101,469 research outputs found
Maximal slicings in spherical symmetry: local existence and construction
We show that any spherically symmetric spacetime locally admits a maximal
spacelike slicing and we give a procedure allowing its construction. The
construction procedure that we have designed is based on purely geometrical
arguments and, in practice, leads to solve a decoupled system of first order
quasi-linear partial differential equations. We have explicitly built up
maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits
further generalizations and efficient computational implementation. As by
product, we suggest some applications of our work in the task of calibrating
Numerical Relativity complex codes, usually written in Cartesian coordinates.Comment: 25 pages, 6 figure
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
Empirical and Strong Coordination via Soft Covering with Polar Codes
We design polar codes for empirical coordination and strong coordination in
two-node networks. Our constructions hinge on the fact that polar codes enable
explicit low-complexity schemes for soft covering. We leverage this property to
propose explicit and low-complexity coding schemes that achieve the capacity
regions of both empirical coordination and strong coordination for sequences of
actions taking value in an alphabet of prime cardinality. Our results improve
previously known polar coding schemes, which (i) were restricted to uniform
distributions and to actions obtained via binary symmetric channels for strong
coordination, (ii) required a non-negligible amount of common randomness for
empirical coordination, and (iii) assumed that the simulation of discrete
memoryless channels could be perfectly implemented. As a by-product of our
results, we obtain a polar coding scheme that achieves channel resolvability
for an arbitrary discrete memoryless channel whose input alphabet has prime
cardinality.Comment: 14 pages, two-column, 5 figures, accepted to IEEE Transactions on
Information Theor
K-user Interference Channels: Achievable Secrecy Rate and Degrees of Freedom
In this work, we consider achievable secrecy rates for symmetric -user () interference channels with confidential messages. We find that nested
lattice codes and layered coding are useful in providing secrecy for these
channels. Achievable secrecy rates are derived for very strong interference. In
addition, we derive the secure degrees of freedom for a range of channel
parameters. As a by-product of our approach, we also demonstrate that nested
lattice codes are useful for K-user symmetric interference channels without
secrecy constraints in that they yield higher degrees of freedom than previous
results.Comment: 5 pages. To appear at IEEE ITW 2009, Volos, June 200
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