30,829 research outputs found
LDPC Codes Based on Algebraic Graphs
In this paper we investigate correcting properties of LDPC codes obtained from families of algebraic graphs. The graphs considered in this article come from the infinite incidence structure. We describe how to construct these codes, choose the parameters and present several simulations, done by using the MAP decoder. We describe how error correcting properties are dependent on the graph structure. We compare our results with the currently used codes, obtained by Guinand and Lodge [1] from the family of graphs D(k; q), which were constructed by Ustimenko and Lazebnik [2]
Error-Correcting codes fromk-resolving sets
We demonstrate a construction of error-correcting codes from graphs by
means of k-resolving sets, and present a decoding algorithm which makes use
of covering designs. Along the way, we determine the k-metric dimension of
grid graphs (i.e., Cartesian products of paths)
The Dynamic Phase Transition for Decoding Algorithms
The state-of-the-art error correcting codes are based on large random
constructions (random graphs, random permutations, ...) and are decoded by
linear-time iterative algorithms. Because of these features, they are
remarkable examples of diluted mean-field spin glasses, both from the static
and from the dynamic points of view. We analyze the behavior of decoding
algorithms using the mapping onto statistical-physics models. This allows to
understand the intrinsic (i.e. algorithm independent) features of this
behavior.Comment: 40 pages, 29 eps figure
Quantum Error Correcting Codes Using Qudit Graph States
Graph states are generalized from qubits to collections of qudits of
arbitrary dimension , and simple graphical methods are used to construct
both additive and nonadditive quantum error correcting codes. Codes of distance
2 saturating the quantum Singleton bound for arbitrarily large and are
constructed using simple graphs, except when is odd and is even.
Computer searches have produced a number of codes with distances 3 and 4, some
previously known and some new. The concept of a stabilizer is extended to
general , and shown to provide a dual representation of an additive graph
code.Comment: Version 4 is almost exactly the same as the published version in
Phys. Rev.
Application of coding theory to interconnection networks
AbstractWe give a few examples of applications of techniques and results borrowed from error-correcting codes to problems in graphs and interconnection networks. The degree and diameter of Cayley graphs with vertex set (Z2Z)r are investigated. The asymptotic case is dealt with in Section 2. The robustness, or fault tolerance, of the n-cube interconnection network is studied in Section 3
Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs
The minimum distance graph of a code has the codewords as vertices and edges
exactly when the Hamming distance between two codewords equals the minimum
distance of the code. A constructive proof for reconstructibility of an
extended perfect binary one-error-correcting code from its minimum distance
graph is presented. Consequently, inequivalent such codes have nonisomorphic
minimum distance graphs. Moreover, it is shown that the automorphism group of a
minimum distance graph is isomorphic to that of the corresponding code.Comment: 4 pages. Accepted for publication in IEEE Transactions on Information
Theor
Wildcard dimensions, coding theory and fault-tolerant meshes and hypercubes
Hypercubes, meshes and tori are well known interconnection networks for parallel computers. The sets of edges in those graphs can be partitioned to dimensions. It is well known that the hypercube can be extended by adding a wildcard dimension resulting in a folded hypercube that has better fault-tolerant and communication capabilities. First we prove that the folded hypercube is optimal in the sense that only a single wildcard dimension can be added to the hypercube. We then investigate the idea of adding wildcard dimensions to d-dimensional meshes and tori. Using techniques from error correcting codes we construct d-dimensional meshes and tori with wildcard dimensions. Finally, we show how these constructions can be used to tolerate edge and node faults in mesh and torus networks
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