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Density Evolution and Convergence Properties of Majority-based Algorithms for Decoding Low-density Parity-check (LDPC) Codes

Pirouz Zarrinkhat, Amir H. Banihashemi

Published Oct 2, 2002• 0 citations
OpenAlex

Abstract

This paper presents a detailed study of a family of binary messagepassing decoding algorithms for low-density parity-check (LDPC) codes, referred to as "majority-based decoding algorithms."Both Gallager's decoding algorithm A and standard majority decoding algorithm belong to this family.These algorithms, which work based on a generalized majority-decision rule, are particularly attractive for their remarkably simple implementation.We investigate the dynamics of these algorithms using density evolution and compute the threshold values for regular LDPC codes decoded by these algorithms.We also present analytical and graphical approaches to study and characterize the convergence properties of majority-based decoding algorithms, including their (convergence) speed.It appears that many of these algorithms enjoy very fast convergence.The analytical and graphical approaches used in this paper are quite general in principle, and with minor changes can be applied to many other algorithms.

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This paper presents a detailed study of a family of binary messagepassing decoding algorithms for low-density parity-check (LDPC) codes, referred to as "majority-based decoding algorithms."Both Gallager's decoding algorithm A and standard majority decoding algorithm belong to this family.These algorithms, which work based on a generalized majority-decision rule, are particularly attractive for their remarkably simple implementation.We investigate the dynamics of these algorithms using density evolution and compute the threshold values for regular LDPC codes decoded by these algorithms.We also present analytical and graphical approaches to study and characterize the convergence properties of majority-based decoding algorithms, including their (convergence) speed.It appears that many of these algorithms enjoy very fast convergence.The analytical and graphical approaches used in this paper are quite general in principle, and with minor changes can be applied to many other algorithms.
This work presents a detailed study of a family of binary message-passing decoding algorithms for low-density parity-check (LDPC) codes, referred to as "majority-based algorithms." Both Gallager's algorithm A (G/sub A/) and the standard majority decoding algorithm belong to this family. These algorithms, which are, in fact, the building blocks of Gallager's algorithm B (G/sub B/), work based on a generalized majority-decision rule and are particularly attractive for their remarkably simple implementation. We investigate the dynamics of these algorithms using density evolution and compute their (noise) threshold values for regular LDPC codes over the binary symmetric channel. For certain ensembles of codes and certain orders of majority-based algorithms, we show that the threshold value can be characterized as the smallest positive root of a polynomial, and thus can be determined analytically. We also study the convergence properties of majority-based algorithms, including their (convergence) speed. Our analysis shows that the stand-alone version of some of these algorithms provides significantly better performance and/or convergence speed compared with G/sub A/. In particular, it is shown that for channel parameters below threshold, while for G/sub A/ the error probability converges to zero exponentially with iteration number, this convergence for other majority-based algorithms is super-exponential.
Time-invariant hybrid (HTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution and finite-length simulation to analyze the performance and the convergence properties of these algorithms. Tight upper bounds on the threshold of MB HTIalgorithms are derived, and it is proven that the asymptotic error probability for these algorithms tends to zero at least exponentially with the number of iterations. We devise optimal MB HTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HTIalgorithms are more robust, and can better cope with unknown channel conditions, and thus can be practically more attractive
Time-invariant hybrid (HscrTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HscrTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB HscrTIalgorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB HscrTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HscrTIalgorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive.
Formulas for updated rules of the belief propagation algorithm applied to low density parity check (LDPC) codes are systematically derived based on the tree and the finite state machine (FSM).The probability measure on the continuous message space is introduced to evaluate the evolution of message densities within iterative decoding rounds,and derivations of the output distributions of common binary symmetric channels and the message density evolution are given in detail.The parameterized estimation of the performance of the algorithm is also discussed.The analytic and calculable means can be used to make systematical analysis for the convergence of BP algorithms independent of channels,and can be of help in designing LDPC codes and analyzing quantization effects for the fast decoding on the purpose of practical applications of LDPC codes to real time communication systems.
We consider a class of message-passing decoders for low-density parity-check (LDPC) codes whose messages are binary valued. We prove that if the channel is symmetric and all codewords are equally likely to be transmitted, an optimum decoding rule (in the sense of minimizing message error rate) should satisfy certain symmetry and isotropy conditions. Using this result, we prove that Gallager's Algorithm B achieves the optimum decoding threshold among all binary message-passing decoding algorithms for regular codes. For irregular codes, we argue that when the nodes of the message-passing decoder do not exploit knowledge of their decoding neighborhood, optimality of Gallager's Algorithm B is preserved. We also consider the problem of designing irregular LDPC codes and find a bound on the achievable rates with Gallager's Algorithm B. Using this bound, we study the case of low error-rate channels and analytically find good degree distributions for them.
We generalize the results about how to compute iterative decoding thresholds over the binary erasure channel of non-binary LDPC code ensembles of [16] to non-binary TLDPC codes [2], [3]. We show in this case how density evolution can be performed in order to calculate iterative decoding thresholds and find several families with a very simple regular structure and thresholds close to the Shannon limit. To check the performances of these codes over other channels we have tested one of the simplest codes over F4which has rate 1/2 on the Gaussian channel. For the (binary) length 1008 for instance, without any optimization on the permutation structure of the code, it matches the performances of the best binary codes of the same length up to the word-error rate 10-3. We also notice that all LDPC codes (binary or not) having at least two symbols of degree 2 per parity-check equation can be represented as a special kind of TLDPC codes. We show that this representation and the associated decoding algorithm leads in the case of cycle codes to a significant reduction of the number of iterations which are needed for iterative decoding.