2,399 research outputs found
Book Review: Finding Jesus in Dharma: Christianity in India
A review of Finding Jesus in Dharma: Christianity in India by Chaturvedi
Book Review: Asian Biblical Hermeneutics and Postcolonialism: Contesting the Interpretations
A review of Asian Biblical Hermeneutics and Postcolonialism: Contesting the Interpretations by R. S. Sugirtharajah
The Treewidth of MDS and Reed-Muller Codes
The constraint complexity of a graphical realization of a linear code is the
maximum dimension of the local constraint codes in the realization. The
treewidth of a linear code is the least constraint complexity of any of its
cycle-free graphical realizations. This notion provides a useful
parametrization of the maximum-likelihood decoding complexity for linear codes.
In this paper, we prove the surprising fact that for maximum distance separable
codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code,
is defined to be the least constraint complexity (or branch complexity) of any
of its trellis realizations. From this, we obtain exact expressions for the
treewidth of these codes, which constitute the only known explicit expressions
for the treewidth of algebraic codes.Comment: This constitutes a major upgrade of previous versions; submitted to
IEEE Transactions on Information Theor
Path Gain Algebraic Formulation for the Scalar Linear Network Coding Problem
In the algebraic view, the solution to a network coding problem is seen as a
variety specified by a system of polynomial equations typically derived by
using edge-to-edge gains as variables. The output from each sink is equated to
its demand to obtain polynomial equations. In this work, we propose a method to
derive the polynomial equations using source-to-sink path gains as the
variables. In the path gain formulation, we show that linear and quadratic
equations suffice; therefore, network coding becomes equivalent to a system of
polynomial equations of maximum degree 2. We present algorithms for generating
the equations in the path gains and for converting path gain solutions to
edge-to-edge gain solutions. Because of the low degree, simplification is
readily possible for the system of equations obtained using path gains. Using
small-sized network coding problems, we show that the path gain approach
results in simpler equations and determines solvability of the problem in
certain cases. On a larger network (with 87 nodes and 161 edges), we show how
the path gain approach continues to provide deterministic solutions to some
network coding problems.Comment: 12 pages, 6 figures. Accepted for publication in IEEE Transactions on
Information Theory (May 2010
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