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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Average Coset Weight Distribution of Multi-Edge Type LDPC Code Ensembles

K. KASAI, Yuji Shimoyama, T. SHIBUYA, K. Sakaniwa

Published Oct 1, 2006• 8 citations
Publisher

Abstract

Multi-Edge type Low-Density Parity-Check codes (MET-LDPC codes) introduced by Richardson and Urbanke are generalized LDPC codes which can be seen as LDPC codes obtained by concatenating several standard (ir)regular LDPC codes. We prove in this paper that MET-LDPC code ensembles possess a certain symmetry with respect to their Average Coset Weight Distributions (ACWD). Using this symmetry, we drive ACWD of MET-LDPC code ensembles from ACWD of their constituent ensembles.

DOI
10.1093/ietfec/e89-a.10.2519

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Multi-Edge type Low-Density Parity-Check codes (MET-LDPC codes) introduced by Richardson and Urbanke are generalized LDPC codes which can be seen as LDPC codes obtained by concatenating several standard (ir)regular LDPC codes. We prove in this paper that MET-LDPC code ensembles possess a certain symmetry with respect to their Average Coset Weight Distributions (ACWD). Using this symmetry, we drive ACWD of MET-LDPC code ensembles from ACWD of their constituent ensembles.
In this paper, the average coset weight distribution (ACWD) of structured ensembles of LDPC (Low-density Parity-Check) matrix, which is called combined ensembles, is discussed. A combined ensemble is composed of a set of simpler ensembles such as a regular bipartite ensemble. Two classes of combined ensembles have prime importance; a stacked ensemble and a concatenated ensemble, which consists of set of stacked matrices and concatenated matrices, respectively. The ACWD formulas of these ensembles is shown in this paper. Such formulas are key tools to evaluate the ACWD of a complex combined ensemble. From the ACWD of an ensemble, we can obtain some detailed properties of a code (e.g., weight of coset leaders) which is not available from an average weight distribution. Moreover, it is shown that the analysis based on the ACWD is indispensable to evaluate the average weight distribution of some classes of combined ensembles.
In this paper, the average coset weight distribution (ACWD) of structured ensembles of low-density parity-check (LDPC) matrices, which are called combined ensembles, is discussed. A combined ensemble is composed of a set of simpler ensembles such as a regular bipartite ensemble. Two classes of combined ensembles have prime importance; a stacked ensemble and a concatenated ensemble. The ACWD formulas of these ensembles are shown in this paper. Such formulas play a key role to evaluate the average weight distribution of some classes of combined ensembles
In this correspondence, the average coset weight distributions of Gallager's low-density parity-check (LDPC) code ensemble are investigated. Gallager's LDPC code ensemble consists of regular mtimesn-LDPC matrices with column weight j and row weight k. The average coset weight distribution can be derived by enumerating the number of parity-check matrices in the ensemble satisfying certain conditions. Based on combinatorial arguments, a formula for the average coset weight distribution will be proved. From the formula, we can show some properties of the average coset weight distributions such as equivalence classes of syndromes, symmetry of the distributions, and a lower bound on coset weight
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.
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.