832 research outputs found
Finite Length Analysis of LDPC Codes
In this paper, we study the performance of finite-length LDPC codes in the
waterfall region. We propose an algorithm to predict the error performance of
finite-length LDPC codes over various binary memoryless channels. Through
numerical results, we find that our technique gives better performance
prediction compared to existing techniques.Comment: Submitted to WCNC 201
Scattered EXIT Charts for Finite Length LDPC Code Design
We introduce the Scattered Extrinsic Information Transfer (S-EXIT) chart as a
tool for optimizing degree profiles of short length Low-Density Parity-Check
(LDPC) codes under iterative decoding. As degree profile optimization is
typically done in the asymptotic length regime, there is space for further
improvement when considering the finite length behavior. We propose to consider
the average extrinsic information as a random variable, exploiting its specific
distribution properties for guiding code design. We explain, step-by-step, how
to generate an S-EXIT chart for short-length LDPC codes. We show that this
approach achieves gains in terms of bit error rate (BER) of 0.5 dB and 0.6 dB
over the additive white Gaussian noise (AWGN) channel for codeword lengths of
128 and 180 bits, respectively, at a target BER of when compared to
conventional Extrinsic Information Transfer (EXIT) chart-based optimization.
Also, a performance gain for the Binary Erasure Channel (BEC) for a block
(i.e., codeword) length of 180 bits is shown.Comment: in IEEE International Conference on Communications (ICC), May 201
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
Polytope of Correct (Linear Programming) Decoding and Low-Weight Pseudo-Codewords
We analyze Linear Programming (LP) decoding of graphical binary codes
operating over soft-output, symmetric and log-concave channels. We show that
the error-surface, separating domain of the correct decoding from domain of the
erroneous decoding, is a polytope. We formulate the problem of finding the
lowest-weight pseudo-codeword as a non-convex optimization (maximization of a
convex function) over a polytope, with the cost function defined by the channel
and the polytope defined by the structure of the code. This formulation
suggests new provably convergent heuristics for finding the lowest weight
pseudo-codewords improving in quality upon previously discussed. The algorithm
performance is tested on the example of the Tanner [155, 64, 20] code over the
Additive White Gaussian Noise (AWGN) channel.Comment: 6 pages, 2 figures, accepted for IEEE ISIT 201
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