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On a Different Perspective and Approach to Implement Adaptive Normalized BP-based Decoding for LDPC Codes
In this paper, we propose an improved version of the min-sum algorithm for low density parity check (LDPC) code decoding, which we call “adaptive normalized BP-based” algorithm. Our decoder provides a compromise solution between the belief propagation and the min-sum algorithms by adding an exponent offset to each variable node’s intrinsic information in the check node update equation. The extrinsic information from the min-sum decoder is then adjusted by applying a negative power of two scale factor, which can be easily implemented by right shifting the min-sum extrinsic information. The difference between our approach and other adaptive normalized min-sum decoders is that we select the normalization scale factor using a clear analytical approach based on underlying principles. Simulation results show that the proposed decoder outperforms the min-sum decoder and performs very close to the BP decoder, but with lower complexity.Keywords: modified min-sum, belief propagation, sum product, min-sum, LDPC codes, iterative decodin
Deriving the Normalized Min-Sum Algorithm from Cooperative Optimization
The normalized min-sum algorithm can achieve near-optimal performance at
decoding LDPC codes. However, it is a critical question to understand the
mathematical principle underlying the algorithm. Traditionally, people thought
that the normalized min-sum algorithm is a good approximation to the
sum-product algorithm, the best known algorithm for decoding LDPC codes and
Turbo codes. This paper offers an alternative approach to understand the
normalized min-sum algorithm. The algorithm is derived directly from
cooperative optimization, a newly discovered general method for
global/combinatorial optimization. This approach provides us another
theoretical basis for the algorithm and offers new insights on its power and
limitation. It also gives us a general framework for designing new decoding
algorithms.Comment: Accepted by IEEE Information Theory Workshop, Chengdu, China, 200
Low-Complexity LP Decoding of Nonbinary Linear Codes
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has
attracted much attention in the research community in the past few years. LP
decoding has been derived for binary and nonbinary linear codes. However, the
most important problem with LP decoding for both binary and nonbinary linear
codes is that the complexity of standard LP solvers such as the simplex
algorithm remains prohibitively large for codes of moderate to large block
length. To address this problem, two low-complexity LP (LCLP) decoding
algorithms for binary linear codes have been proposed by Vontobel and Koetter,
henceforth called the basic LCLP decoding algorithm and the subgradient LCLP
decoding algorithm.
In this paper, we generalize these LCLP decoding algorithms to nonbinary
linear codes. The computational complexity per iteration of the proposed
nonbinary LCLP decoding algorithms scales linearly with the block length of the
code. A modified BCJR algorithm for efficient check-node calculations in the
nonbinary basic LCLP decoding algorithm is also proposed, which has complexity
linear in the check node degree.
Several simulation results are presented for nonbinary LDPC codes defined
over Z_4, GF(4), and GF(8) using quaternary phase-shift keying and
8-phase-shift keying, respectively, over the AWGN channel. It is shown that for
some group-structured LDPC codes, the error-correcting performance of the
nonbinary LCLP decoding algorithms is similar to or better than that of the
min-sum decoding algorithm.Comment: To appear in IEEE Transactions on Communications, 201
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
Fourier Domain Decoding Algorithm of Non-Binary LDPC codes for Parallel Implementation
For decoding non-binary low-density parity check (LDPC) codes,
logarithm-domain sum-product (Log-SP) algorithms were proposed for reducing
quantization effects of SP algorithm in conjunction with FFT. Since FFT is not
applicable in the logarithm domain, the computations required at check nodes in
the Log-SP algorithms are computationally intensive. What is worth, check nodes
usually have higher degree than variable nodes. As a result, most of the time
for decoding is used for check node computations, which leads to a bottleneck
effect. In this paper, we propose a Log-SP algorithm in the Fourier domain.
With this algorithm, the role of variable nodes and check nodes are switched.
The intensive computations are spread over lower-degree variable nodes, which
can be efficiently calculated in parallel. Furthermore, we develop a fast
calculation method for the estimated bits and syndromes in the Fourier domain.Comment: To appear in IEICE Trans. Fundamentals, vol.E93-A, no.11 November
201
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