On Universal Properties of Capacity-Approaching LDPC Ensembles

TL;DR

This paper derives universal information-theoretic bounds on LDPC code ensembles approaching capacity over MBIOS channels, applicable to finite and infinite lengths, and valid for various decoding algorithms.

Contribution

It introduces universal bounds on LDPC degree distributions and graphical complexity that hold across all MBIOS channels with a given capacity, regardless of decoding method.

Findings

Bounds are valid for any decoding algorithm.

Bounds are applicable to finite and infinite block lengths.

Bounds are easy to compute and compare with capacity-approaching ensembles.

Abstract

This paper is focused on the derivation of some universal properties of capacity-approaching low-density parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information-theoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and some others are proved under belief propagation (BP) decoding. Proving these bounds under a certain decoding algorithm, validates them automatically also under any sub-optimal decoding algorithm. A proper modification of these bounds makes them universal for the set of all MBIOS channels which exhibit a given capacity. Bounds on the degree distributions and graphical complexity apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. The bounds are compared with capacity-approaching LDPC code ensembles under BP decoding, and they are shown to be informative and are easy to calculate. Finally, some interesting open problems are considered.