Average Coset Weight Distribution of Combined LDPC Matrix Ensemble

TL;DR

This paper derives formulas for the average coset weight distribution of combined LDPC matrix ensembles, enabling detailed analysis of code properties beyond traditional weight distributions.

Contribution

It introduces ACWD formulas for combined LDPC ensembles, including stacked and concatenated types, facilitating analysis of complex code structures.

Findings

ACWD formulas for stacked and concatenated ensembles

ACWD enables detailed code property analysis

Analysis based on ACWD is essential for complex ensembles

Abstract

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.