Analytical Solution of Covariance Evolution for Regular LDPC Codes

TL;DR

This paper provides an analytical solution to the covariance evolution equations for regular LDPC codes, enabling precise estimation of block error probabilities without relying on numerical methods.

Contribution

It introduces an exact analytical solution to the covariance evolution system, improving the understanding and analysis of finite-length LDPC codes.

Findings

Derived explicit formulas for covariance evolution

Enhanced accuracy in block error probability estimation

Reduced computational complexity compared to numerical methods

Abstract

The covariance evolution is a system of differential equations with respect to the covariance of the number of edges connecting to the nodes of each residual degree. Solving the covariance evolution, we can derive distributions of the number of check nodes of residual degree 1, which helps us to estimate the block error probability for finite-length LDPC code. Amraoui et al.\ resorted to numerical computations to solve the covariance evolution. In this paper, we give the analytical solution of the covariance evolution.