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International Symposium on Turbo Codes and Iterative Information Processing

Analysis of weight distributions of two-edge type LDPC codes by Hayman approximation

Tomoharu Awano, Kenta Kasai, Tomoharu Shibuya, Kohichi Sakaniwa

Published Sep 1, 2008• 2 citations
Publisher

Abstract

Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with small linear weight codewords exponentially decreasing in code length. Nakasendo et al. derived the condition in which code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. Our conclusion is the same as that of Nakasendo et al., and although Nakasendopsilas method is difficult to apply to more than three edge-types, there is a possibility that our method using Hayman approximation as Di et al. do derives the condition for multi-edge type LDPC codes ensembles.

DOI
10.1109/turbocoding.2008.4658667

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Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with small linear weight codewords exponentially decreasing in code length. Nakasendo et al. derived the condition in which code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. Our conclusion is the same as that of Nakasendo et al., and although Nakasendopsilas method is difficult to apply to more than three edge-types, there is a possibility that our method using Hayman approximation as Di et al. do derives the condition for multi-edge type LDPC codes ensembles.
Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. Nakasendo et al. derived the condition that code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. In this paper, we derive the condition for three-edge type LDPC code ensembles whose edge-types does not share any variable node and does share all of the check nodes with exponentially decreasing small linear weight codewords. The condition is necessary for the existence of the average relative minimum distance of ensembles. Our method is expected to derive the condition for multi-edge type LDPC codes.
For a (lambda(x); rho(x)) standard irregular LDPC code ensemble, the growth rate of the average weight distribution for small relative weight omega is given by log(lambda'(0)rho'(1))omega + O(omega2) in the limit of code length n. If lambda'(0)rho'(1) < 1, there exist exponentially few code words of small linear weight, as n tends to infinity. It is known that the condition coincides with the stability condition of density evolution over the erasure channels with the erasure probability 1. In this paper, we show that this is also the case with multi-edge type LDPC (MET-LDPC) codes. MET-LDPC codes are generalized structured LDPC codes introduced by Richardson and Urbanke. The parameter corresponding lambda'(0)rho'(1) appearing in the conditions for MET-LDPC codes is given by the spectral radius of the matrix defined by extended degree distributions.
Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.