was presented in part at the IEEE International Symposium
on Information Theory (ISIT 2012). submitted to IEEE transactions on
information theor
Explicit MDS Codes for Optimal Repair Bandwidth
MDS codes are erasure-correcting codes that can correct the maximum number of
erasures for a given number of redundancy or parity symbols. If an MDS code has
parities and no more than erasures occur, then by transmitting all the
remaining data in the code, the original information can be recovered. However,
it was shown that in order to recover a single symbol erasure, only a fraction
of of the information needs to be transmitted. This fraction is called
the repair bandwidth (fraction). Explicit code constructions were given in
previous works. If we view each symbol in the code as a vector or a column over
some field, then the code forms a 2D array and such codes are especially widely
used in storage systems. In this paper, we address the following question:
given the length of the column , number of parities , can we construct
high-rate MDS array codes with optimal repair bandwidth of , whose code
length is as long as possible? In this paper, we give code constructions such
that the code length is .Comment: 17 page
A Framework of Constructions of Minimal Storage Regenerating Codes with the Optimal Access/Update Property
In this paper, we present a generic framework for constructing systematic
minimum storage regenerating codes with two parity nodes based on the invariant
subspace technique. Codes constructed in our framework not only contain some
best known codes as special cases, but also include some new codes with key
properties such as the optimal access property and the optimal update property.
In particular, for a given storage capacity of an individual node, one of the
new codes has the largest number of systematic nodes and two of the new codes
have the largest number of systematic nodes with the optimal update property.Comment: Accepted for publication in IEEE Transactions on Information Theor
Long MDS Codes for Optimal Repair Bandwidth
MDS codes are erasure-correcting codes that can
correct the maximum number of erasures given the number of
redundancy or parity symbols. If an MDS code has r parities
and no more than r erasures occur, then by transmitting all
the remaining data in the code one can recover the original
information. However, it was shown that in order to recover a
single symbol erasure, only a fraction of 1/r of the information
needs to be transmitted. This fraction is called the repair
bandwidth (fraction). Explicit code constructions were given in
previous works. If we view each symbol in the code as a vector
or a column, then the code forms a 2D array and such codes
are especially widely used in storage systems. In this paper, we
ask the following question: given the length of the column l, can
we construct high-rate MDS array codes with optimal repair
bandwidth of 1/r, whose code length is as long as possible? In
this paper, we give code constructions such that the code length
is (r + 1)log_r l
On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes
has recently motivated a new class of codes, called Regenerating Codes, that
optimally trade off storage cost for repair bandwidth. In this paper, we
address bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any
failed node to be repaired exactly with access to arbitrary d survivor nodes,
where k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve
minimum repair bandwidth (matching the cutset lower bound) for arbitrary
admissible (n,k,d), i.e., k<n and k<=d<=n-1. Our approach is based on
interference alignment techniques and uses vector linear codes which allow to
split symbols into arbitrarily small subsymbols.Comment: 20 pages, 6 figure
Interference Alignment in Regenerating Codes for Distributed Storage: Necessity and Code Constructions
Regenerating codes are a class of recently developed codes for distributed
storage that, like Reed-Solomon codes, permit data recovery from any arbitrary
k of n nodes. However regenerating codes possess in addition, the ability to
repair a failed node by connecting to any arbitrary d nodes and downloading an
amount of data that is typically far less than the size of the data file. This
amount of download is termed the repair bandwidth. Minimum storage regenerating
(MSR) codes are a subclass of regenerating codes that require the least amount
of network storage; every such code is a maximum distance separable (MDS) code.
Further, when a replacement node stores data identical to that in the failed
node, the repair is termed as exact.
The four principal results of the paper are (a) the explicit construction of
a class of MDS codes for d = n-1 >= 2k-1 termed the MISER code, that achieves
the cut-set bound on the repair bandwidth for the exact-repair of systematic
nodes, (b) proof of the necessity of interference alignment in exact-repair MSR
codes, (c) a proof showing the impossibility of constructing linear,
exact-repair MSR codes for d < 2k-3 in the absence of symbol extension, and (d)
the construction, also explicit, of MSR codes for d = k+1. Interference
alignment (IA) is a theme that runs throughout the paper: the MISER code is
built on the principles of IA and IA is also a crucial component to the
non-existence proof for d < 2k-3. To the best of our knowledge, the
constructions presented in this paper are the first, explicit constructions of
regenerating codes that achieve the cut-set bound.Comment: 38 pages, 12 figures, submitted to the IEEE Transactions on
Information Theory;v3 - The title has been modified to better reflect the
contributions of the submission. The paper is extensively revised with
several carefully constructed figures and example
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