202 research outputs found
Low-Complexity Codes for Random and Clustered High-Order Failures in Storage Arrays
RC (Random/Clustered) codes are a new efficient array-code family for recovering from 4-erasures. RC codes correct most 4-erasures, and essentially all 4-erasures that are clustered. Clustered erasures are introduced as a new erasure model for storage arrays. This model draws its motivation from correlated device failures, that are caused by physical proximity of devices, or by age proximity of endurance-limited solid-state drives. The reliability of storage arrays that employ RC codes is analyzed and compared to known codes. The new RC code is significantly more efficient, in all practical implementation factors, than the best known 4-erasure correcting MDS code. These factors include: small-write update-complexity, full-device update-complexity, decoding complexity and number of supported devices in the array
X-code: MDS array codes with optimal encoding
We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns
Optimal Rebuilding of Multiple Erasures in MDS Codes
MDS array codes are widely used in storage systems due to their
computationally efficient encoding and decoding procedures. An MDS code with
redundancy nodes can correct any node erasures by accessing all the
remaining information in the surviving nodes. However, in practice,
erasures is a more likely failure event, for . Hence, a natural
question is how much information do we need to access in order to rebuild
storage nodes? We define the rebuilding ratio as the fraction of remaining
information accessed during the rebuilding of erasures. In our previous
work we constructed MDS codes, called zigzag codes, that achieve the optimal
rebuilding ratio of for the rebuilding of any systematic node when ,
however, all the information needs to be accessed for the rebuilding of the
parity node erasure.
The (normalized) repair bandwidth is defined as the fraction of information
transmitted from the remaining nodes during the rebuilding process. For codes
that are not necessarily MDS, Dimakis et al. proposed the regenerating codes
framework where any erasures can be corrected by accessing some of the
remaining information, and any erasure can be rebuilt from some subsets
of surviving nodes with optimal repair bandwidth.
In this work, we study 3 questions on rebuilding of codes: (i) We show a
fundamental trade-off between the storage size of the node and the repair
bandwidth similar to the regenerating codes framework, and show that zigzag
codes achieve the optimal rebuilding ratio of for MDS codes, for any
. (ii) We construct systematic codes that achieve optimal
rebuilding ratio of , for any systematic or parity node erasure. (iii) We
present error correction algorithms for zigzag codes, and in particular
demonstrate how these codes can be corrected beyond their minimum Hamming
distances.Comment: There is an overlap of this work with our two previous submissions:
Zigzag Codes: MDS Array Codes with Optimal Rebuilding; On Codes for Optimal
Rebuilding Access. arXiv admin note: text overlap with arXiv:1112.037
MDS array codes for correcting a signle criss-cross error
We present a family of maximum-distance separable (MDS) array codes of size (p-1)×(p-1), p a prime number, and minimum criss-cross distance 3, i.e., the code is capable of correcting any row or column in error, without a priori knowledge of what type of error occurred. The complexity of the encoding and decoding algorithms is lower than that of known codes with the same error-correcting power, since our algorithms are based on exclusive-OR operations over lines of different slopes, as opposed to algebraic operations over a finite field. We also provide efficient encoding and decoding algorithms for errors and erasures
Zigzag Codes: MDS Array Codes with Optimal Rebuilding
MDS array codes are widely used in storage systems to protect data against
erasures. We address the \emph{rebuilding ratio} problem, namely, in the case
of erasures, what is the fraction of the remaining information that needs to be
accessed in order to rebuild \emph{exactly} the lost information? It is clear
that when the number of erasures equals the maximum number of erasures that an
MDS code can correct then the rebuilding ratio is 1 (access all the remaining
information). However, the interesting and more practical case is when the
number of erasures is smaller than the erasure correcting capability of the
code. For example, consider an MDS code that can correct two erasures: What is
the smallest amount of information that one needs to access in order to correct
a single erasure? Previous work showed that the rebuilding ratio is bounded
between 1/2 and 3/4, however, the exact value was left as an open problem. In
this paper, we solve this open problem and prove that for the case of a single
erasure with a 2-erasure correcting code, the rebuilding ratio is 1/2. In
general, we construct a new family of -erasure correcting MDS array codes
that has optimal rebuilding ratio of in the case of erasures,
. Our array codes have efficient encoding and decoding
algorithms (for the case they use a finite field of size 3) and an
optimal update property.Comment: 23 pages, 5 figures, submitted to IEEE transactions on information
theor
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
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