82,403 research outputs found
Average Stopping Set Weight Distribution of Redundant Random Matrix Ensembles
In this paper, redundant random matrix ensembles (abbreviated as redundant
random ensembles) are defined and their stopping set (SS) weight distributions
are analyzed. A redundant random ensemble consists of a set of binary matrices
with linearly dependent rows. These linearly dependent rows (redundant rows)
significantly reduce the number of stopping sets of small size. An upper and
lower bound on the average SS weight distribution of the redundant random
ensembles are shown. From these bounds, the trade-off between the number of
redundant rows (corresponding to decoding complexity of BP on BEC) and the
critical exponent of the asymptotic growth rate of SS weight distribution
(corresponding to decoding performance) can be derived. It is shown that, in
some cases, a dense matrix with linearly dependent rows yields asymptotically
(i.e., in the regime of small erasure probability) better performance than
regular LDPC matrices with comparable parameters.Comment: 14 pages, 7 figures, Conference version to appear at the 2007 IEEE
International Symposium on Information Theory, Nice, France, June 200
Parsing a sequence of qubits
We develop a theoretical framework for frame synchronization, also known as
block synchronization, in the quantum domain which makes it possible to attach
classical and quantum metadata to quantum information over a noisy channel even
when the information source and sink are frame-wise asynchronous. This
eliminates the need of frame synchronization at the hardware level and allows
for parsing qubit sequences during quantum information processing. Our
framework exploits binary constant-weight codes that are self-synchronizing.
Possible applications may include asynchronous quantum communication such as a
self-synchronizing quantum network where one can hop into the channel at any
time, catch the next coming quantum information with a label indicating the
sender, and reply by routing her quantum information with control qubits for
quantum switches all without assuming prior frame synchronization between
users.Comment: 11 pages, 2 figures, 1 table. Final accepted version for publication
in the IEEE Transactions on Information Theor
Reed-Muller codes for random erasures and errors
This paper studies the parameters for which Reed-Muller (RM) codes over
can correct random erasures and random errors with high probability,
and in particular when can they achieve capacity for these two classical
channels. Necessarily, the paper also studies properties of evaluations of
multi-variate polynomials on random sets of inputs.
For erasures, we prove that RM codes achieve capacity both for very high rate
and very low rate regimes. For errors, we prove that RM codes achieve capacity
for very low rate regimes, and for very high rates, we show that they can
uniquely decode at about square root of the number of errors at capacity.
The proofs of these four results are based on different techniques, which we
find interesting in their own right. In particular, we study the following
questions about , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in that define a submatrix having full column
rank (resp. full row rank) with high probability? We obtain tight bounds for
very small (resp. very large) degrees , which we use to show that RM codes
achieve capacity for erasures in these regimes.
Our decoding from random errors follows from the following novel reduction.
For every linear code of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
Specializing this to RM codes and using our results for erasures imply our
result on unique decoding of RM codes at high rate.
Finally, two of our capacity achieving results require tight bounds on the
weight distribution of RM codes. We obtain such bounds extending the recent
\cite{KLP} bounds from constant degree to linear degree polynomials
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