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International Symposium on Turbo Codes and Iterative Information Processing

Analysis of LDPC code syndrome entropy based on subgraphs

David Matas, Meritxell Lamarca

Published Sep 1, 2016• 3 citations
Publisher

Abstract

We propose a method to bound the syndrome entropy of linear block codes from their factor graph representation. It is specially suited for sparse graphs such as those of low density parity check codes. It is based on the chain rule decomposition of the entropy and the confinement of dependencies within code subgraphs. After forcing or assuming the subgraphs to have a tree structure, the computation is done by means of density evolution as for a belief propagation analysis. We employ this method to compute upper bounds of the LDPC code syndrome entropy, which allows us to obtain asymptotic MAP upper bounds that match the ones obtained by the generalized area theorem.

DOI
10.1109/istc.2016.7593128

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We propose a method to bound the syndrome entropy of linear block codes from their factor graph representation. It is specially suited for sparse graphs such as those of low density parity check codes. It is based on the chain rule decomposition of the entropy and the confinement of dependencies within code subgraphs. After forcing or assuming the subgraphs to have a tree structure, the computation is done by means of density evolution as for a belief propagation analysis. We employ this method to compute upper bounds of the LDPC code syndrome entropy, which allows us to obtain asymptotic MAP upper bounds that match the ones obtained by the generalized area theorem.
This paper aims at computing tight upper bounds for the maximum a posteriori threshold of low-density parity check codes in the asymptotic blocklength regime for the transmission over binary-input memoryless symmetric-output channels. While these bounds are already known, we propose a novel derivation based on a completely different approach: based solely on the concept of the chain rule and the conditional entropy, resorting to the concentration theorem for the code ensemble to compute the syndrome entropy with low complexity employing density evolution.
Low-density parity-check (LDPC) codes on symmetric memoryless channels have been analyzed using statistical physics by several authors. In this paper, statistical mechanical analysis of LDPC codes is performed for asymmetric memoryless channels and general Markov channels. It is shown that the saddle point equations of the replica symmetric solution for a Markov channel is equivalent to the density evolution of the belief propagation on the factor graph representing LDPC codes on the Markov channel. The derivation uses the method of types for Markov chain.
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.
The design of optimal linear block codes capable of being efficiently decoded is of major concern, especially for short block lengths. As near capacity-approaching codes, Low-Density Parity-Check (LDPC) codes possess several advantages over other families of codes, the most notable being its efficient decoding via Belief Propagation. While many LDPC code design methods exist, the development of efficient sparse codes that meet the constraints of modern short code lengths and accommodate new channel models remains a challenge. In this work, we propose for the first time a gradient-based data-driven approach for the design of sparse codes. We develop locally optimal codes with respect to Belief Propagation decoding via the learning of the Factor graph under channel noise simulations. This is performed via a novel complete graph tensor representation of the Belief Propagation algorithm, optimized over finite fields via backpropagation and coupled with an efficient line-search method. The proposed approach is shown to outperform the decoding performance of existing popular codes by orders of magnitude and demonstrates the power of data-driven approaches for code design.
Low Density Parity Check(LDPC) codes is one of the high topics of the channel coding theory and has been paid more and more attention.This paper begins with a brief review of channel coding theories.Then the iterative decoding algorithm of LDPC codes,so-called as Sum-Product algorithm or Belief Propagation algorithm,is represented in detail.And the construction of LDPC codes by means of random bipartite graph and algebraic constructions,and the performance analysis by Density Evolution theory,Gaussian Approximation method and Extrinsic Information Transfer(EXIT) charts etc.,are respectively discussed.Finally,the latest study results of LDPC codes are presented and some open questions about LDPC codes are provided.
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sum-product algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for short-to-intermediate codelengths compared to Arikan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.