316 research outputs found
Codes for Graph Erasures
Motivated by systems where the information is represented by a graph, such as
neural networks, associative memories, and distributed systems, we present in
this work a new class of codes, called codes over graphs. Under this paradigm,
the information is stored on the edges of an undirected graph, and a code over
graphs is a set of graphs. A node failure is the event where all edges in the
neighborhood of the failed node have been erased. We say that a code over
graphs can tolerate node failures if it can correct the erased edges of
any failed nodes in the graph. While the construction of such codes can
be easily accomplished by MDS codes, their field size has to be at least
, when is the number of nodes in the graph. In this work we present
several constructions of codes over graphs with smaller field size. In
particular, we present optimal codes over graphs correcting two node failures
over the binary field, when the number of nodes in the graph is a prime number.
We also present a construction of codes over graphs correcting node
failures for all over a field of size at least , and show how
to improve this construction for optimal codes when .Comment: To appear in IEEE International Symposium 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
A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes
Spatially-coupled LDPC codes are known to have excellent asymptotic
properties. Much less is known regarding their finite-length performance. We
propose a scaling law to predict the error probability of finite-length
spatially-coupled ensembles when transmission takes place over the binary
erasure channel. We discuss how the parameters of the scaling law are connected
to fundamental quantities appearing in the asymptotic analysis of these
ensembles and we verify that the predictions of the scaling law fit well to the
data derived from simulations over a wide range of parameters. The ultimate
goal of this line of research is to develop analytic tools for the design of
spatially-coupled LDPC codes under practical constraints
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
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