1,207 research outputs found
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
Repair-Optimal MDS Array Codes over GF(2)
Maximum-distance separable (MDS) array codes with high rate and an optimal
repair property were introduced recently. These codes could be applied in
distributed storage systems, where they minimize the communication and disk
access required for the recovery of failed nodes. However, the encoding and
decoding algorithms of the proposed codes use arithmetic over finite fields of
order greater than 2, which could result in a complex implementation.
In this work, we present a construction of 2-parity MDS array codes, that
allow for optimal repair of a failed information node using XOR operations
only. The reduction of the field order is achieved by allowing more parity bits
to be updated when a single information bit is being changed by the user.Comment: 5 pages, submitted to ISIT 201
MDS Array Codes with Optimal Rebuilding
MDS array codes are widely used in storage systems
to protect data against erasures. We address the rebuilding ratio
problem, namely, in the case of erasures, what is the the fraction
of the remaining information that needs to be accessed in order
to rebuild 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 r-erasure correcting
MDS array codes that has optimal rebuilding ratio of 1/r
in the
case of a single erasure. Our array codes have efficient encoding
and decoding algorithms (for the case r = 2 they use a finite field
of size 3) and an optimal update property
Error correction based on partial information
We consider the decoding of linear and array codes from errors when we are
only allowed to download a part of the codeword. More specifically, suppose
that we have encoded data symbols using an code with code length
and dimension During storage, some of the codeword coordinates might
be corrupted by errors. We aim to recover the original data by reading the
corrupted codeword with a limit on the transmitting bandwidth, namely, we can
only download an proportion of the corrupted codeword. For a given
our objective is to design a code and a decoding scheme such that we
can recover the original data from the largest possible number of errors. A
naive scheme is to read coordinates of the codeword. This method
used in conjunction with MDS codes guarantees recovery from any errors. In this paper we show that we can instead read an
proportion from each of the codeword's coordinates. For a
well-designed MDS code, this method can guarantee recovery from errors, which is times more than the naive
method, and is also the maximum number of errors that an code can
correct by downloading only an proportion of the codeword. We present
two families of such optimal constructions and decoding schemes. One is a
Reed-Solomon code with evaluation points in a subfield and the other is based
on Folded Reed-Solomon codes. We further show that both code constructions
attain asymptotically optimal list decoding radius when downloading only a part
of the corrupted codeword. We also construct an ensemble of random codes that
with high probability approaches the upper bound on the number of correctable
errors when the decoder downloads an proportion of the corrupted
codeword.Comment: Extended version of the conference paper in ISIT 201
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
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