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Further Progress on the Finite-Length Analysis of LDPC Codes on the Binary Erasure Channel

Tom Richardson, Rudiger L. Urbanke

Published Oct 2, 2002• 0 citations
OpenAlex

Abstract

In this paper we derive density evolution equations for finite-length ensembles when transmitting over the binary erasure channel.Further we show how to determine the performance of suitably chosen expurgated ensembles.Finally, we investigate the distribution of the number of required iterations in an iterative coding system.

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IEEE International Symposium on Information Theory (ISIT)•Jun 25, 2003•50 citations
Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution and it is well known that a difference between the bit error probability at finite iteration number for finite block length nnn and for infinite block length is asymptotically α/nα/nα/n, where ααα is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating ααα for regular ensembles. The approximation using ααα is accurate for (2,r)(2,r)(2,r)-regular ensembles even in small block length.
In this paper we present a thorough analysis of non binary LDPC codes over the binary erasure channel. First, the decoding of non binary LDPC codes is investigated. The proposed algorithm performs ldquoon-the flyrdquo decoding, i.e. it starts decoding as soon as the first symbols are received, which generalizes the erasure decoding of binary LDPC codes. Next, we evaluate the asymptotical performance of ensembles of non binary LDPC codes, by using the density evolution method. Density evolution equations are derived by taking into consideration both the irregularity of the bipartite graph and the probability distribution of the graph edge labels. Finally, infinite-length performance of some ensembles of non binary LDPC codes for different edge label distributions are shown.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area.
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. Furthermore, a gap between the bit error probability after finite number of iterations for finite block length n and that for infinite block length is asymptotically α/n, where α denotes a speci..c constant determined by a degree distribution, a number of iterations and erasure probability. Our main result is to derive an ef..cient algorithm for calculating α for regular ensembles.
We investigate the average erasure probability of the belief propagation algorithm over the binary erasure channel (BEC) for various finite-length low-density parity-check (LDPC) ensembles. In particular, we give tight upper bounds on the "error floor", i.e., on the contribution to the erasure probability stemming from relatively small deficiencies in the graph. We also define and analyse "expurgated" ensembles and give upper bounds on the error floor of "typical" codes. Finally, we show that typical codes of properly chosen ensembles have an erasure correcting radius which grows linearly in the block length. The presented method provides valuable insight into how finite-length codes should be designed. Although the standard ensemble can achieve capacity on the BEC, the construction of good finite-length codes requires careful control of the structure of the degree two variable nodes. In this paper, we investigate the standard ensemble, the Poisson ensemble, an ensemble in which degree two nodes are added in a cycle, and an ensemble in which different degree two variable nodes have distinct neighboring check nodes.