144 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
Optimal Linear and Cyclic Locally Repairable Codes over Small Fields
We consider locally repairable codes over small fields and propose
constructions of optimal cyclic and linear codes in terms of the dimension for
a given distance and length. Four new constructions of optimal linear codes
over small fields with locality properties are developed. The first two
approaches give binary cyclic codes with locality two. While the first
construction has availability one, the second binary code is characterized by
multiple available repair sets based on a binary Simplex code. The third
approach extends the first one to q-ary cyclic codes including (binary)
extension fields, where the locality property is determined by the properties
of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear
codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem,
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