604 research outputs found
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
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
Mutual Information-Maximizing Quantized Belief Propagation Decoding of Regular LDPC Codes
In mutual information-maximizing lookup table (MIM-LUT) decoding of
low-density parity-check (LDPC) codes, table lookup operations are used to
replace arithmetic operations. In practice, large tables need to be decomposed
into small tables to save the memory consumption, at the cost of degraded error
performance. In this paper, we propose a method, called mutual
information-maximizing quantized belief propagation (MIM-QBP) decoding, to
remove the lookup tables used for MIM-LUT decoding. Our method leads to a very
efficient decoder, namely the MIM-QBP decoder, which can be implemented based
only on simple mappings and fixed-point additions. Simulation results show that
the MIM-QBP decoder can always considerably outperform the state-of-the-art
MIM-LUT decoder, mainly because it can avoid the performance loss due to table
decomposition. Furthermore, the MIM-QBP decoder with only 3 bits per message
can outperform the floating-point belief propagation (BP) decoder at high
signal-to-noise ratio (SNR) regions when testing on high-rate codes with a
maximum of 10-30 iterations
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