1,046 research outputs found
Design of Non-Binary Quasi-Cyclic LDPC Codes by ACE Optimization
An algorithm for constructing Tanner graphs of non-binary irregular
quasi-cyclic LDPC codes is introduced. It employs a new method for selection of
edge labels allowing control over the code's non-binary ACE spectrum and
resulting in low error-floor. The efficiency of the algorithm is demonstrated
by generating good codes of short to moderate length over small fields,
outperforming codes generated by the known methods.Comment: Accepted to 2013 IEEE Information Theory Worksho
Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel
Using combinatorial arguments, we determine an upper bound on achievable
rates of stabilizer codes used over the quantum erasure channel. This allows us
to recover the no-cloning bound on the capacity of the quantum erasure channel,
R is below 1-2p, for stabilizer codes: we also derive an improved upper bound
of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for
0 < p < 1/2 and for any family of stabilizer codes whose generators have
weights bounded from above by a constant - low density stabilizer codes.
We obtain an application to percolation theory for a family of self-dual
tilings of the hyperbolic plane. We associate a family of low density
stabilizer codes with appropriate finite quotients of these tilings. We then
relate the probability of percolation to the probability of a decoding error
for these codes on the quantum erasure channel. The application of our upper
bound on achievable rates of low density stabilizer codes gives rise to an
upper bound on the critical probability for these tilings.Comment: 32 page
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
Deriving Good LDPC Convolutional Codes from LDPC Block Codes
Low-density parity-check (LDPC) convolutional codes are capable of achieving
excellent performance with low encoding and decoding complexity. In this paper
we discuss several graph-cover-based methods for deriving families of
time-invariant and time-varying LDPC convolutional codes from LDPC block codes
and show how earlier proposed LDPC convolutional code constructions can be
presented within this framework. Some of the constructed convolutional codes
significantly outperform the underlying LDPC block codes. We investigate some
possible reasons for this "convolutional gain," and we also discuss the ---
mostly moderate --- decoder cost increase that is incurred by going from LDPC
block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010;
revised August 2010, revised November 2010 (essentially final version).
(Besides many small changes, the first and second revised versions contain
corrected entries in Tables I and II.
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