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IEEE Transactions on Information Theory

Average Coset Weight Distributions of Gallager's LDPC Code Ensemble

Tadashi Wadayama

Published Jun 28, 2005• 5 citations
Publisher

Abstract

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

DOI
10.1109/tit.2005.850144

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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
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
The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.