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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Detailed Evolution of Degree Distributions in Residual Graphs with Joint Degree Distributions

T. Nozaki, K. KASAI, T. SHIBUYA, K. Sakaniwa

Published Oct 1, 2008• 0 citations
Publisher

Abstract

Luby et al. derived evolution of degree distributions in residual graphs for irregular LDPC code ensembles. Evolution of degree distributions in residual graphs is important characteristic which is used for finite-length analysis of the expected block and bit error probability over the binary erasure channel. In this paper, we derive detailed evolution of degree distributions in residual graphs for irregular LDPC code ensembles with joint degree distributions.

DOI
10.1093/ietfec/e91-a.10.2737

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Luby et al. derived evolution of degree distributions in residual graphs for irregular LDPC code ensembles. Evolution of degree distributions in residual graphs is important characteristic which is used for finite-length analysis of the expected block and bit error probability over the binary erasure channel. In this paper, we derive detailed evolution of degree distributions in residual graphs for irregular LDPC code ensembles with joint degree distributions.
Decoding performance of LDPC (Low-Density Parity-Check) codes is highly dependent on the degree distributions of the Tanner graphs which define the LDPC codes. We compare two LDPC code ensembles, one has a uniform degree distribution and the other a non-uniform one over a BEC (Binary Erasure Channel) and a BSC (Binary Symmetric Channel) thorough DE (Density Evolution). We then derive sufficient conditions on the erasure probability of a BEC and the error probability of a BSC, under which the LDPC code ensembles with uniform degree distributions outperform those with non-uniform degree distributions.
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.
We investigate the finite-length scaling methodology for irregular LDPC code ensembles when transmission takes place over the binary erasure channel (BEC). We first show how the necessary computations, namely the covariance evolution and the computation of the finite-length shift, can be accomplished in the irregular case. We then investigate how the obtained approximation can be used to predict the performance of irregular code ensembles and to optimize the degree distributions for finite-length codes.