256 research outputs found
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
We introduce the notion of the stopping redundancy hierarchy of a linear
block code as a measure of the trade-off between performance and complexity of
iterative decoding for the binary erasure channel. We derive lower and upper
bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and
Bonferroni-type inequalities, and specialize them for codes with cyclic
parity-check matrices. Based on the observed properties of parity-check
matrices with good stopping redundancy characteristics, we develop a novel
decoding technique, termed automorphism group decoding, that combines iterative
message passing and permutation decoding. We also present bounds on the
smallest number of permutations of an automorphism group decoder needed to
correct any set of erasures up to a prescribed size. Simulation results
demonstrate that for a large number of algebraic codes, the performance of the
new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on
Information Theor
From rubber bands to rational maps: A research report
This research report outlines work, partially joint with Jeremy Kahn and
Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal
surfaces with boundary. One one hand, this lets us tell when one rubber band
network is looser than another, and on the other hand tell when one conformal
surface embeds in another.
We apply this to give a new characterization of hyperbolic critically finite
rational maps among branched self-coverings of the sphere, by a positive
criterion: a branched covering is equivalent to a hyperbolic rational map if
and only if there is an elastic graph with a particular "self-embedding"
property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example
A functional central limit theorem for interacting particle systems on transitive graphs
A finite range interacting particle system on a transitive graph is
considered. Assuming that the dynamics and the initial measure are invariant,
the normalized empirical distribution process converges in distribution to a
centered diffusion process. As an application, a central limit theorem for
certain hitting times, interpreted as failure times of a coherent system in
reliability, is derived.Comment: 35 page
Heegaard Floer invariants of Legendrian knots in contact three--manifolds
We define invariants of null--homologous Legendrian and transverse knots in
contact 3--manifolds. The invariants are determined by elements of the knot
Floer homology of the underlying smooth knot. We compute these invariants, and
show that they do not vanish for certain non--loose knots in overtwisted
3--spheres. Moreover, we apply the invariants to find transversely non--simple
knot types in many overtwisted contact 3--manifolds.Comment: 70 pages, 30 figure
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