On the Design of Universal LDPC Codes

TL;DR

This paper proposes a universal LDPC code design approach that decomposes channels into basis channels, enabling codes to perform well across multiple channels with the same capacity, backed by theoretical proofs and empirical results.

Contribution

It introduces a method to decompose symmetric channels into basis channels of equal capacity, facilitating the design of universal LDPC codes applicable to various channels.

Findings

Codes designed on basis channels outperform existing codes across multiple channels.

The stability condition extends to the convex hull of channels, ensuring convergence.

The proposed codes achieve significant coding gains in multi-channel scenarios.

Abstract

Low-density parity-check (LDPC) coding for a multitude of equal-capacity channels is studied. First, based on numerous observations, a conjecture is stated that when the belief propagation decoder converges on a set of equal-capacity channels, it would also converge on any convex combination of those channels. Then, it is proved that when the stability condition is satisfied for a number of channels, it is also satisfied for any channel in their convex hull. For the purpose of code design, a method is proposed which can decompose every symmetric channel with capacity C into a set of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any channel with capacity C. Such codes are found and in comparison with existing LDPC codes that are designed for specific channels, our codes obtain considerable coding gains when used across a multitude of channels.